Problems / erdos-1050 / erdos-1050--h1-v2--h2

Integrality: b(n)z - a(n) = 9^n W(n) E(n)

erdos-1050--h1-v2--h2

proved

Hole integrality of erdos-1050--h1-v2 (split by 20260924T123700Z-t0924-1050-mcp-partial.lean).

$\displaystyle \exists\, a,b:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad b(n)\,z-a(n)\;=\;9^n\,W(n)\,E(n)$

There are two sequences of integers $a(n)$, $b(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, where, with $c=8/3$, $z=\sum_{j\ge 0}\bigl(1-c\,2^{j+1}\bigr)^{-1}$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$ and $E(n)=\sum_{j\ge 0}\;-\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$. Here $n-2$ is natural-number subtraction (so $(n-2)!=0!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponents $2n+j$, $k-n-j$ and $k+n+j$ in $E(n)$ are integers (so $2^{k-n-j}=1/2^{n+j-k}$, and the product over $k$ is empty at $n=1$), and both infinite sums are Lean's ∑' over $j\ge 0$, which is $0$ for a non-summable family. Nothing is required of $a(0)$ or $b(0)$, and $a(n)$, $b(n)$ are not required to be nonzero.

unverified · history

Lean

theorem erdos_1050__h1_v2__h2 : ∃ (a : ℕ → ℤ) (b : ℕ → ℤ),
  ∀ (n : ℕ),
    (1 : ℕ) ≤ n →
      (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) =
        (9 : ℝ) ^ n *
            (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) *
              ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) *
          ∑' (j : ℕ),
            -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ *
              ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)),
                ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) *
                  ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ := by
  sorry

Why does it end in sorry?

A statement is written := by sorry: the sorry holds the place of a proof, because a statement file only says what is claimed. A proof the gate checked is its own file, Proof.lean. How to read a proof page →

Explainer

Informal account, unverified — unverified prose about a kernel-checked proof; shown section by section, each labelled with who drafted, wrote or verified it (D-3 v3.31).

Unverified: explainer, its words from explainer/658a25aec9a9….md, written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent, 2026-10-06; explainer/d31708683eee….md, written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent, 2026-10-06.

What is proved, and the plan

Draft · drafted, by the model named above

Fix $c = 8/3$ and write $z=\sum_{j\ge0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$. For $n\ge1$ and $1\le k\le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k=-\frac{\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{D_k},\qquad W(n)=(n-2)!\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr)\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),$$ and $E(n)$ for the infinite sum on the right of the statement, so the statement asks for integers with $b(n)z-a(n)=9^nW(n)E(n)$ for $n\ge1$. Put $$\beta_0=\sum_{k=1}^{n}A_k,\qquad S_1=\sum_{k=1}^{n}A_k\sum_{i=1}^{k}\bigl(1-c\,2^i\bigr)^{-1},\qquad S_2=\sum_{i=1}^{n-1}\frac{B_i}{2^i-1},\quad B_i=\sum_{k=1}^{n}A_k\,(c\,2^k)^{-i}.$$

The proof has four parts, then one line:

(1) numerator_integral: $9^nW(n)(S_1-S_2)$ is an integer $a(n)$.

(2) hden: $9^nW(n)\beta_0$ is an integer $b(n)$.

(3) hpf: a partial-fraction identity for the summand of $E(n)$.

(4) hres: summing it, $E(n)=\beta_0\,z-(S_1-S_2)$.

The final line takes these $a$ and $b$ and checks $b(n)z-a(n)=9^nW(n)E(n)$ by linear_combination from the three equations. The comments in the proof point to annexes 959e0d5b… and a03db62f… for the informal argument (Borwein's construction). The outline lists no graph node as used: this proof proves part 1 in full itself and does not call the node erdos-1050--h1-v2--h2--h1, although that node states the same fact and is listed as a dependency.

Read it step by step (5 more sections)

Steps

8 steps

h0 have ∀ (s : Finset ℕ), s.Nonempty → ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (1 : ℝ)
hdvd have ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k)
hgeo have ∀ (m : ℕ), (1 : ℕ) ≤ m → ∀ (k : ℕ), ∑ i ∈ Finset.Icc (1 : ℕ) k, ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ))
hpoly have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j
hrec have ∀ (s : Finset ℕ) (a : ℕ) (f : ℕ → ℝ), a ∈ s → ∑ k ∈ s, (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
hsumZ have ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
hV have ∀ (m : ℕ) (s : Finset ℕ), m + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ)
hW have ∀ (N : ℕ) (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))

Shared lemmas

Draft · drafted, by the model named above

These eight steps are stated once at the top and used by parts 1 and 2. With $D_s(k)=\prod_{l\in s,\,l\ne k}(1-2^{\,l-k})$ for a finite set $s$ of naturals:

hsumZ: a finite sum of terms that are each integers is an integer.

h0: $\sum_{k\in s}1/D_s(k)=1$ for nonempty $s$ (Mathlib's Lagrange basis for the nodes $2^{-k}$, evaluated at $0$).

hrec: for $a\in s$, $\sum_{k\in s}2^kf(k)/D_s(k)=\sum_{k\in s\setminus\{a\}}2^kf(k)/D_{s\setminus\{a\}}(k)+2^a\sum_{k\in s}f(k)/D_s(k)$.

hW: $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer for every $e\ge0$ (induction on $|s|+e$ using h0 and hrec). Write $H_e$ for this sum with $s=\{1,\dots,n\}$.

hV: $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$ when $m+1<|s|$.

hgeo: $\sum_{i=1}^{k}2^{-im}=(1-2^{-km})/(2^m-1)$ for $m\ge1$.

hdvd: $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$ for $1\le i\le n-1$ (it divides $2^{k_0}-1$ for the multiple $k_0=i\lfloor n/i\rfloor$ of $i$, which lies in that range).

hpoly: $\prod_{t=1}^{n-1}(1-2^ty)=\sum_{j<n}\alpha_jy^j$ with integers $\alpha_j$.

Steps

numerator_integral have ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
s10 obtain ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))

Part 1: the numerator

Draft · drafted, by the model named above

numerator_integral states exactly what the node erdos-1050--h1-v2--h2--h1 states, and is proved here by the same argument, whose own steps are residue_integral, polypart_integral, head_integral and hmer nested inside it:

$S_1$ is split by partial fractions in the parameter $c$ into a residue part $\sum_{i=1}^n b_i\,(1-c\,2^i)^{-1}$, with $b_i=\sum_{k=i}^nA_k|_{c\to2^{-i}}$, and a remainder. Each $b_i$ is an integer: every term of it is an integer, using that Gaussian binomial coefficients at $q=2$ are integers. The remainder times $3^n\prod_{k=\lceil n/2\rceil}^n(1-2^k)$ is an integer, by hW, hgeo and hdvd.

$3^{\,n-1-i}B_i$ is an integer for $1\le i\le n-1$. Expanding $A_k$ with hpoly, the terms of negative degree vanish by hV and the others are multiples of the $H_e$. The factor $2^i-1$ is cancelled by hdvd.

s10 unpacks the resulting sequence $a$.

Steps

hden have ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
s12 obtain ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))

Part 2: the denominator

Draft · drafted, by the model named above

hden produces integers $b(n)=9^nW(n)\beta_0$ for $n\ge1$. Its step key shows that $3^{\,n-1}\beta_0$ is an integer. Expanding the numerator of $A_k$ with hpoly gives $$\beta_0=-\sum_{j<n}\alpha_j\,c^{\,j}H_j,$$ and $3^{\,n-1}c^{\,j}=3^{\,n-1-j}8^{\,j}$ for $j\le n-1$, so $3^{\,n-1}\beta_0=-\sum_j\alpha_j3^{\,n-1-j}8^jH_j$, an integer by hW. Then, writing $9^n=3\cdot3^{\,n-1}\cdot3^n$ and $3^n\prod_{k=1}^n(1-c\,2^k)=\prod_{k=1}^n(3-8\cdot2^k)$, the proof defines $$b(n)=3\,(n-2)!\prod_{k=1}^n(3-8\cdot2^k)\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)\cdot\bigl(3^{\,n-1}\beta_0\bigr)$$ for $n\ge1$ (and $0$ at $n=0$). s12 unpacks it.

Steps

hpf have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹

Part 3: partial fractions at $x=2^m$

Draft · drafted, by the model named above

hpf: for $n\ge1$ and $m\ge1$, the summand of $E(n)$ evaluated at $x=2^m$, that is $$-\bigl(1-c\,2^{\,m+n}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-m}\bigr)\bigl(1-c\,2^{\,k+m}\bigr)^{-1},$$ equals $$\sum_{k=1}^{n}A_k\,\bigl(1-c\,2^{\,k+m}\bigr)^{-1}+\sum_{i=1}^{n-1}B_i\,(2^m)^{-i}.$$

The step lag is a general partial-fraction formula from Mathlib's Lagrange interpolation. For distinct nonzero $p_k$ ($k\in s$) a polynomial $Q$ of degree less than $|s|$, and $x$ with every $1-p_kx\ne0$,

$$Q(x)\prod_{k\in s}(1-p_kx)^{-1}=\sum_{k\in s}Q(1/p_k)\prod_{j\ne k}(1-p_j/p_k)^{-1}(1-p_kx)^{-1}.$$

It is applied with $p_k=c\,2^k$ for $k=1,\dots,n$ and $Q(X)=-\prod_{t=1}^{n-1}(X-2^t)$, which has degree $n-1$.

The left side is rewritten into this shape (the factor $(2^m)^{-(n-1)}$ comes from $1-2^{k-m}=2^{-m}(2^m-2^k)$).

The coefficients are matched with $A_k\,(c\,2^k)^{-(n-1)}$.

A finite geometric identity, $(1-u)^{-1}+\sum_{i=1}^{N}u^{-i}=u^{-N}(1-u)^{-1}$ with $u=c\,2^{k}2^{m}$ and $N=n-1$, splits each term into the two sums shown. The calc chain at the end of hpf strings these equalities together.

Steps

hres have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))

Part 4: summing over $m$

Draft · drafted, by the model named above

hres: for $n\ge1$, $E(n)=\beta_0\,z-(S_1-S_2)$.

Re-indexing. The terms of $E(n)$ are the summand above at $m=n+j$, $j\ge0$. The terms at $m=1,\dots,n-1$ are $0$, because the factor $k=m$ of $\prod_k(1-2^{\,k-m})$ vanishes. So $E(n)$ is the sum of the summand over all $m\ge1$.

Summability. The proof establishes it for every piece: $(1-c\,2^{\,k+m+1})^{-1}$ is bounded in size by $(1/2)^m$, and $\sum_m (2^{m+1})^{-i}$ is geometric. The full summand is then summable because it equals, by hpf, a sum of summable pieces.

Summing term by term. The proof applies hpf for every $m\ge1$ and sums. On the first part it uses $\sum_{m\ge1}(1-c\,2^{\,k+m})^{-1}=z-\sum_{i=1}^{k}(1-c\,2^i)^{-1}$, the tail of $z$. On the second it uses $\sum_{m\ge1}2^{-mi}=1/(2^i-1)$ for $i\ge1$. The result is $\beta_0z-S_1+S_2$.

The proof then concludes with the single linear_combination described in the first section.

Improve these words →

History: 2 versions on record

2026-10-06 · written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent · not signed · explainer/d31708683eee….md · the first version of its chain
Read this version
What is proved, and the plan
Fix $c = 8/3$ and write $z=\sum_{j\ge0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$. For $n\ge1$ and $1\le k\le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k=-\frac{\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{D_k},\qquad W(n)=(n-2)!\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr)\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),$$ and $E(n)$ for the infinite sum on the right of the statement, so the statement asks for integers with $b(n)z-a(n)=9^nW(n)E(n)$ for $n\ge1$. Put $$\beta_0=\sum_{k=1}^{n}A_k,\qquad S_1=\sum_{k=1}^{n}A_k\sum_{i=1}^{k}\bigl(1-c\,2^i\bigr)^{-1},\qquad S_2=\sum_{i=1}^{n-1}\frac{B_i}{2^i-1},\quad B_i=\sum_{k=1}^{n}A_k\,(c\,2^k)^{-i}.$$

The proof has four parts, then one line:
1. numerator_integral: $9^nW(n)(S_1-S_2)$ is an integer $a(n)$.
2. hden: $9^nW(n)\beta_0$ is an integer $b(n)$.
3. hpf: a partial-fraction identity for the summand of $E(n)$.
4. hres: summing it, $E(n)=\beta_0\,z-(S_1-S_2)$.
The final line takes these $a$ and $b$ and checks $b(n)z-a(n)=9^nW(n)E(n)$ by linear_combination from the three equations. The comments in the proof point to annexes 959e0d5b… and a03db62f… for the informal argument (Borwein's construction). The outline lists no graph node as used: this proof proves part 1 in full itself and does not call the node erdos-1050--h1-v2--h2--h1, although that node states the same fact and is listed as a dependency.
Shared lemmas (h0 hdvd hgeo hpoly hrec hsumZ hV hW)
These eight steps are stated once at the top and used by parts 1 and 2. With $D_s(k)=\prod_{l\in s,\,l\ne k}(1-2^{\,l-k})$ for a finite set $s$ of naturals:
- hsumZ: a finite sum of terms that are each integers is an integer.
- h0: $\sum_{k\in s}1/D_s(k)=1$ for nonempty $s$ (Mathlib's Lagrange basis for the nodes $2^{-k}$, evaluated at $0$).
- hrec: for $a\in s$, $\sum_{k\in s}2^kf(k)/D_s(k)=\sum_{k\in s\setminus\{a\}}2^kf(k)/D_{s\setminus\{a\}}(k)+2^a\sum_{k\in s}f(k)/D_s(k)$.
- hW: $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer for every $e\ge0$ (induction on $|s|+e$ using h0 and hrec). Write $H_e$ for this sum with $s=\{1,\dots,n\}$.
- hV: $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$ when $m+1<|s|$.
- hgeo: $\sum_{i=1}^{k}2^{-im}=(1-2^{-km})/(2^m-1)$ for $m\ge1$.
- hdvd: $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$ for $1\le i\le n-1$ (it divides $2^{k_0}-1$ for the multiple $k_0=i\lfloor n/i\rfloor$ of $i$, which lies in that range).
- hpoly: $\prod_{t=1}^{n-1}(1-2^ty)=\sum_{j<n}\alpha_jy^j$ with integers $\alpha_j$.
Part 1: the numerator (numerator_integral s10)
numerator_integral states exactly what the node erdos-1050--h1-v2--h2--h1 states, and is proved here by the same argument, whose own steps are residue_integral, polypart_integral, head_integral and hmer nested inside it:
- $S_1$ is split by partial fractions in the parameter $c$ into a residue part $\sum_{i=1}^n b_i\,(1-c\,2^i)^{-1}$, with $b_i=\sum_{k=i}^nA_k|_{c\to2^{-i}}$, and a remainder. Each $b_i$ is an integer: every term of it is an integer, using that Gaussian binomial coefficients at $q=2$ are integers. The remainder times $3^n\prod_{k=\lceil n/2\rceil}^n(1-2^k)$ is an integer, by hW, hgeo and hdvd.
- $3^{\,n-1-i}B_i$ is an integer for $1\le i\le n-1$. Expanding $A_k$ with hpoly, the terms of negative degree vanish by hV and the others are multiples of the $H_e$. The factor $2^i-1$ is cancelled by hdvd.
s10 unpacks the resulting sequence $a$.
Part 2: the denominator (hden s12)
hden produces integers $b(n)=9^nW(n)\beta_0$ for $n\ge1$. Its step key shows that $3^{\,n-1}\beta_0$ is an integer. Expanding the numerator of $A_k$ with hpoly gives $$\beta_0=-\sum_{j<n}\alpha_j\,c^{\,j}H_j,$$ and $3^{\,n-1}c^{\,j}=3^{\,n-1-j}8^{\,j}$ for $j\le n-1$, so $3^{\,n-1}\beta_0=-\sum_j\alpha_j3^{\,n-1-j}8^jH_j$, an integer by hW. Then, writing $9^n=3\cdot3^{\,n-1}\cdot3^n$ and $3^n\prod_{k=1}^n(1-c\,2^k)=\prod_{k=1}^n(3-8\cdot2^k)$, the proof defines $$b(n)=3\,(n-2)!\prod_{k=1}^n(3-8\cdot2^k)\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)\cdot\bigl(3^{\,n-1}\beta_0\bigr)$$ for $n\ge1$ (and $0$ at $n=0$). s12 unpacks it.
Part 3: partial fractions at $x=2^m$ (hpf)
hpf: for $n\ge1$ and $m\ge1$, the summand of $E(n)$ evaluated at $x=2^m$, that is $$-\bigl(1-c\,2^{\,m+n}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-m}\bigr)\bigl(1-c\,2^{\,k+m}\bigr)^{-1},$$ equals $$\sum_{k=1}^{n}A_k\,\bigl(1-c\,2^{\,k+m}\bigr)^{-1}+\sum_{i=1}^{n-1}B_i\,(2^m)^{-i}.$$
- The step lag is a general partial-fraction formula from Mathlib's Lagrange interpolation. For distinct nonzero $p_k$ ($k\in s$) a polynomial $Q$ of degree less than $|s|$, and $x$ with every $1-p_kx\ne0$,
$$Q(x)\prod_{k\in s}(1-p_kx)^{-1}=\sum_{k\in s}Q(1/p_k)\prod_{j\ne k}(1-p_j/p_k)^{-1}(1-p_kx)^{-1}.$$
- It is applied with $p_k=c\,2^k$ for $k=1,\dots,n$ and $Q(X)=-\prod_{t=1}^{n-1}(X-2^t)$, which has degree $n-1$.
- The left side is rewritten into this shape (the factor $(2^m)^{-(n-1)}$ comes from $1-2^{k-m}=2^{-m}(2^m-2^k)$).
- The coefficients are matched with $A_k\,(c\,2^k)^{-(n-1)}$.
- A finite geometric identity, $(1-u)^{-1}+\sum_{i=1}^{N}u^{-i}=u^{-N}(1-u)^{-1}$ with $u=c\,2^{k}2^{m}$ and $N=n-1$, splits each term into the two sums shown. The calc chain at the end of hpf strings these equalities together.
Part 4: summing over $m$ (hres)
hres: for $n\ge1$, $E(n)=\beta_0\,z-(S_1-S_2)$.
- Re-indexing. The terms of $E(n)$ are the summand above at $m=n+j$, $j\ge0$. The terms at $m=1,\dots,n-1$ are $0$, because the factor $k=m$ of $\prod_k(1-2^{\,k-m})$ vanishes. So $E(n)$ is the sum of the summand over all $m\ge1$.
- Summability. The proof establishes it for every piece: $(1-c\,2^{\,k+m+1})^{-1}$ is bounded in size by $(1/2)^m$, and $\sum_m (2^{m+1})^{-i}$ is geometric. The full summand is then summable because it equals, by hpf, a sum of summable pieces.
- Summing term by term. The proof applies hpf for every $m\ge1$ and sums. On the first part it uses $\sum_{m\ge1}(1-c\,2^{\,k+m})^{-1}=z-\sum_{i=1}^{k}(1-c\,2^i)^{-1}$, the tail of $z$. On the second it uses $\sum_{m\ge1}2^{-mi}=1/(2^i-1)$ for $i\ge1$. The result is $\beta_0z-S_1+S_2$.
The proof then concludes with the single linear_combination described in the first section.
2026-10-06 · written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent · not signed · current · explainer/658a25aec9a9….md · diff to its predecessor
Read this version
What is proved, and the plan
Fix $c = 8/3$ and write $z=\sum_{j\ge0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$. For $n\ge1$ and $1\le k\le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k=-\frac{\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{D_k},\qquad W(n)=(n-2)!\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr)\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),$$ and $E(n)$ for the infinite sum on the right of the statement, so the statement asks for integers with $b(n)z-a(n)=9^nW(n)E(n)$ for $n\ge1$. Put $$\beta_0=\sum_{k=1}^{n}A_k,\qquad S_1=\sum_{k=1}^{n}A_k\sum_{i=1}^{k}\bigl(1-c\,2^i\bigr)^{-1},\qquad S_2=\sum_{i=1}^{n-1}\frac{B_i}{2^i-1},\quad B_i=\sum_{k=1}^{n}A_k\,(c\,2^k)^{-i}.$$

The proof has four parts, then one line:

(1) numerator_integral: $9^nW(n)(S_1-S_2)$ is an integer $a(n)$.

(2) hden: $9^nW(n)\beta_0$ is an integer $b(n)$.

(3) hpf: a partial-fraction identity for the summand of $E(n)$.

(4) hres: summing it, $E(n)=\beta_0\,z-(S_1-S_2)$.

The final line takes these $a$ and $b$ and checks $b(n)z-a(n)=9^nW(n)E(n)$ by linear_combination from the three equations. The comments in the proof point to annexes 959e0d5b… and a03db62f… for the informal argument (Borwein's construction). The outline lists no graph node as used: this proof proves part 1 in full itself and does not call the node erdos-1050--h1-v2--h2--h1, although that node states the same fact and is listed as a dependency.
Shared lemmas (h0 hdvd hgeo hpoly hrec hsumZ hV hW)
These eight steps are stated once at the top and used by parts 1 and 2. With $D_s(k)=\prod_{l\in s,\,l\ne k}(1-2^{\,l-k})$ for a finite set $s$ of naturals:

hsumZ: a finite sum of terms that are each integers is an integer.

h0: $\sum_{k\in s}1/D_s(k)=1$ for nonempty $s$ (Mathlib's Lagrange basis for the nodes $2^{-k}$, evaluated at $0$).

hrec: for $a\in s$, $\sum_{k\in s}2^kf(k)/D_s(k)=\sum_{k\in s\setminus\{a\}}2^kf(k)/D_{s\setminus\{a\}}(k)+2^a\sum_{k\in s}f(k)/D_s(k)$.

hW: $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer for every $e\ge0$ (induction on $|s|+e$ using h0 and hrec). Write $H_e$ for this sum with $s=\{1,\dots,n\}$.

hV: $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$ when $m+1<|s|$.

hgeo: $\sum_{i=1}^{k}2^{-im}=(1-2^{-km})/(2^m-1)$ for $m\ge1$.

hdvd: $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$ for $1\le i\le n-1$ (it divides $2^{k_0}-1$ for the multiple $k_0=i\lfloor n/i\rfloor$ of $i$, which lies in that range).

hpoly: $\prod_{t=1}^{n-1}(1-2^ty)=\sum_{j<n}\alpha_jy^j$ with integers $\alpha_j$.
Part 1: the numerator (numerator_integral s10)
numerator_integral states exactly what the node erdos-1050--h1-v2--h2--h1 states, and is proved here by the same argument, whose own steps are residue_integral, polypart_integral, head_integral and hmer nested inside it:

$S_1$ is split by partial fractions in the parameter $c$ into a residue part $\sum_{i=1}^n b_i\,(1-c\,2^i)^{-1}$, with $b_i=\sum_{k=i}^nA_k|_{c\to2^{-i}}$, and a remainder. Each $b_i$ is an integer: every term of it is an integer, using that Gaussian binomial coefficients at $q=2$ are integers. The remainder times $3^n\prod_{k=\lceil n/2\rceil}^n(1-2^k)$ is an integer, by hW, hgeo and hdvd.

$3^{\,n-1-i}B_i$ is an integer for $1\le i\le n-1$. Expanding $A_k$ with hpoly, the terms of negative degree vanish by hV and the others are multiples of the $H_e$. The factor $2^i-1$ is cancelled by hdvd.

s10 unpacks the resulting sequence $a$.
Part 2: the denominator (hden s12)
hden produces integers $b(n)=9^nW(n)\beta_0$ for $n\ge1$. Its step key shows that $3^{\,n-1}\beta_0$ is an integer. Expanding the numerator of $A_k$ with hpoly gives $$\beta_0=-\sum_{j<n}\alpha_j\,c^{\,j}H_j,$$ and $3^{\,n-1}c^{\,j}=3^{\,n-1-j}8^{\,j}$ for $j\le n-1$, so $3^{\,n-1}\beta_0=-\sum_j\alpha_j3^{\,n-1-j}8^jH_j$, an integer by hW. Then, writing $9^n=3\cdot3^{\,n-1}\cdot3^n$ and $3^n\prod_{k=1}^n(1-c\,2^k)=\prod_{k=1}^n(3-8\cdot2^k)$, the proof defines $$b(n)=3\,(n-2)!\prod_{k=1}^n(3-8\cdot2^k)\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)\cdot\bigl(3^{\,n-1}\beta_0\bigr)$$ for $n\ge1$ (and $0$ at $n=0$). s12 unpacks it.
Part 3: partial fractions at $x=2^m$ (hpf)
hpf: for $n\ge1$ and $m\ge1$, the summand of $E(n)$ evaluated at $x=2^m$, that is $$-\bigl(1-c\,2^{\,m+n}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-m}\bigr)\bigl(1-c\,2^{\,k+m}\bigr)^{-1},$$ equals $$\sum_{k=1}^{n}A_k\,\bigl(1-c\,2^{\,k+m}\bigr)^{-1}+\sum_{i=1}^{n-1}B_i\,(2^m)^{-i}.$$

The step lag is a general partial-fraction formula from Mathlib's Lagrange interpolation. For distinct nonzero $p_k$ ($k\in s$) a polynomial $Q$ of degree less than $|s|$, and $x$ with every $1-p_kx\ne0$,

$$Q(x)\prod_{k\in s}(1-p_kx)^{-1}=\sum_{k\in s}Q(1/p_k)\prod_{j\ne k}(1-p_j/p_k)^{-1}(1-p_kx)^{-1}.$$

It is applied with $p_k=c\,2^k$ for $k=1,\dots,n$ and $Q(X)=-\prod_{t=1}^{n-1}(X-2^t)$, which has degree $n-1$.

The left side is rewritten into this shape (the factor $(2^m)^{-(n-1)}$ comes from $1-2^{k-m}=2^{-m}(2^m-2^k)$).

The coefficients are matched with $A_k\,(c\,2^k)^{-(n-1)}$.

A finite geometric identity, $(1-u)^{-1}+\sum_{i=1}^{N}u^{-i}=u^{-N}(1-u)^{-1}$ with $u=c\,2^{k}2^{m}$ and $N=n-1$, splits each term into the two sums shown. The calc chain at the end of hpf strings these equalities together.
Part 4: summing over $m$ (hres)
hres: for $n\ge1$, $E(n)=\beta_0\,z-(S_1-S_2)$.

Re-indexing. The terms of $E(n)$ are the summand above at $m=n+j$, $j\ge0$. The terms at $m=1,\dots,n-1$ are $0$, because the factor $k=m$ of $\prod_k(1-2^{\,k-m})$ vanishes. So $E(n)$ is the sum of the summand over all $m\ge1$.

Summability. The proof establishes it for every piece: $(1-c\,2^{\,k+m+1})^{-1}$ is bounded in size by $(1/2)^m$, and $\sum_m (2^{m+1})^{-i}$ is geometric. The full summand is then summable because it equals, by hpf, a sum of summable pieces.

Summing term by term. The proof applies hpf for every $m\ge1$ and sums. On the first part it uses $\sum_{m\ge1}(1-c\,2^{\,k+m})^{-1}=z-\sum_{i=1}^{k}(1-c\,2^i)^{-1}$, the tail of $z$. On the second it uses $\sum_{m\ge1}2^{-mi}=1/(2^i-1)$ for $i\ge1$. The result is $\beta_0z-S_1+S_2$.

The proof then concludes with the single linear_combination described in the first section.

Diff of 658a25aec9a9 against d31708683eee

@@ -8,8 +8,11 @@
 The proof has four parts, then one line:
 
-1. `numerator_integral`: $9^nW(n)(S_1-S_2)$ is an integer $a(n)$.
-2. `hden`: $9^nW(n)\beta_0$ is an integer $b(n)$.
-3. `hpf`: a partial-fraction identity for the summand of $E(n)$.
-4. `hres`: summing it, $E(n)=\beta_0\,z-(S_1-S_2)$.
+(1) `numerator_integral`: $9^nW(n)(S_1-S_2)$ is an integer $a(n)$.
+
+(2) `hden`: $9^nW(n)\beta_0$ is an integer $b(n)$.
+
+(3) `hpf`: a partial-fraction identity for the summand of $E(n)$.
+
+(4) `hres`: summing it, $E(n)=\beta_0\,z-(S_1-S_2)$.
 
 The final line takes these $a$ and $b$ and checks $b(n)z-a(n)=9^nW(n)E(n)$ by `linear_combination` from the three equations. The comments in the proof point to annexes 959e0d5b… and a03db62f… for the informal argument (Borwein's construction). The outline lists no graph node as used: this proof proves part 1 in full itself and does not call the node erdos-1050--h1-v2--h2--h1, although that node states the same fact and is listed as a dependency.
@@ -19,12 +22,19 @@
 These eight steps are stated once at the top and used by parts 1 and 2. With $D_s(k)=\prod_{l\in s,\,l\ne k}(1-2^{\,l-k})$ for a finite set $s$ of naturals:
 
-- `hsumZ`: a finite sum of terms that are each integers is an integer.
-- `h0`: $\sum_{k\in s}1/D_s(k)=1$ for nonempty $s$ (Mathlib's Lagrange basis for the nodes $2^{-k}$, evaluated at $0$).
-- `hrec`: for $a\in s$, $\sum_{k\in s}2^kf(k)/D_s(k)=\sum_{k\in s\setminus\{a\}}2^kf(k)/D_{s\setminus\{a\}}(k)+2^a\sum_{k\in s}f(k)/D_s(k)$.
-- `hW`: $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer for every $e\ge0$ (induction on $|s|+e$ using `h0` and `hrec`). Write $H_e$ for this sum with $s=\{1,\dots,n\}$.
-- `hV`: $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$ when $m+1<|s|$.
-- `hgeo`: $\sum_{i=1}^{k}2^{-im}=(1-2^{-km})/(2^m-1)$ for $m\ge1$.
-- `hdvd`: $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$ for $1\le i\le n-1$ (it divides $2^{k_0}-1$ for the multiple $k_0=i\lfloor n/i\rfloor$ of $i$, which lies in that range).
-- `hpoly`: $\prod_{t=1}^{n-1}(1-2^ty)=\sum_{j<n}\alpha_jy^j$ with integers $\alpha_j$.
+`hsumZ`: a finite sum of terms that are each integers is an integer.
+
+`h0`: $\sum_{k\in s}1/D_s(k)=1$ for nonempty $s$ (Mathlib's Lagrange basis for the nodes $2^{-k}$, evaluated at $0$).
+
+`hrec`: for $a\in s$, $\sum_{k\in s}2^kf(k)/D_s(k)=\sum_{k\in s\setminus\{a\}}2^kf(k)/D_{s\setminus\{a\}}(k)+2^a\sum_{k\in s}f(k)/D_s(k)$.
+
+`hW`: $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer for every $e\ge0$ (induction on $|s|+e$ using `h0` and `hrec`). Write $H_e$ for this sum with $s=\{1,\dots,n\}$.
+
+`hV`: $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$ when $m+1<|s|$.
+
+`hgeo`: $\sum_{i=1}^{k}2^{-im}=(1-2^{-km})/(2^m-1)$ for $m\ge1$.
+
+`hdvd`: $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$ for $1\le i\le n-1$ (it divides $2^{k_0}-1$ for the multiple $k_0=i\lfloor n/i\rfloor$ of $i$, which lies in that range).
+
+`hpoly`: $\prod_{t=1}^{n-1}(1-2^ty)=\sum_{j<n}\alpha_jy^j$ with integers $\alpha_j$.
 
 ## Part 1: the numerator {steps: numerator_integral s10}
@@ -32,6 +42,7 @@
 `numerator_integral` states exactly what the node erdos-1050--h1-v2--h2--h1 states, and is proved here by the same argument, whose own steps are `residue_integral`, `polypart_integral`, `head_integral` and `hmer` nested inside it:
 
-- $S_1$ is split by partial fractions in the parameter $c$ into a residue part $\sum_{i=1}^n b_i\,(1-c\,2^i)^{-1}$, with $b_i=\sum_{k=i}^nA_k|_{c\to2^{-i}}$, and a remainder. Each $b_i$ is an integer: every term of it is an integer, using that Gaussian binomial coefficients at $q=2$ are integers. The remainder times $3^n\prod_{k=\lceil n/2\rceil}^n(1-2^k)$ is an integer, by `hW`, `hgeo` and `hdvd`.
-- $3^{\,n-1-i}B_i$ is an integer for $1\le i\le n-1$. Expanding $A_k$ with `hpoly`, the terms of negative degree vanish by `hV` and the others are multiples of the $H_e$. The factor $2^i-1$ is cancelled by `hdvd`.
+$S_1$ is split by partial fractions in the parameter $c$ into a residue part $\sum_{i=1}^n b_i\,(1-c\,2^i)^{-1}$, with $b_i=\sum_{k=i}^nA_k|_{c\to2^{-i}}$, and a remainder. Each $b_i$ is an integer: every term of it is an integer, using that Gaussian binomial coefficients at $q=2$ are integers. The remainder times $3^n\prod_{k=\lceil n/2\rceil}^n(1-2^k)$ is an integer, by `hW`, `hgeo` and `hdvd`.
+
+$3^{\,n-1-i}B_i$ is an integer for $1\le i\le n-1$. Expanding $A_k$ with `hpoly`, the terms of negative degree vanish by `hV` and the others are multiples of the $H_e$. The factor $2^i-1$ is cancelled by `hdvd`.
 
 `s10` unpacks the resulting sequence $a$.
@@ -52,10 +63,15 @@
 $$\sum_{k=1}^{n}A_k\,\bigl(1-c\,2^{\,k+m}\bigr)^{-1}+\sum_{i=1}^{n-1}B_i\,(2^m)^{-i}.$$
 
-- The step `lag` is a general partial-fraction formula from Mathlib's Lagrange interpolation. For distinct nonzero $p_k$ ($k\in s$) a polynomial $Q$ of degree less than $|s|$, and $x$ with every $1-p_kx\ne0$,
+The step `lag` is a general partial-fraction formula from Mathlib's Lagrange interpolation. For distinct nonzero $p_k$ ($k\in s$) a polynomial $Q$ of degree less than $|s|$, and $x$ with every $1-p_kx\ne0$,
+
 $$Q(x)\prod_{k\in s}(1-p_kx)^{-1}=\sum_{k\in s}Q(1/p_k)\prod_{j\ne k}(1-p_j/p_k)^{-1}(1-p_kx)^{-1}.$$
-- It is applied with $p_k=c\,2^k$ for $k=1,\dots,n$ and $Q(X)=-\prod_{t=1}^{n-1}(X-2^t)$, which has degree $n-1$.
-- The left side is rewritten into this shape (the factor $(2^m)^{-(n-1)}$ comes from $1-2^{k-m}=2^{-m}(2^m-2^k)$).
-- The coefficients are matched with $A_k\,(c\,2^k)^{-(n-1)}$.
-- A finite geometric identity, $(1-u)^{-1}+\sum_{i=1}^{N}u^{-i}=u^{-N}(1-u)^{-1}$ with $u=c\,2^{k}2^{m}$ and $N=n-1$, splits each term into the two sums shown. The `calc` chain at the end of `hpf` strings these equalities together.
+
+It is applied with $p_k=c\,2^k$ for $k=1,\dots,n$ and $Q(X)=-\prod_{t=1}^{n-1}(X-2^t)$, which has degree $n-1$.
+
+The left side is rewritten into this shape (the factor $(2^m)^{-(n-1)}$ comes from $1-2^{k-m}=2^{-m}(2^m-2^k)$).
+
+The coefficients are matched with $A_k\,(c\,2^k)^{-(n-1)}$.
+
+A finite geometric identity, $(1-u)^{-1}+\sum_{i=1}^{N}u^{-i}=u^{-N}(1-u)^{-1}$ with $u=c\,2^{k}2^{m}$ and $N=n-1$, splits each term into the two sums shown. The `calc` chain at the end of `hpf` strings these equalities together.
 
 ## Part 4: summing over $m$ {steps: hres}
@@ -63,7 +79,9 @@
 `hres`: for $n\ge1$, $E(n)=\beta_0\,z-(S_1-S_2)$.
 
-- **Re-indexing.** The terms of $E(n)$ are the summand above at $m=n+j$, $j\ge0$. The terms at $m=1,\dots,n-1$ are $0$, because the factor $k=m$ of $\prod_k(1-2^{\,k-m})$ vanishes. So $E(n)$ is the sum of the summand over all $m\ge1$.
-- **Summability.** The proof establishes it for every piece: $(1-c\,2^{\,k+m+1})^{-1}$ is bounded in size by $(1/2)^m$, and $\sum_m (2^{m+1})^{-i}$ is geometric. The full summand is then summable because it equals, by `hpf`, a sum of summable pieces.
-- **Summing term by term.** The proof applies `hpf` for every $m\ge1$ and sums. On the first part it uses $\sum_{m\ge1}(1-c\,2^{\,k+m})^{-1}=z-\sum_{i=1}^{k}(1-c\,2^i)^{-1}$, the tail of $z$. On the second it uses $\sum_{m\ge1}2^{-mi}=1/(2^i-1)$ for $i\ge1$. The result is $\beta_0z-S_1+S_2$.
+Re-indexing. The terms of $E(n)$ are the summand above at $m=n+j$, $j\ge0$. The terms at $m=1,\dots,n-1$ are $0$, because the factor $k=m$ of $\prod_k(1-2^{\,k-m})$ vanishes. So $E(n)$ is the sum of the summand over all $m\ge1$.
+
+Summability. The proof establishes it for every piece: $(1-c\,2^{\,k+m+1})^{-1}$ is bounded in size by $(1/2)^m$, and $\sum_m (2^{m+1})^{-i}$ is geometric. The full summand is then summable because it equals, by `hpf`, a sum of summable pieces.
+
+Summing term by term. The proof applies `hpf` for every $m\ge1$ and sums. On the first part it uses $\sum_{m\ge1}(1-c\,2^{\,k+m})^{-1}=z-\sum_{i=1}^{k}(1-c\,2^i)^{-1}$, the tail of $z$. On the second it uses $\sum_{m\ge1}2^{-mi}=1/(2^i-1)$ for $i\ge1$. The result is $\beta_0z-S_1+S_2$.
 
 The proof then concludes with the single `linear_combination` described in the first section.

Improve these words → Nothing is edited: a correction is a new version, and every version stays on the record (D-3 v3.30).

How it was proved

Proved by Proof.lean, checked by the kernel. Its attestation, 000335, records the gate’s run.

The proof step by step (outline)

Checked by the kernel — an outline of the attested proof below: attestations/000335.json, merge commit b16ab29fc51b.

Outline of Proof.lean, extracted by the gate at 82834c4684b8 from the proof’s elaboration: each step is a claim the proof establishes and the goal it leaves. Open a step for what it uses and its Lean lines.

hsumZ have ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
Uses library Finset, Finset.induction_on.
Lean, lines 23–31
```
  have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
    intro s g
    induction s using Finset.induction_on with
    | empty => intro _; exact ⟨0, by simp⟩
    | insert a s ha ih =>
      intro h
      obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
      obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
      exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 23–31 at b16ab29fc51b ↗
1. hsumZ.s1 case printed form did not read back: shown as its Lean lines only
  | empty => intro _; exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 26 at b16ab29fc51b ↗
2. hsumZ.s2 case
  Leaves (∀ i ∈ insert a s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with a : ℕ, s : Finset ℕ, ha : a ∉ s, ih : (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Exists.intro, Finset.sum_insert.
  Lean, lines 27–31
  | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 27–31 at b16ab29fc51b ↗
  hsumZ.s2.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = g a
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with za : ℤ, hza : (↑za : ℝ) = g a.
  Uses library Finset.mem_insert_self.
  Lean, line 29
  obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 29 at b16ab29fc51b ↗
  hsumZ.s2.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with zs : ℤ, hzs : (↑zs : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset.mem_insert_of_mem.
  Lean, line 30
  obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 30 at b16ab29fc51b ↗
h0 have ∀ (s : Finset ℕ), s.Nonempty → ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (1 : ℝ)
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with h0 : ∀ (s : Finset ℕ), s.Nonempty → ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (1 : ℝ).
Uses library Finset, Polynomial.eval_finset_sum, Polynomial.eval_one, Eq.trans, Finset.sum_congr, rfl, Lagrange.basis, Polynomial.eval_prod, one_div, Finset.prod_inv_distrib, Finset.prod_congr, Lagrange.basisDivisor, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_sub, Polynomial.eval_X, zpow_sub₀, two_ne_zero, zpow_natCast, mul_inv, zero_sub, mul_assoc, inv_mul_cancel₀, neg_ne_zero, inv_ne_zero, mul_one.
Lean, lines 33–49
```
  have h0 : ∀ s : Finset ℕ, s.Nonempty → ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = 1 := by
    intro s hs
    have hv : Set.InjOn (fun k : ℕ => ((2 : ℝ) ^ k)⁻¹) (s : Set ℕ) := by
      intro a _ b _ hab
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (inv_injective hab)
    have hsum := congrArg (Polynomial.eval (0 : ℝ)) (Lagrange.sum_basis hv hs)
    rw [Polynomial.eval_finset_sum, Polynomial.eval_one] at hsum
    refine Eq.trans (Finset.sum_congr rfl (fun k hk => ?_)) hsum
    unfold Lagrange.basis
    rw [Polynomial.eval_prod, one_div, ← Finset.prod_inv_distrib]
    refine Finset.prod_congr rfl (fun l hl => ?_)
    have ha : (2 : ℝ) ^ k ≠ 0 := by positivity
    have hb : (2 : ℝ) ^ l ≠ 0 := by positivity
    simp only [Lagrange.basisDivisor, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_sub, Polynomial.eval_X]
    rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast,
      show ((2 : ℝ) ^ k)⁻¹ - ((2 : ℝ) ^ l)⁻¹ = (1 - (2 : ℝ) ^ l / (2 : ℝ) ^ k) * (-((2 : ℝ) ^ l)⁻¹) by field_simp; ring,
      mul_inv, zero_sub, mul_assoc, inv_mul_cancel₀ (neg_ne_zero.mpr (inv_ne_zero hb)), mul_one]
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 33–49 at b16ab29fc51b ↗
1. h0.hv have Set.InjOn (fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) (↑s : Set ℕ)
  Leaves ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (1 : ℝ), with hv : Set.InjOn (fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) (↑s : Set ℕ).
  Uses library Set.InjOn, Set, pow_right_injective₀, inv_injective.
  Lean, lines 35–37
  have hv : Set.InjOn (fun k : ℕ => ((2 : ℝ) ^ k)⁻¹) (s : Set ℕ) := by intro a _ b _ hab exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (inv_injective hab)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 35–37 at b16ab29fc51b ↗
2. h0.hsum have Polynomial.eval (0 : ℝ) (∑ j ∈ s, Lagrange.basis s (fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) j) = Polynomial.eval (0 : ℝ) (1 : Polynomial ℝ)
  Leaves ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (1 : ℝ), with hsum : Polynomial.eval (0 : ℝ) (∑ j ∈ s, Lagrange.basis s (fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) j) = Polynomial.eval (0 : ℝ) (1 : Polynomial ℝ).
  Uses library congrArg, Polynomial.eval, Lagrange.sum_basis.
  Lean, line 38
  have hsum := congrArg (Polynomial.eval (0 : ℝ)) (Lagrange.sum_basis hv hs)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 38 at b16ab29fc51b ↗
3. h0.ha have (2 : ℝ) ^ k ≠ (0 : ℝ) routine: positivity
  Leaves ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))⁻¹ = Polynomial.eval (0 : ℝ) (Lagrange.basisDivisor ((fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) k) ((fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) l)), with ha : (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 44
  have ha : (2 : ℝ) ^ k ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 44 at b16ab29fc51b ↗
4. h0.hb have (2 : ℝ) ^ l ≠ (0 : ℝ) routine: positivity
  Leaves ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))⁻¹ = Polynomial.eval (0 : ℝ) (Lagrange.basisDivisor ((fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) k) ((fun (k : ℕ) => ((2 : ℝ) ^ k)⁻¹) l)), with hb : (2 : ℝ) ^ l ≠ (0 : ℝ).
  Lean, line 45
  have hb : (2 : ℝ) ^ l ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 45 at b16ab29fc51b ↗
hrec have ∀ (s : Finset ℕ) (a : ℕ) (f : ℕ → ℝ), a ∈ s → ∑ k ∈ s, (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hrec : ∀ (s : Finset ℕ) (a : ℕ) (f : ℕ → ℝ), a ∈ s → ∑ k ∈ s, (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Uses library Finset, Finset.add_sum_erase, mul_add, Finset.mul_sum, add_left_comm, Finset.sum_add_distrib.
Lean, lines 51–73
```
  have hrec : ∀ (s : Finset ℕ) (a : ℕ) (f : ℕ → ℝ), a ∈ s → ∑ k ∈ s, (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
    intro s a f ha
    rw [← Finset.add_sum_erase s (fun k => f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) ha, ← Finset.add_sum_erase s (fun k => (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) ha, mul_add, Finset.mul_sum, add_left_comm, ← Finset.sum_add_distrib]
    congr 1
    · ring
    · refine Finset.sum_congr rfl (fun k hk => ?_)
      have hka : k ≠ a := Finset.ne_of_mem_erase hk
      have hmem : a ∈ s.erase k := Finset.mem_erase.mpr ⟨Ne.symm hka, ha⟩
      have hsplit := Finset.mul_prod_erase (s.erase k) (fun l => (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) hmem
      rw [Finset.erase_right_comm] at hsplit
      rw [← hsplit]
      have hρ : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) ≠ 0 := by
        intro h1
        have h2 : (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) = 2 ^ (0 : ℤ) := by rw [zpow_zero]; linarith
        have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2
        omega
      have hq : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) * (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)))⁻¹ = 1 := mul_inv_cancel₀ hρ
      have hak : (2 : ℝ) ^ a = (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) * 2 ^ k := by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      simp only [div_eq_mul_inv, mul_inv]
      rw [hak]
      linear_combination ((2 : ℝ) ^ k * f k * (∏ l ∈ (s.erase a).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hq
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 51–73 at b16ab29fc51b ↗
1. hrec.s1 case routine: ring
  Leaves (2 : ℝ) ^ a * f a / ∏ l ∈ s.erase a, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑a : ℤ))) = (2 : ℝ) ^ a * (f a / ∏ l ∈ s.erase a, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑a : ℤ)))).
  Lean, line 55
  · ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 55 at b16ab29fc51b ↗
2. hrec.s2 case
  Leaves ∑ x ∈ s.erase a, (2 : ℝ) ^ x * f x / ∏ l ∈ s.erase x, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑x : ℤ))) = ∑ x ∈ s.erase a, ((2 : ℝ) ^ x * f x / ∏ l ∈ (s.erase a).erase x, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑x : ℤ))) + (2 : ℝ) ^ a * (f x / ∏ l ∈ s.erase x, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑x : ℤ))))).
  Uses library Finset.sum_congr, rfl, Finset.erase_right_comm, div_eq_mul_inv, mul_inv.
  Lean, lines 56–73
  · refine Finset.sum_congr rfl (fun k hk => ?_) have hka : k ≠ a := Finset.ne_of_mem_erase hk have hmem : a ∈ s.erase k := Finset.mem_erase.mpr ⟨Ne.symm hka, ha⟩ have hsplit := Finset.mul_prod_erase (s.erase k) (fun l => (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) hmem rw [Finset.erase_right_comm] at hsplit rw [← hsplit] have hρ : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) ≠ 0 := by intro h1 have h2 : (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) = 2 ^ (0 : ℤ) := by rw [zpow_zero]; linarith have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2 omega have hq : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) * (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)))⁻¹ = 1 := mul_inv_cancel₀ hρ have hak : (2 : ℝ) ^ a = (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) * 2 ^ k := by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp simp only [div_eq_mul_inv, mul_inv] rw [hak] linear_combination ((2 : ℝ) ^ k * f k * (∏ l ∈ (s.erase a).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hq
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 56–73 at b16ab29fc51b ↗
  hrec.s2.hka have k ≠ a
  Leaves (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hka : k ≠ a.
  Uses library Finset.ne_of_mem_erase.
  Lean, line 57
  have hka : k ≠ a := Finset.ne_of_mem_erase hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 57 at b16ab29fc51b ↗
  hrec.s2.hmem have a ∈ s.erase k
  Leaves (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hmem : a ∈ s.erase k.
  Uses library Finset.mem_erase, And.intro, Ne.symm.
  Lean, line 58
  have hmem : a ∈ s.erase k := Finset.mem_erase.mpr ⟨Ne.symm hka, ha⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 58 at b16ab29fc51b ↗
  hrec.s2.hsplit have ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase k).erase a, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))) = ∏ x ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ)))
  Leaves (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hsplit : ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase k).erase a, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))) = ∏ x ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))).
  Uses library Finset.mul_prod_erase.
  Lean, line 59
  have hsplit := Finset.mul_prod_erase (s.erase k) (fun l => (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) hmem
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 59 at b16ab29fc51b ↗
  hrec.s2.h_x3c1 (hρ) have (1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) ≠ (0 : ℝ)
  Leaves (2 : ℝ) ^ k * f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ)))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))))), with hρ : (1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) ≠ (0 : ℝ).
  Lean, lines 62–66
  have hρ : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) ≠ 0 := by intro h1 have h2 : (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) = 2 ^ (0 : ℤ) := by rw [zpow_zero]; linarith have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2 omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 62–66 at b16ab29fc51b ↗
  hrec.s2.h_x3c1.h2 have (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) = (2 : ℝ) ^ (0 : ℤ)
  Leaves False, with h2 : (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) = (2 : ℝ) ^ (0 : ℤ).
  Uses library zpow_zero.
  Lean, line 64
  have h2 : (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) = 2 ^ (0 : ℤ) := by rw [zpow_zero]; linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 64 at b16ab29fc51b ↗
  hrec.s2.h_x3c1.h3 have (↑a : ℤ) - (↑k : ℤ) = (0 : ℤ)
  Leaves False, with h3 : (↑a : ℤ) - (↑k : ℤ) = (0 : ℤ).
  Uses library zpow_right_injective₀.
  Lean, line 65
  have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 65 at b16ab29fc51b ↗
  hrec.s2.hq have ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)))⁻¹ = (1 : ℝ)
  Leaves (2 : ℝ) ^ k * f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ)))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))))), with hq : ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)))⁻¹ = (1 : ℝ).
  Uses library mul_inv_cancel₀.
  Lean, line 67
  have hq : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) * (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)))⁻¹ = 1 := mul_inv_cancel₀ hρ
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 67 at b16ab29fc51b ↗
  hrec.s2.hak have (2 : ℝ) ^ a = (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) * (2 : ℝ) ^ k
  Leaves (2 : ℝ) ^ k * f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ)))) = (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * (f k / (((1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ))) * ∏ x ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑x : ℤ) - (↑k : ℤ))))), with hak : (2 : ℝ) ^ a = (2 : ℝ) ^ ((↑a : ℤ) - (↑k : ℤ)) * (2 : ℝ) ^ k.
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast.
  Lean, lines 68–70
  have hak : (2 : ℝ) ^ a = (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) * 2 ^ k := by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 68–70 at b16ab29fc51b ↗
hW have ∀ (N : ℕ) (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hW : ∀ (N : ℕ) (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Uses library Finset.
Lean, lines 75–98
```
  have hW : ∀ N : ℕ, ∀ (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ z : ℤ, (z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
    intro N
    induction N with
    | zero =>
      intro s e h
      have hs : s = ∅ := Finset.card_eq_zero.mp (by omega)
      subst hs
      exact ⟨0, by simp⟩
    | succ N ih =>
      intro s e h
      rcases s.eq_empty_or_nonempty with rfl | hs
      · exact ⟨0, by simp⟩
      rcases Nat.eq_zero_or_pos e with rfl | he
      · exact ⟨1, by rw [Int.cast_one]; exact (h0 s hs).symm.trans (Finset.sum_congr rfl (fun k _ => by rw [pow_zero]))⟩
      obtain ⟨e', rfl⟩ : ∃ e', e = e' + 1 := ⟨e - 1, by omega⟩
      obtain ⟨a, ha⟩ := hs
      obtain ⟨z1, hz1⟩ := ih (s.erase a) (e' + 1) (by rw [Finset.card_erase_of_mem ha]; have := Finset.card_pos.mpr ⟨a, ha⟩; omega)
      obtain ⟨z2, hz2⟩ := ih s e' (by omega)
      refine ⟨z1 + 2 ^ a * z2, ?_⟩
      have hr := hrec s a (fun k => ((2 : ℝ) ^ k) ^ e') ha
      simp only [← pow_succ'] at hr
      push_cast
      rw [hz1, hz2]
      exact hr.symm
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 75–98 at b16ab29fc51b ↗
1. hW.s1 case
  Leaves ∀ (s : Finset ℕ) (e : ℕ), s.card + e = (0 : ℕ) → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Nat.zero, Exists.intro.
  Lean, lines 78–82
  | zero => intro s e h have hs : s = ∅ := Finset.card_eq_zero.mp (by omega) subst hs exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 78–82 at b16ab29fc51b ↗
  hW.s1.hs have s = ∅
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hs : s = ∅.
  Uses library Finset.card_eq_zero.
  Lean, line 80
  have hs : s = ∅ := Finset.card_eq_zero.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 80 at b16ab29fc51b ↗
2. hW.s2 case
  Leaves ∀ (s : Finset ℕ) (e : ℕ), s.card + e = N + (1 : ℕ) → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with N : ℕ, ih : ∀ (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Nat.succ, Nat.eq_zero_or_pos, Exists.intro, pow_succ'.
  Lean, lines 83–98
  | succ N ih => intro s e h rcases s.eq_empty_or_nonempty with rfl | hs · exact ⟨0, by simp⟩ rcases Nat.eq_zero_or_pos e with rfl | he · exact ⟨1, by rw [Int.cast_one]; exact (h0 s hs).symm.trans (Finset.sum_congr rfl (fun k _ => by rw [pow_zero]))⟩ obtain ⟨e', rfl⟩ : ∃ e', e = e' + 1 := ⟨e - 1, by omega⟩ obtain ⟨a, ha⟩ := hs obtain ⟨z1, hz1⟩ := ih (s.erase a) (e' + 1) (by rw [Finset.card_erase_of_mem ha]; have := Finset.card_pos.mpr ⟨a, ha⟩; omega) obtain ⟨z2, hz2⟩ := ih s e' (by omega) refine ⟨z1 + 2 ^ a * z2, ?_⟩ have hr := hrec s a (fun k => ((2 : ℝ) ^ k) ^ e') ha simp only [← pow_succ'] at hr push_cast rw [hz1, hz2] exact hr.symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 83–98 at b16ab29fc51b ↗
  hW.s2.s1 case
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ ∅, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ ∅.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Exists.intro.
  Lean, line 86
  · exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 86 at b16ab29fc51b ↗
  hW.s2.s2 case
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (0 : ℕ) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Exists.intro, Int.cast_one, Finset.sum_congr, rfl, pow_zero.
  Lean, line 88
  · exact ⟨1, by rw [Int.cast_one]; exact (h0 s hs).symm.trans (Finset.sum_congr rfl (fun k _ => by rw [pow_zero]))⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 88 at b16ab29fc51b ↗
  hW.s2.s3 obtain ∃ (e' : ℕ), e = e' + (1 : ℕ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with e' : ℕ, h : s.card + (e' + (1 : ℕ)) = N + (1 : ℕ), he : e' + (1 : ℕ) > (0 : ℕ).
  Uses library Exists.intro.
  Lean, line 89
  obtain ⟨e', rfl⟩ : ∃ e', e = e' + 1 := ⟨e - 1, by omega⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 89 at b16ab29fc51b ↗
  hW.s2.s4 obtain s.Nonempty
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with a : ℕ, ha : a ∈ s.
  Lean, line 90
  obtain ⟨a, ha⟩ := hs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 90 at b16ab29fc51b ↗
  hW.s2.s5 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s.erase a, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with z1 : ℤ, hz1 : (↑z1 : ℝ) = ∑ k ∈ s.erase a, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.card_erase_of_mem.
  Lean, line 91
  obtain ⟨z1, hz1⟩ := ih (s.erase a) (e' + 1) (by rw [Finset.card_erase_of_mem ha]; have := Finset.card_pos.mpr ⟨a, ha⟩; omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 91 at b16ab29fc51b ↗
  hW.s2.s5.this have (0 : ℕ) < s.card
  Leaves s.card - (1 : ℕ) + (e' + (1 : ℕ)) = N, with this : (0 : ℕ) < s.card.
  Uses library Finset.card_pos, Exists.intro.
  Lean, line 91
  obtain ⟨z1, hz1⟩ := ih (s.erase a) (e' + 1) (by rw [Finset.card_erase_of_mem ha]; have := Finset.card_pos.mpr ⟨a, ha⟩; omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 91 at b16ab29fc51b ↗
  hW.s2.s6 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with z2 : ℤ, hz2 : (↑z2 : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Lean, line 92
  obtain ⟨z2, hz2⟩ := ih s e' (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 92 at b16ab29fc51b ↗
  hW.s2.hr have ∑ k ∈ s, (2 : ℝ) ^ k * ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves (↑(z1 + (2 : ℤ) ^ a * z2) : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ (e' + (1 : ℕ)) / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hr : ∑ k ∈ s, (2 : ℝ) ^ k * ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e' / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Lean, line 94
  have hr := hrec s a (fun k => ((2 : ℝ) ^ k) ^ e') ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 94 at b16ab29fc51b ↗
hV have ∀ (m : ℕ) (s : Finset ℕ), m + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ)
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hV : ∀ (m : ℕ) (s : Finset ℕ), m + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
Uses library Finset.
Lean, lines 100–127
```
  have hV : ∀ m : ℕ, ∀ s : Finset ℕ, m + 1 < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = 0 := by
    intro m
    induction m with
    | zero =>
      intro s hs
      obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
      have hne : (s.erase a).Nonempty := Finset.card_pos.mp (by rw [Finset.card_erase_of_mem ha]; omega)
      have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹) ha
      have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ = 1 := fun k => by
        rw [zero_add, pow_one]; exact mul_inv_cancel₀ (by positivity)
      simp only [e] at h
      rw [h0 s ⟨a, ha⟩, h0 _ hne] at h
      have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
      rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5
      · exact absurd h5 h2a
      · exact h5
    | succ m ih =>
      intro s hs
      obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
      have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹) ha
      have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ = (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ := fun k => by
        rw [pow_succ' _ (m + 1), mul_inv, ← mul_assoc, mul_inv_cancel₀ (by positivity), one_mul]
      simp only [e] at h
      rw [ih s (by omega), ih (s.erase a) (by rw [Finset.card_erase_of_mem ha]; omega)] at h
      have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
      rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5
      · exact absurd h5 h2a
      · exact h5
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 100–127 at b16ab29fc51b ↗
1. hV.s1 case
  Leaves ∀ (s : Finset ℕ), (0 : ℕ) + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Uses library Nat.zero, Exists.intro, mul_eq_zero.
  Lean, lines 103–115
  | zero => intro s hs obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega) have hne : (s.erase a).Nonempty := Finset.card_pos.mp (by rw [Finset.card_erase_of_mem ha]; omega) have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹) ha have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ = 1 := fun k => by rw [zero_add, pow_one]; exact mul_inv_cancel₀ (by positivity) simp only [e] at h rw [h0 s ⟨a, ha⟩, h0 _ hne] at h have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5 · exact absurd h5 h2a · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 103–115 at b16ab29fc51b ↗
  hV.s1.s1 obtain s.Nonempty
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with a : ℕ, ha : a ∈ s.
  Uses library Finset.card_pos.
  Lean, line 105
  obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 105 at b16ab29fc51b ↗
  hV.s1.hne have (s.erase a).Nonempty
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with hne : (s.erase a).Nonempty.
  Uses library Finset.card_pos, Finset.card_erase_of_mem.
  Lean, line 106
  have hne : (s.erase a).Nonempty := Finset.card_pos.mp (by rw [Finset.card_erase_of_mem ha]; omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 106 at b16ab29fc51b ↗
  hV.s1.h have ∑ k ∈ s, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with h : ∑ k ∈ s, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Lean, line 107
  have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹) ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 107 at b16ab29fc51b ↗
  hV.s1.e have ∀ (k : ℕ), (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ = (1 : ℝ)
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with e : ∀ (k : ℕ), (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ = (1 : ℝ).
  Uses library zero_add, pow_one, mul_inv_cancel₀.
  Lean, lines 108–109
  have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ = 1 := fun k => by rw [zero_add, pow_one]; exact mul_inv_cancel₀ (by positivity)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 108–109 at b16ab29fc51b ↗
  hV.s1.h2a have (2 : ℝ) ^ a ≠ (0 : ℝ) routine: positivity
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with h2a : (2 : ℝ) ^ a ≠ (0 : ℝ).
  Lean, line 112
  have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 112 at b16ab29fc51b ↗
  hV.s1.s6 case
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Uses library absurd.
  Lean, line 114
  · exact absurd h5 h2a
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 114 at b16ab29fc51b ↗
  hV.s1.s7 case
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ ((0 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Lean, line 115
  · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 115 at b16ab29fc51b ↗
2. hV.s2 case
  Leaves ∀ (s : Finset ℕ), m + (1 : ℕ) + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with m : ℕ, ih : ∀ (s : Finset ℕ), m + (1 : ℕ) < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Uses library Nat.succ, Finset.card_erase_of_mem, mul_eq_zero.
  Lean, lines 116–127
  | succ m ih => intro s hs obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega) have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹) ha have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ = (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ := fun k => by rw [pow_succ' _ (m + 1), mul_inv, ← mul_assoc, mul_inv_cancel₀ (by positivity), one_mul] simp only [e] at h rw [ih s (by omega), ih (s.erase a) (by rw [Finset.card_erase_of_mem ha]; omega)] at h have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5 · exact absurd h5 h2a · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 116–127 at b16ab29fc51b ↗
  hV.s2.s1 obtain s.Nonempty
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with a : ℕ, ha : a ∈ s.
  Uses library Finset.card_pos.
  Lean, line 118
  obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 118 at b16ab29fc51b ↗
  hV.s2.h have ∑ k ∈ s, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with h : ∑ k ∈ s, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (s.erase a).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Lean, line 119
  have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹) ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 119 at b16ab29fc51b ↗
  hV.s2.e have ∀ (k : ℕ), (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ = (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with e : ∀ (k : ℕ), (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ = (((2 : ℝ) ^ k) ^ (m + (1 : ℕ)))⁻¹.
  Uses library pow_succ', mul_inv, mul_assoc, mul_inv_cancel₀, one_mul.
  Lean, lines 120–121
  have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ = (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ := fun k => by rw [pow_succ' _ (m + 1), mul_inv, ← mul_assoc, mul_inv_cancel₀ (by positivity), one_mul]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 120–121 at b16ab29fc51b ↗
  hV.s2.h2a have (2 : ℝ) ^ a ≠ (0 : ℝ) routine: positivity
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ), with h2a : (2 : ℝ) ^ a ≠ (0 : ℝ).
  Lean, line 124
  have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 124 at b16ab29fc51b ↗
  hV.s2.s5 case
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Uses library absurd.
  Lean, line 126
  · exact absurd h5 h2a
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 126 at b16ab29fc51b ↗
  hV.s2.s6 case
  Leaves ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ s.erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (0 : ℝ).
  Lean, line 127
  · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 127 at b16ab29fc51b ↗
hgeo have ∀ (m : ℕ), (1 : ℕ) ≤ m → ∀ (k : ℕ), ∑ i ∈ Finset.Icc (1 : ℕ) k, ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hgeo : ∀ (m : ℕ), (1 : ℕ) ≤ m → ∀ (k : ℕ), ∑ i ∈ Finset.Icc (1 : ℕ) k, ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)).
Uses library Finset.Icc.
Lean, lines 129–145
```
  have hgeo : ∀ m : ℕ, 1 ≤ m → ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (((2 : ℝ) ^ i)⁻¹) ^ m = (1 - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - 1) := by
    intro m hm k
    have hB : (2 : ℝ) ^ m - 1 ≠ 0 := by
      have : (2 : ℝ) ≤ 2 ^ m := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
      linarith
    induction k with
    | zero => simp
    | succ k ih =>
      rw [Finset.sum_Icc_succ_top (by omega), ih, inv_pow, pow_succ, mul_pow]
      have hX : ((2 : ℝ) ^ k) ^ m ≠ 0 := by positivity
      have hY : (2 : ℝ) ^ m ≠ 0 := by positivity
      generalize ((2 : ℝ) ^ k) ^ m = X at hX ⊢
      generalize (2 : ℝ) ^ m = Y at hY hB ⊢
      field_simp
      ring
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 129–145 at b16ab29fc51b ↗
1. hgeo.hB have (2 : ℝ) ^ m - (1 : ℝ) ≠ (0 : ℝ)
  Leaves ∑ i ∈ Finset.Icc (1 : ℕ) k, ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)), with hB : (2 : ℝ) ^ m - (1 : ℝ) ≠ (0 : ℝ).
  Lean, lines 131–135
  have hB : (2 : ℝ) ^ m - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ m := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 131–135 at b16ab29fc51b ↗
  hgeo.hB.this have (2 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves (2 : ℝ) ^ m - (1 : ℝ) ≠ (0 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ m.
  Lean, lines 132–134
  have : (2 : ℝ) ≤ 2 ^ m := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 132–134 at b16ab29fc51b ↗
  hgeo.hB.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 133
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 133 at b16ab29fc51b ↗
  hgeo.hB.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ m
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 134
  _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 134 at b16ab29fc51b ↗
2. hgeo.s2 case routine: simp
  Leaves ∑ i ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ (0 : ℕ)) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)).
  Uses library Nat.zero.
  Lean, line 137
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 137 at b16ab29fc51b ↗
3. hgeo.s3 case
  Leaves ∑ i ∈ Finset.Icc (1 : ℕ) (k + (1 : ℕ)), ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ (k + (1 : ℕ))) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)), with k : ℕ, ih : ∑ i ∈ Finset.Icc (1 : ℕ) k, ((2 : ℝ) ^ i)⁻¹ ^ m = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)).
  Uses library Nat.succ, Finset.sum_Icc_succ_top, inv_pow, pow_succ, mul_pow.
  Lean, lines 138–145
  | succ k ih => rw [Finset.sum_Icc_succ_top (by omega), ih, inv_pow, pow_succ, mul_pow] have hX : ((2 : ℝ) ^ k) ^ m ≠ 0 := by positivity have hY : (2 : ℝ) ^ m ≠ 0 := by positivity generalize ((2 : ℝ) ^ k) ^ m = X at hX ⊢ generalize (2 : ℝ) ^ m = Y at hY hB ⊢ field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 138–145 at b16ab29fc51b ↗
  hgeo.s3.hX have ((2 : ℝ) ^ k) ^ m ≠ (0 : ℝ) routine: positivity
  Leaves ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)) + (((2 : ℝ) ^ k) ^ m * (2 : ℝ) ^ m)⁻¹ = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m * (2 : ℝ) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)), with hX : ((2 : ℝ) ^ k) ^ m ≠ (0 : ℝ).
  Lean, line 140
  have hX : ((2 : ℝ) ^ k) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 140 at b16ab29fc51b ↗
  hgeo.s3.hY have (2 : ℝ) ^ m ≠ (0 : ℝ) routine: positivity
  Leaves ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)) + (((2 : ℝ) ^ k) ^ m * (2 : ℝ) ^ m)⁻¹ = ((1 : ℝ) - (((2 : ℝ) ^ k) ^ m * (2 : ℝ) ^ m)⁻¹) / ((2 : ℝ) ^ m - (1 : ℝ)), with hY : (2 : ℝ) ^ m ≠ (0 : ℝ).
  Lean, line 141
  have hY : (2 : ℝ) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 141 at b16ab29fc51b ↗
hdvd have ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k)
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hdvd : ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k).
Uses library Finset.Icc, Finset.dvd_prod_of_mem.
Lean, lines 146–160
```
  have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by
    intro n i hi1 hi2
    obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
    have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
    have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
    have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
    have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
    have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by
      have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
      rw [one_pow, ← pow_mul, ← hk0] at this
      exact this
    have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by
      have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
      rw [this]; exact (dvd_neg).mpr h1
    exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem)
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 146–160 at b16ab29fc51b ↗
1. hdvd.s1 obtain ∃ (k0 : ℕ), k0 = i * (n / i)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with k0 : ℕ, hk0 : k0 = i * (n / i).
  Uses library Exists.intro, rfl.
  Lean, line 148
  obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 148 at b16ab29fc51b ↗
2. hdvd.hle have k0 ≤ n
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hle : k0 ≤ n.
  Uses library mul_comm, Nat.div_mul_le_self.
  Lean, line 149
  have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 149 at b16ab29fc51b ↗
3. hdvd.hlt have n < k0 + i
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hlt : n < k0 + i.
  Lean, line 150
  have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 150 at b16ab29fc51b ↗
  hdvd.hlt.this have n < i * (n / i + (1 : ℕ))
  Leaves n < i * (n / i) + i, with this : n < i * (n / i + (1 : ℕ)).
  Uses library Nat.lt_mul_div_succ.
  Lean, line 150
  have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 150 at b16ab29fc51b ↗
4. hdvd.hge have i ≤ k0
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hge : i ≤ k0.
  Uses library Nat.le_mul_of_pos_right, Nat.div_pos.
  Lean, line 151
  have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 151 at b16ab29fc51b ↗
5. hdvd.hmem have k0 ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hmem : k0 ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n.
  Uses library Finset.Icc, Finset.mem_Icc, And.intro.
  Lean, line 152
  have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 152 at b16ab29fc51b ↗
6. hdvd.h1 have (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with h1 : (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ).
  Uses library one_pow, pow_mul.
  Lean, lines 153–156
  have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i) rw [one_pow, ← pow_mul, ← hk0] at this exact this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 153–156 at b16ab29fc51b ↗
  hdvd.h1.this have (2 : ℤ) ^ i - (1 : ℤ) ∣ ((2 : ℤ) ^ i) ^ (n / i) - (1 : ℤ) ^ (n / i)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ), with this : (2 : ℤ) ^ i - (1 : ℤ) ∣ ((2 : ℤ) ^ i) ^ (n / i) - (1 : ℤ) ^ (n / i).
  Uses library sub_dvd_pow_sub_pow.
  Lean, line 154
  have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 154 at b16ab29fc51b ↗
7. hdvd.h2 have (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with h2 : (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0.
  Uses library dvd_neg.
  Lean, lines 157–159
  have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring rw [this]; exact (dvd_neg).mpr h1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 157–159 at b16ab29fc51b ↗
  hdvd.h2.this have (1 : ℤ) - (2 : ℤ) ^ k0 = -((2 : ℤ) ^ k0 - (1 : ℤ)) routine: ring
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0, with this : (1 : ℤ) - (2 : ℤ) ^ k0 = -((2 : ℤ) ^ k0 - (1 : ℤ)).
  Lean, line 158
  have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 158 at b16ab29fc51b ↗
hpoly have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hpoly : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j.
Uses library Finset.Icc, Finset.range, Exists.intro, Polynomial.C, Polynomial.X, Finset.sum_congr, rfl, Polynomial.coeff_map.
Lean, lines 161–178
```
  have hpoly : ∀ n : ℕ, 1 ≤ n → ∃ α : ℕ → ℤ, ∀ y : ℝ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (α j : ℝ) * y ^ j := by
    intro n hn
    refine ⟨fun j => (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X)).coeff j, fun y => ?_⟩
    have hdeg : (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))).natDegree < n := by
      refine lt_of_le_of_lt (Polynomial.natDegree_map_le) ?_
      refine lt_of_le_of_lt (Polynomial.natDegree_prod_le _ _) ?_
      refine lt_of_le_of_lt (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)) ?_
      · compute_degree
      · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
    have hev := Polynomial.eval_eq_sum_range' hdeg y
    have hev2 : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ t * y) = Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))) := by
      rw [Polynomial.map_prod, Polynomial.eval_prod]
      refine Finset.prod_congr rfl (fun t _ => ?_)
      simp
    rw [hev2, hev]
    refine Finset.sum_congr rfl (fun j _ => ?_)
    rw [Polynomial.coeff_map]
    rfl
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 161–178 at b16ab29fc51b ↗
1. hpoly.hdeg have (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))).natDegree < n
  Leaves ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑((fun (j : ℕ) => (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X)).coeff j) j) : ℝ) * y ^ j, with hdeg : (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))).natDegree < n.
  Uses library Polynomial.map, Int.castRingHom, Finset.Icc, Polynomial.C, Polynomial.X, lt_of_le_of_lt, Polynomial.natDegree_map_le, Polynomial.natDegree_prod_le, Finset.sum_le_sum.
  Lean, lines 164–169
  have hdeg : (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))).natDegree < n := by refine lt_of_le_of_lt (Polynomial.natDegree_map_le) ?_ refine lt_of_le_of_lt (Polynomial.natDegree_prod_le _ _) ?_ refine lt_of_le_of_lt (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)) ?_ · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 164–169 at b16ab29fc51b ↗
  hpoly.hdeg.s1 case
  Leaves ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X).natDegree ≤ (1 : ℕ).
  Lean, line 168
  · compute_degree
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 168 at b16ab29fc51b ↗
  hpoly.hdeg.s2 case routine: simp, omega
  Leaves ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (1 : ℕ) < n.
  Uses library Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc.
  Lean, line 169
  · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 169 at b16ab29fc51b ↗
2. hpoly.hev have Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))) = ∑ i ∈ Finset.range n, (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))).coeff i * y ^ i
  Leaves ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑((fun (j : ℕ) => (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X)).coeff j) j) : ℝ) * y ^ j, with hev : Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))) = ∑ i ∈ Finset.range n, (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))).coeff i * y ^ i.
  Uses library Polynomial.eval_eq_sum_range'.
  Lean, line 170
  have hev := Polynomial.eval_eq_sum_range' hdeg y
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 170 at b16ab29fc51b ↗
3. hpoly.hev2 have ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X)))
  Leaves ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑((fun (j : ℕ) => (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X)).coeff j) j) : ℝ) * y ^ j, with hev2 : ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X))).
  Uses library Finset.Icc, Polynomial.eval, Polynomial.map, Int.castRingHom, Polynomial.C, Polynomial.X, Polynomial.map_prod, Polynomial.eval_prod, Finset.prod_congr, rfl.
  Lean, lines 171–174
  have hev2 : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ t * y) = Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))) := by rw [Polynomial.map_prod, Polynomial.eval_prod] refine Finset.prod_congr rfl (fun t _ => ?_) simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 171–174 at b16ab29fc51b ↗
numerator_integral have ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with numerator_integral : ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Uses library Nat.factorial, Finset.Icc, Exists.intro.
Lean, lines 183–561
```
  have numerator_integral : ∃ a : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    have residue_integral : ∀ n i : ℕ, 1 ≤ i → i ≤ n → ∃ z : ℤ, (z : ℝ) = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      have hQne : ∀ m : ℕ, (∏ s ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ s)) ≠ 0 := by
        intro m
        refine Finset.prod_ne_zero_iff.mpr (fun s hs => ?_)
        have h1 := (Finset.mem_Icc.mp hs).1
        have h2 : (2 : ℤ) ≤ 2 ^ s := by
          calc (2 : ℤ) = 2 ^ 1 := by norm_num
            _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1
        omega
      have hG : ∀ N a b : ℕ, a + b = N → ∃ z : ℤ, (∏ s ∈ Finset.Icc 1 (a + b), ((1 : ℤ) - 2 ^ s)) = z * ((∏ s ∈ Finset.Icc 1 a, ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 b, ((1 : ℤ) - 2 ^ s))) := by
        intro N
        induction N with
        | zero =>
          intro a b h
          obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega
          exact ⟨1, by simp⟩
        | succ N ih =>
          intro a b h
          rcases Nat.eq_zero_or_pos a with rfl | ha
          · exact ⟨1, by simp⟩
          rcases Nat.eq_zero_or_pos b with rfl | hb
          · exact ⟨1, by simp⟩
          obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩
          obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩
          obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega)
          obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega)
          refine ⟨2 ^ (b' + 1) * z1 + z2, ?_⟩
          have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega
          have e2 : a' + 1 + b' = a' + (b' + 1) := by omega
          have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          beta_reduce at hA hB hC
          rw [e2, hA] at h2
          rw [hB] at h1
          rw [e1, hC, hA, hB]
          linear_combination (2 ^ (b' + 1) * (1 - 2 ^ (a' + 1))) * h1 + (1 - 2 ^ (b' + 1)) * h2
      have hN : ∀ m d : ℕ, (∏ s ∈ Finset.Icc 1 d, ((1 : ℤ) - 2 ^ s)) * (∏ t ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ (t + d))) = ∏ s ∈ Finset.Icc 1 (d + m), ((1 : ℤ) - 2 ^ s) := by
        intro m d
        induction m with
        | zero => simp
        | succ m ih =>
          rw [Finset.prod_Icc_succ_top (by omega), show d + (m + 1) = (d + m) + 1 by omega, Finset.prod_Icc_succ_top (by omega), ← mul_assoc, ih,
            show m + 1 + d = d + m + 1 by omega]
      have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      intro n i hi hin
      refine hsumZ _ _ (fun k hk => ?_)
      have hk' := Finset.mem_Icc.mp hk
      obtain ⟨d, rfl⟩ : ∃ d, k = i + d := ⟨k - i, by omega⟩
      -- numerator
      have hNr : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) := by
        push_cast
        refine Finset.prod_congr rfl (fun t _ => ?_)
        have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
        rw [show t + (i + d) = i + (t + d) by omega, pow_add, ← mul_assoc, inv_mul_cancel₀ h2i, one_mul]
      -- denominator
      have hset : (Finset.Icc 1 n).erase (i + d) = Finset.Icc 1 (i + d - 1) ∪ Finset.Icc (i + d + 1) n := by
        ext l
        simp only [Finset.mem_erase, Finset.mem_Icc, Finset.mem_union]
        omega
      have hdisj : Disjoint (Finset.Icc 1 (i + d - 1)) (Finset.Icc (i + d + 1) n) := by
        rw [Finset.disjoint_left]
        intro l h1 h2
        simp only [Finset.mem_Icc] at h1 h2
        omega
      have hD1 : (∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        have hrefl : ∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ∏ s ∈ Finset.Icc 1 (i + d - 1), (1 - ((2 : ℝ) ^ s)⁻¹) := by
          refine Finset.prod_nbij' (fun l => i + d - l) (fun s => i + d - s) ?_ ?_ ?_ ?_ ?_
          · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl; omega
          · intro l hl
            simp only [Finset.mem_Icc] at hl
            rw [← zpow_natCast, ← zpow_neg]
            congr 2
            push_cast [Nat.cast_sub (show l ≤ i + d by omega)]
            ring
        rw [hrefl]
        push_cast
        rw [← Finset.prod_mul_distrib]
        refine Finset.prod_congr rfl (fun s _ => ?_)
        have h2s : (2 : ℝ) ^ s ≠ 0 := by positivity
        field_simp
        ring
      have hD2 : ∏ l ∈ Finset.Icc (i + d + 1) n, (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        push_cast
        refine Finset.prod_nbij' (fun l => l - (i + d)) (fun s => s + (i + d)) ?_ ?_ ?_ ?_ ?_
        · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl; omega
        · intro l hl
          simp only [Finset.mem_Icc] at hl
          rw [← zpow_natCast]
          congr 2
          push_cast [Nat.cast_sub (show i + d ≤ l by omega)]
          ring
      obtain ⟨z1, hz1⟩ := hG _ d (n - 1) rfl
      obtain ⟨z2, hz2⟩ := hG _ (i + d - 1) (n - (i + d)) rfl
      rw [show i + d - 1 + (n - (i + d)) = n - 1 by omega] at hz2
      have hNZ : (∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d))) = z1 * z2 * ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s))) := by
        apply mul_left_cancel₀ (hQne d)
        rw [hN (n - 1) d, hz1, hz2]
        ring
      have hNZ' : ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) = (z1 : ℝ) * z2 * (((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ)) := by
        exact_mod_cast hNZ
      have hQ1 : ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
      have hQ2 : ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
      have hDr : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        rw [hset, Finset.prod_union hdisj, hD2, ← hD1]
        ring
      have hDne : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) ≠ 0 := by
        intro h0
        rw [h0, zero_mul] at hDr
        exact mul_ne_zero hQ1 hQ2 hDr.symm
      refine ⟨-(z1 * z2 * (∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s))), ?_⟩
      rw [hNr, eq_div_iff hDne, hNZ']
      rw [Int.cast_neg, Int.cast_mul, Int.cast_mul]
      linear_combination (-((z1 : ℝ) * z2)) * hDr
    have polypart_integral : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
      intro n hn
      obtain ⟨α, hα⟩ := hpoly n hn
      obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ k, D k = ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := ⟨fun k => _, fun k => rfl⟩
      have hA : ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc 1 (n - 1), (1 - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * (x * 2 ^ k) ^ j := by
        intro x k
        rw [← hα]
        refine Finset.prod_congr rfl (fun t _ => ?_)
        rw [pow_add]; ring
      simp only [← hD, hA]
      -- (a) one double sum
      have hswap : ∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
        simp only [Finset.sum_mul]
        refine Finset.sum_comm' (fun i k => ?_)
        simp only [Finset.mem_Icc]
        omega
      rw [hswap]
      simp only [Finset.mul_sum]
      rw [← Finset.sum_sub_distrib]
      simp only [← Finset.sum_sub_distrib, ← sub_mul]
      -- (b) per (k, i)
      have hb : ∀ (k i : ℕ), 1 ≤ i → (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s))) / D k := by
        intro k i hi
        have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
        have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by
          have : (2 : ℝ) ≤ 2 ^ i := by
            calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
              _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
          intro h0
          linarith
        have hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ := by
          rw [show (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) = ((2 : ℝ) ^ i)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) by field_simp, mul_assoc, mul_inv_cancel₀ hne, mul_one]
        have hj : ∀ j : ℕ, ((((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s) := by
          intro j
          have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j
          have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by
            linear_combination hg
          rw [hg', mul_assoc, hch, Finset.sum_mul]
          refine Finset.sum_congr rfl (fun s hs => ?_)
          have hs' := Finset.mem_range.mp hs
          rw [mul_assoc, ← pow_succ, show j - 1 - s + 1 = j - s by omega]
        have e1 : (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j - ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) / D k := by
          ring
        rw [e1, ← Finset.sum_sub_distrib, Finset.sum_mul]
        congr 1
        refine Finset.sum_congr rfl (fun j _ => ?_)
        rw [← Finset.mul_sum, ← hj j, mul_pow, mul_pow]
        ring
      -- (c) sum over i
      have hc : ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - 1) := by
        intro k
        rw [Finset.sum_congr rfl (fun i hi => hb k i (Finset.mem_Icc.mp hi).1), ← Finset.sum_div, Finset.sum_comm, Finset.sum_div]
        refine Finset.sum_congr rfl (fun j _ => ?_)
        rw [Finset.sum_comm, Finset.sum_div]
        refine Finset.sum_congr rfl (fun s hs => ?_)
        have hs' := Finset.mem_range.mp hs
        rw [Finset.sum_congr rfl (fun i _ => (mul_assoc _ _ _).symm), ← Finset.mul_sum, hgeo (j - s) (by omega) k]
        have hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s) := by
          rw [← pow_add]; congr 1; omega
        have hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = 1 := mul_inv_cancel₀ (by positivity)
        rw [hjs]
        linear_combination (-((α j : ℝ) * ((2 : ℝ) ^ k) ^ s * (8 / 3 : ℝ) ^ s / ((2 : ℝ) ^ (j - s) - 1) / D k)) * hZ
      rw [Finset.sum_congr rfl (fun k _ => hc k), Finset.sum_comm, Finset.mul_sum]
      refine hsumZ _ _ (fun j hj => ?_)
      rw [Finset.sum_comm, Finset.mul_sum]
      refine hsumZ _ _ (fun s hs => ?_)
      have hj' := Finset.mem_range.mp hj
      have hs' := Finset.mem_range.mp hs
      obtain ⟨wj, hwj⟩ := hW _ (Finset.Icc 1 n) j rfl
      obtain ⟨ws, hws⟩ := hW _ (Finset.Icc 1 n) s rfl
      obtain ⟨q, hq⟩ := hdvd n (j - s) (by omega) (by omega)
      have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ (j - s) - 1) * q := by exact_mod_cast hq
      have hB : (2 : ℝ) ^ (j - s) - 1 ≠ 0 := by
        have : (2 : ℝ) ≤ 2 ^ (j - s) := by
          calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
            _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega)
        linarith
      have h3 : (3 : ℝ) ^ n = 3 ^ (n - s) * 3 ^ s := by rw [← pow_add]; congr 1; omega
      have h83 : (8 / 3 : ℝ) ^ s * 3 ^ s = 8 ^ s := by rw [← mul_pow]; norm_num
      refine ⟨3 ^ (n - s) * 8 ^ s * q * α j * (wj - ws), ?_⟩
      simp only [← hD] at hwj hws
      push_cast
      rw [hwj, hws, hqR, h3, ← h83, ← Finset.sum_sub_distrib]
      rw [Finset.mul_sum, Finset.mul_sum]
      refine Finset.sum_congr rfl (fun k _ => ?_)
      field_simp
    have head_integral : ∃ a₁ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₁ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
      have hsumZ0 : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
        intro n hn
        obtain ⟨zE, hzE⟩ := polypart_integral n hn
        have hR : ∃ zR : ℤ, (zR : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) := by
          rw [Finset.mul_sum]
          refine hsumZ0 _ _ (fun i hi => ?_)
          have hi' := Finset.mem_Icc.mp hi
          obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2
          refine ⟨zb * (3 * ∏ j ∈ (Finset.Icc 1 n).erase i, ((3 : ℤ) - 8 * 2 ^ j)), ?_⟩
          have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by
            have : (2 : ℝ) ≤ 2 ^ i := by
              calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
                _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
            intro h0
            linarith
          have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) :=
            (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm
          have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by
            rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
              Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
          have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by
            rw [← pow_succ']; congr 1; omega
          have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity
          obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩
          rw [← hb] at hzb ⊢
          push_cast
          rw [hzb, hsplit, he, h3]
          have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by
            intro h0
            apply hne
            linear_combination (1 / 3 : ℝ) * h0
          field_simp
        obtain ⟨zR, hzR⟩ := hR
        have hPZ : ∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) := by
          rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
            Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
          have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
          field_simp
        have h9 : (9 : ℝ) ^ n = 3 ^ n * 3 ^ n := by
          rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow]
        refine ⟨(Nat.factorial (n - 2) : ℤ) * ((∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * zE + 3 ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * zR), ?_⟩
        push_cast
        rw [hzE, hzR, hPZ, h9]
        ring
      refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩
      simp only [dif_pos hn']
      exact Classical.choose_spec (hn n hn')
    obtain ⟨a₁, h₁⟩ := head_integral
    have hmer : ∃ a₂ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₂ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
      have key : ∀ n : ℕ, 1 ≤ n → ∀ i : ℕ, 1 ≤ i → i ≤ n - 1 → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1 - i) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) := by
        intro n hn i hi1 hi2
        obtain ⟨α, hα⟩ := hpoly n hn
        have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by
          intro k
          rw [← hα]
          refine Finset.prod_congr rfl (fun t _ => ?_)
          rw [pow_add]; ring
        simp only [hA]
        have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by
          rw [Finset.sum_comm]
          refine Finset.sum_congr rfl (fun k _ => ?_)
          rw [neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, ← Finset.sum_neg_distrib]
          refine Finset.sum_congr rfl (fun j _ => ?_)
          ring
        rw [e1, Finset.mul_sum]
        refine hsumZ _ _ (fun j hj => ?_)
        have hj' := Finset.mem_range.mp hj
        rw [← Finset.mul_sum]
        have hcX : ∀ k : ℕ, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ≠ 0 := fun k => by positivity
        rcases lt_or_ge j i with hji | hji
        · refine ⟨0, ?_⟩
          have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
            rw [Finset.mul_sum]
            refine Finset.sum_congr rfl (fun k _ => ?_)
            have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by
              rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega,
                mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv]
            rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega]
          rw [hvan, hV (i - j - 1) (Finset.Icc 1 n) (by rw [Nat.card_Icc]; omega)]
          simp
        · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl
          refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ (j - i) * w, ?_⟩
          have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
            rw [Finset.mul_sum]
            refine Finset.sum_congr rfl (fun k _ => ?_)
            have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by
              rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega,
                mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow]
            rw [hk, mul_div_assoc]
          have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega
          have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num
          push_cast
          rw [hpos, hw, h3, ← h83]
          ring
      have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
        intro n
        rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
          Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
      have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by
        intro n i hi1 hi2
        obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
        have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
        have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
        have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
        have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
        have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by
          have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
          rw [one_pow, ← pow_mul, ← hk0] at this
          exact this
        have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by
          have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
          rw [this]; exact (dvd_neg).mpr h1
        exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem)
      have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
        intro n hn
        rw [Finset.mul_sum]
        refine hsumZ _ _ (fun i hi => ?_)
        have hi' := Finset.mem_Icc.mp hi
        obtain ⟨z1, hz1⟩ := key n hn i hi'.1 hi'.2
        obtain ⟨q, hq⟩ := hdvd n i hi'.1 hi'.2
        have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ i - 1) * q := by exact_mod_cast hq
        have h2i : (2 : ℝ) ^ i - 1 ≠ 0 := by
          have : (2 : ℝ) ≤ 2 ^ i := by
            calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
              _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
          linarith
        have h9 : (9 : ℝ) ^ n = 3 ^ (i + 1) * 3 ^ (n - 1 - i) * 3 ^ n := by
          rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow]
          congr 1
          rw [← pow_add]
          congr 1
          omega
        refine ⟨3 ^ (i + 1) * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * q * z1, ?_⟩
        push_cast
        rw [hz1, hW3 n, hqR, h9]
        have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
        field_simp
      refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩
      simp only [dif_pos hn']
      exact Classical.choose_spec (hn n hn')
    obtain ⟨a₂, h₂⟩ := hmer
    refine ⟨fun n => a₁ n - a₂ n, fun n hn => ?_⟩
    push_cast
    rw [h₁ n hn, h₂ n hn]
    ring
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 183–561 at b16ab29fc51b ↗
1. numerator_integral.residue_integral have ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with residue_integral : ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Exists.intro, eq_div_iff, Int.cast_neg, Int.cast_mul.
  Lean, lines 184–311
  have residue_integral : ∀ n i : ℕ, 1 ≤ i → i ≤ n → ∃ z : ℤ, (z : ℝ) = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by have hQne : ∀ m : ℕ, (∏ s ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ s)) ≠ 0 := by intro m refine Finset.prod_ne_zero_iff.mpr (fun s hs => ?_) have h1 := (Finset.mem_Icc.mp hs).1 have h2 : (2 : ℤ) ≤ 2 ^ s := by calc (2 : ℤ) = 2 ^ 1 := by norm_num _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1 omega have hG : ∀ N a b : ℕ, a + b = N → ∃ z : ℤ, (∏ s ∈ Finset.Icc 1 (a + b), ((1 : ℤ) - 2 ^ s)) = z * ((∏ s ∈ Finset.Icc 1 a, ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 b, ((1 : ℤ) - 2 ^ s))) := by intro N induction N with | zero => intro a b h obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega exact ⟨1, by simp⟩ | succ N ih => intro a b h rcases Nat.eq_zero_or_pos a with rfl | ha · exact ⟨1, by simp⟩ rcases Nat.eq_zero_or_pos b with rfl | hb · exact ⟨1, by simp⟩ obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩ obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩ obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega) obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega) refine ⟨2 ^ (b' + 1) * z1 + z2, ?_⟩ have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega have e2 : a' + 1 + b' = a' + (b' + 1) := by omega have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) beta_reduce at hA hB hC rw [e2, hA] at h2 rw [hB] at h1 rw [e1, hC, hA, hB] linear_combination (2 ^ (b' + 1) * (1 - 2 ^ (a' + 1))) * h1 + (1 - 2 ^ (b' + 1)) * h2 have hN : ∀ m d : ℕ, (∏ s ∈ Finset.Icc 1 d, ((1 : ℤ) - 2 ^ s)) * (∏ t ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ (t + d))) = ∏ s ∈ Finset.Icc 1 (d + m), ((1 : ℤ) - 2 ^ s) := by intro m d induction m with | zero => simp | succ m ih => rw [Finset.prod_Icc_succ_top (by omega), show d + (m + 1) = (d + m) + 1 by omega, Finset.prod_Icc_succ_top (by omega), ← mul_assoc, ih, show m + 1 + d = d + m + 1 by omega] have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩ intro n i hi hin refine hsumZ _ _ (fun k hk => ?_) have hk' := Finset.mem_Icc.mp hk obtain ⟨d, rfl⟩ : ∃ d, k = i + d := ⟨k - i, by omega⟩ -- numerator have hNr : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) := by push_cast refine Finset.prod_congr rfl (fun t _ => ?_) have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity rw [show t + (i + d) = i + (t + d) by omega, pow_add, ← mul_assoc, inv_mul_cancel₀ h2i, one_mul] -- denominator have hset : (Finset.Icc 1 n).erase (i + d) = Finset.Icc 1 (i + d - 1) ∪ Finset.Icc (i + d + 1) n := by ext l simp only [Finset.mem_erase, Finset.mem_Icc, Finset.mem_union] omega have hdisj : Disjoint (Finset.Icc 1 (i + d - 1)) (Finset.Icc (i + d + 1) n) := by rw [Finset.disjoint_left] intro l h1 h2 simp only [Finset.mem_Icc] at h1 h2 omega have hD1 : (∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by have hrefl : ∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ∏ s ∈ Finset.Icc 1 (i + d - 1), (1 - ((2 : ℝ) ^ s)⁻¹) := by refine Finset.prod_nbij' (fun l => i + d - l) (fun s => i + d - s) ?_ ?_ ?_ ?_ ?_ · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast, ← zpow_neg] congr 2 push_cast [Nat.cast_sub (show l ≤ i + d by omega)] ring rw [hrefl] push_cast rw [← Finset.prod_mul_distrib] refine Finset.prod_congr rfl (fun s _ => ?_) have h2s : (2 : ℝ) ^ s ≠ 0 := by positivity field_simp ring have hD2 : ∏ l ∈ Finset.Icc (i + d + 1) n, (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by push_cast refine Finset.prod_nbij' (fun l => l - (i + d)) (fun s => s + (i + d)) ?_ ?_ ?_ ?_ ?_ · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast] congr 2 push_cast [Nat.cast_sub (show i + d ≤ l by omega)] ring obtain ⟨z1, hz1⟩ := hG _ d (n - 1) rfl obtain ⟨z2, hz2⟩ := hG _ (i + d - 1) (n - (i + d)) rfl rw [show i + d - 1 + (n - (i + d)) = n - 1 by omega] at hz2 have hNZ : (∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d))) = z1 * z2 * ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s))) := by apply mul_left_cancel₀ (hQne d) rw [hN (n - 1) d, hz1, hz2] ring have hNZ' : ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) = (z1 : ℝ) * z2 * (((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ)) := by exact_mod_cast hNZ have hQ1 : ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _ have hQ2 : ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _ have hDr : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by rw [hset, Finset.prod_union hdisj, hD2, ← hD1] ring have hDne : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) ≠ 0 := by intro h0 rw [h0, zero_mul] at hDr exact mul_ne_zero hQ1 hQ2 hDr.symm refine ⟨-(z1 * z2 * (∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s))), ?_⟩ rw [hNr, eq_div_iff hDne, hNZ'] rw [Int.cast_neg, Int.cast_mul, Int.cast_mul] linear_combination (-((z1 : ℝ) * z2)) * hDr
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 184–311 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQne have ∀ (m : ℕ), ∏ s ∈ Finset.Icc (1 : ℕ) m, ((1 : ℤ) - (2 : ℤ) ^ s) ≠ (0 : ℤ)
  Leaves ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hQne : ∀ (m : ℕ), ∏ s ∈ Finset.Icc (1 : ℕ) m, ((1 : ℤ) - (2 : ℤ) ^ s) ≠ (0 : ℤ).
  Uses library Finset.Icc, Finset.prod_ne_zero_iff.
  Lean, lines 185–192
  have hQne : ∀ m : ℕ, (∏ s ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ s)) ≠ 0 := by intro m refine Finset.prod_ne_zero_iff.mpr (fun s hs => ?_) have h1 := (Finset.mem_Icc.mp hs).1 have h2 : (2 : ℤ) ≤ 2 ^ s := by calc (2 : ℤ) = 2 ^ 1 := by norm_num _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1 omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 185–192 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQne.h1 have (1 : ℕ) ≤ s
  Leaves (1 : ℤ) - (2 : ℤ) ^ s ≠ (0 : ℤ), with h1 : (1 : ℕ) ≤ s.
  Uses library Finset.mem_Icc.
  Lean, line 188
  have h1 := (Finset.mem_Icc.mp hs).1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 188 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQne.h2 have (2 : ℤ) ≤ (2 : ℤ) ^ s
  Leaves (1 : ℤ) - (2 : ℤ) ^ s ≠ (0 : ℤ), with h2 : (2 : ℤ) ≤ (2 : ℤ) ^ s.
  Lean, lines 189–191
  have h2 : (2 : ℤ) ≤ 2 ^ s := by calc (2 : ℤ) = 2 ^ 1 := by norm_num _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 189–191 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQne.h2.s1 calc (2 : ℤ) = (2 : ℤ) ^ (1 : ℕ) routine: norm_num
  Closes the goal.
  Lean, line 190
  calc (2 : ℤ) = 2 ^ 1 := by norm_num
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 190 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQne.h2.s2 calc (2 : ℤ) ^ (1 : ℕ) ≤ (2 : ℤ) ^ s
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 191
  _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 191 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG have ∀ (N a b : ℕ), a + b = N → ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hG : ∀ (N a b : ℕ), a + b = N → ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Finset.Icc.
  Lean, lines 193–220
  have hG : ∀ N a b : ℕ, a + b = N → ∃ z : ℤ, (∏ s ∈ Finset.Icc 1 (a + b), ((1 : ℤ) - 2 ^ s)) = z * ((∏ s ∈ Finset.Icc 1 a, ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 b, ((1 : ℤ) - 2 ^ s))) := by intro N induction N with | zero => intro a b h obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega exact ⟨1, by simp⟩ | succ N ih => intro a b h rcases Nat.eq_zero_or_pos a with rfl | ha · exact ⟨1, by simp⟩ rcases Nat.eq_zero_or_pos b with rfl | hb · exact ⟨1, by simp⟩ obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩ obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩ obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega) obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega) refine ⟨2 ^ (b' + 1) * z1 + z2, ?_⟩ have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega have e2 : a' + 1 + b' = a' + (b' + 1) := by omega have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) beta_reduce at hA hB hC rw [e2, hA] at h2 rw [hB] at h1 rw [e1, hC, hA, hB] linear_combination (2 ^ (b' + 1) * (1 - 2 ^ (a' + 1))) * h1 + (1 - 2 ^ (b' + 1)) * h2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 193–220 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s1 case
  Leaves ∀ (a b : ℕ), a + b = (0 : ℕ) → ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Nat.zero, Exists.intro.
  Lean, lines 196–199
  | zero => intro a b h obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega exact ⟨1, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 196–199 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s1.s1 obtain a = (0 : ℕ) ∧ b = (0 : ℕ) routine: omega
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) ((0 : ℕ) + (0 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℤ) - (2 : ℤ) ^ s)), with h : (0 : ℕ) + (0 : ℕ) = (0 : ℕ).
  Lean, line 198
  obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 198 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2 case
  Leaves ∀ (a b : ℕ), a + b = N + (1 : ℕ) → ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)), with N : ℕ, ih : ∀ (a b : ℕ), a + b = N → ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Nat.succ, Nat.eq_zero_or_pos, Exists.intro.
  Lean, lines 200–220
  | succ N ih => intro a b h rcases Nat.eq_zero_or_pos a with rfl | ha · exact ⟨1, by simp⟩ rcases Nat.eq_zero_or_pos b with rfl | hb · exact ⟨1, by simp⟩ obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩ obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩ obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega) obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega) refine ⟨2 ^ (b' + 1) * z1 + z2, ?_⟩ have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega have e2 : a' + 1 + b' = a' + (b' + 1) := by omega have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s) beta_reduce at hA hB hC rw [e2, hA] at h2 rw [hB] at h1 rw [e1, hC, hA, hB] linear_combination (2 ^ (b' + 1) * (1 - 2 ^ (a' + 1))) * h1 + (1 - 2 ^ (b' + 1)) * h2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 200–220 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s1 case
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) ((0 : ℕ) + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Exists.intro.
  Lean, line 203
  · exact ⟨1, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 203 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s2 case
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a + (0 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Exists.intro.
  Lean, line 205
  · exact ⟨1, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 205 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s3 obtain ∃ (a' : ℕ), a = a' + (1 : ℕ)
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + b), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b, ((1 : ℤ) - (2 : ℤ) ^ s)), with a' : ℕ, h : a' + (1 : ℕ) + b = N + (1 : ℕ), ha : a' + (1 : ℕ) > (0 : ℕ).
  Uses library Exists.intro.
  Lean, line 206
  obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 206 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s4 obtain ∃ (b' : ℕ), b = b' + (1 : ℕ)
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with b' : ℕ, hb : b' + (1 : ℕ) > (0 : ℕ), h : a' + (1 : ℕ) + (b' + (1 : ℕ)) = N + (1 : ℕ).
  Uses library Exists.intro.
  Lean, line 207
  obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 207 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s5 obtain ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) a', ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with z1 : ℤ, h1 : ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z1 * ((∏ s ∈ Finset.Icc (1 : ℕ) a', ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)).
  Lean, line 208
  obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 208 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.s6 obtain ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + b'), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b', ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with z2 : ℤ, h2 : ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + b'), ((1 : ℤ) - (2 : ℤ) ^ s) = z2 * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) b', ((1 : ℤ) - (2 : ℤ) ^ s)).
  Lean, line 209
  obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 209 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.e1 have a' + (1 : ℕ) + (b' + (1 : ℕ)) = a' + (b' + (1 : ℕ)) + (1 : ℕ) routine: omega
  Leaves ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = ((2 : ℤ) ^ (b' + (1 : ℕ)) * z1 + z2) * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with e1 : a' + (1 : ℕ) + (b' + (1 : ℕ)) = a' + (b' + (1 : ℕ)) + (1 : ℕ).
  Lean, line 211
  have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 211 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.e2 have a' + (1 : ℕ) + b' = a' + (b' + (1 : ℕ)) routine: omega
  Leaves ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = ((2 : ℤ) ^ (b' + (1 : ℕ)) * z1 + z2) * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with e2 : a' + (1 : ℕ) + b' = a' + (b' + (1 : ℕ)).
  Lean, line 212
  have e2 : a' + 1 + b' = a' + (b' + 1) := by omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 212 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.hA have ∏ k ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) a', ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (a' + (1 : ℕ)))
  Leaves ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = ((2 : ℤ) ^ (b' + (1 : ℕ)) * z1 + z2) * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with hA : ∏ k ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) a', ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (a' + (1 : ℕ))).
  Uses library Finset.prod_Icc_succ_top.
  Lean, line 213
  have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 213 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.hB have ∏ k ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) b', ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (b' + (1 : ℕ)))
  Leaves ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = ((2 : ℤ) ^ (b' + (1 : ℕ)) * z1 + z2) * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with hB : ∏ k ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) b', ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (b' + (1 : ℕ))).
  Uses library Finset.prod_Icc_succ_top.
  Lean, line 214
  have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 214 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hG.s2.hC have ∏ k ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ)) + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (a' + (b' + (1 : ℕ)) + (1 : ℕ)))
  Leaves ∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ) + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = ((2 : ℤ) ^ (b' + (1 : ℕ)) * z1 + z2) * ((∏ s ∈ Finset.Icc (1 : ℕ) (a' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (b' + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)), with hC : ∏ k ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ)) + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) (a' + (b' + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ k)) * ((1 : ℤ) - (2 : ℤ) ^ (a' + (b' + (1 : ℕ)) + (1 : ℕ))).
  Uses library Finset.prod_Icc_succ_top.
  Lean, line 215
  have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 215 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hN have ∀ (m d : ℕ), (∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ t ∈ Finset.Icc (1 : ℕ) m, ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = ∏ s ∈ Finset.Icc (1 : ℕ) (d + m), ((1 : ℤ) - (2 : ℤ) ^ s)
  Leaves ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hN : ∀ (m d : ℕ), (∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ t ∈ Finset.Icc (1 : ℕ) m, ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = ∏ s ∈ Finset.Icc (1 : ℕ) (d + m), ((1 : ℤ) - (2 : ℤ) ^ s).
  Uses library Finset.Icc.
  Lean, lines 221–227
  have hN : ∀ m d : ℕ, (∏ s ∈ Finset.Icc 1 d, ((1 : ℤ) - 2 ^ s)) * (∏ t ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ (t + d))) = ∏ s ∈ Finset.Icc 1 (d + m), ((1 : ℤ) - 2 ^ s) := by intro m d induction m with | zero => simp | succ m ih => rw [Finset.prod_Icc_succ_top (by omega), show d + (m + 1) = (d + m) + 1 by omega, Finset.prod_Icc_succ_top (by omega), ← mul_assoc, ih, show m + 1 + d = d + m + 1 by omega]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 221–227 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hN.s1 case routine: simp
  Leaves (∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ t ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = ∏ s ∈ Finset.Icc (1 : ℕ) (d + (0 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s).
  Uses library Nat.zero.
  Lean, line 224
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 224 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hN.s2 case
  Leaves (∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ t ∈ Finset.Icc (1 : ℕ) (m + (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = ∏ s ∈ Finset.Icc (1 : ℕ) (d + (m + (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s), with m : ℕ, ih : (∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ t ∈ Finset.Icc (1 : ℕ) m, ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = ∏ s ∈ Finset.Icc (1 : ℕ) (d + m), ((1 : ℤ) - (2 : ℤ) ^ s).
  Uses library Nat.succ, Finset.prod_Icc_succ_top, mul_assoc.
  Lean, lines 225–227
  | succ m ih => rw [Finset.prod_Icc_succ_top (by omega), show d + (m + 1) = (d + m) + 1 by omega, Finset.prod_Icc_succ_top (by omega), ← mul_assoc, ih, show m + 1 + d = d + m + 1 by omega]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 225–227 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hsumZ have ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset, Finset.induction_on.
  Lean, lines 228–236
  have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 228–236 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hsumZ.s1 case printed form did not read back: shown as its Lean lines only
  | empty => intro _; exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 231 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hsumZ.s2 case
  Leaves (∀ i ∈ insert a s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with a : ℕ, s : Finset ℕ, ha : a ∉ s, ih : (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Exists.intro, Finset.sum_insert.
  Lean, lines 232–236
  | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 232–236 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hsumZ.s2.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = g a
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with za : ℤ, hza : (↑za : ℝ) = g a.
  Uses library Finset.mem_insert_self.
  Lean, line 234
  obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 234 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hsumZ.s2.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with zs : ℤ, hzs : (↑zs : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset.mem_insert_of_mem.
  Lean, line 235
  obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 235 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hk' have i ≤ k ∧ k ≤ n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hk' : i ≤ k ∧ k ≤ n.
  Uses library Finset.mem_Icc.
  Lean, line 239
  have hk' := Finset.mem_Icc.mp hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 239 at b16ab29fc51b ↗
  numerator_integral.residue_integral.s6 obtain ∃ (d : ℕ), k = i + d
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with d : ℕ, hk : i + d ∈ Finset.Icc i n, hk' : i ≤ i + d ∧ i + d ≤ n.
  Uses library Exists.intro.
  Lean, line 240
  obtain ⟨d, rfl⟩ : ∃ d, k = i + d := ⟨k - i, by omega⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 240 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hNr have ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = (↑(∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d))) : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hNr : ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = (↑(∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d))) : ℝ).
  Uses library Finset.Icc, Finset.prod_congr, rfl, pow_add, mul_assoc, inv_mul_cancel₀, one_mul.
  Lean, lines 242–246
  have hNr : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) := by push_cast refine Finset.prod_congr rfl (fun t _ => ?_) have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity rw [show t + (i + d) = i + (t + d) by omega, pow_add, ← mul_assoc, inv_mul_cancel₀ h2i, one_mul]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 242–246 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hNr.h2i have (2 : ℝ) ^ i ≠ (0 : ℝ) routine: positivity
  Leaves (1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)) = (1 : ℝ) - (2 : ℝ) ^ (t + d), with h2i : (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, line 245
  have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 245 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hset have (Finset.Icc (1 : ℕ) n).erase (i + d) = Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)) ∪ Finset.Icc (i + d + (1 : ℕ)) n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hset : (Finset.Icc (1 : ℕ) n).erase (i + d) = Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)) ∪ Finset.Icc (i + d + (1 : ℕ)) n.
  Uses library Finset.Icc, Finset.mem_erase, Finset.mem_Icc, Finset.mem_union.
  Lean, lines 248–251
  have hset : (Finset.Icc 1 n).erase (i + d) = Finset.Icc 1 (i + d - 1) ∪ Finset.Icc (i + d + 1) n := by ext l simp only [Finset.mem_erase, Finset.mem_Icc, Finset.mem_union] omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 248–251 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hdisj have Disjoint (Finset.Icc (1 : ℕ) (i + d - (1 : ℕ))) (Finset.Icc (i + d + (1 : ℕ)) n)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hdisj : Disjoint (Finset.Icc (1 : ℕ) (i + d - (1 : ℕ))) (Finset.Icc (i + d + (1 : ℕ)) n).
  Uses library Disjoint, Finset.Icc, Finset.disjoint_left, Finset.mem_Icc.
  Lean, lines 252–256
  have hdisj : Disjoint (Finset.Icc 1 (i + d - 1)) (Finset.Icc (i + d + 1) n) := by rw [Finset.disjoint_left] intro l h1 h2 simp only [Finset.mem_Icc] at h1 h2 omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 252–256 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1 have (∏ l ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ)))) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), -(2 : ℤ) ^ s) : ℝ) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hD1 : (∏ l ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ)))) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), -(2 : ℤ) ^ s) : ℝ) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ).
  Uses library Finset.Icc, Finset.prod_mul_distrib, Finset.prod_congr, rfl.
  Lean, lines 257–276
  have hD1 : (∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by have hrefl : ∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ∏ s ∈ Finset.Icc 1 (i + d - 1), (1 - ((2 : ℝ) ^ s)⁻¹) := by refine Finset.prod_nbij' (fun l => i + d - l) (fun s => i + d - s) ?_ ?_ ?_ ?_ ?_ · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast, ← zpow_neg] congr 2 push_cast [Nat.cast_sub (show l ≤ i + d by omega)] ring rw [hrefl] push_cast rw [← Finset.prod_mul_distrib] refine Finset.prod_congr rfl (fun s _ => ?_) have h2s : (2 : ℝ) ^ s ≠ 0 := by positivity field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 257–276 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl have ∏ l ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) = ∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ s)⁻¹)
  Leaves (∏ l ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ)))) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), -(2 : ℤ) ^ s) : ℝ) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ), with hrefl : ∏ l ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) = ∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ s)⁻¹).
  Uses library Finset.Icc, Finset.prod_nbij'.
  Lean, lines 258–269
  have hrefl : ∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ∏ s ∈ Finset.Icc 1 (i + d - 1), (1 - ((2 : ℝ) ^ s)⁻¹) := by refine Finset.prod_nbij' (fun l => i + d - l) (fun s => i + d - s) ?_ ?_ ?_ ?_ ?_ · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast, ← zpow_neg] congr 2 push_cast [Nat.cast_sub (show l ≤ i + d by omega)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 258–269 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl.s1 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), i + d - a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)).
  Uses library Finset.mem_Icc.
  Lean, line 260
  · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 260 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl.s2 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), i + d - a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)).
  Uses library Finset.mem_Icc.
  Lean, line 261
  · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 261 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl.s3 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), i + d - (i + d - a) = a.
  Uses library Finset.mem_Icc.
  Lean, line 262
  · intro l hl; simp only [Finset.mem_Icc] at hl; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 262 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl.s4 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), i + d - (i + d - a) = a.
  Uses library Finset.mem_Icc.
  Lean, line 263
  · intro l hl; simp only [Finset.mem_Icc] at hl; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 263 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.hrefl.s5 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), (1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - (↑(i + d) : ℤ)) = (1 : ℝ) - ((2 : ℝ) ^ (i + d - a))⁻¹.
  Uses library Finset.mem_Icc, zpow_natCast, zpow_neg, Nat.cast_sub.
  Lean, lines 264–269
  · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast, ← zpow_neg] congr 2 push_cast [Nat.cast_sub (show l ≤ i + d by omega)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 264–269 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD1.h2s have (2 : ℝ) ^ s ≠ (0 : ℝ) routine: positivity
  Leaves ((1 : ℝ) - ((2 : ℝ) ^ s)⁻¹) * -(2 : ℝ) ^ s = (1 : ℝ) - (2 : ℝ) ^ s, with h2s : (2 : ℝ) ^ s ≠ (0 : ℝ).
  Lean, line 274
  have h2s : (2 : ℝ) ^ s ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 274 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2 have ∏ l ∈ Finset.Icc (i + d + (1 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hD2 : ∏ l ∈ Finset.Icc (i + d + (1 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ).
  Uses library Finset.Icc, Finset.prod_nbij'.
  Lean, lines 277–289
  have hD2 : ∏ l ∈ Finset.Icc (i + d + 1) n, (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by push_cast refine Finset.prod_nbij' (fun l => l - (i + d)) (fun s => s + (i + d)) ?_ ?_ ?_ ?_ ?_ · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl; simp only [Finset.mem_Icc] at hl; omega · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast] congr 2 push_cast [Nat.cast_sub (show i + d ≤ l by omega)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 277–289 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2.s1 case
  Leaves ∀ a ∈ Finset.Icc (i + d + (1 : ℕ)) n, a - (i + d) ∈ Finset.Icc (1 : ℕ) (n - (i + d)).
  Uses library Finset.mem_Icc.
  Lean, line 280
  · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 280 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2.s2 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (n - (i + d)), a + (i + d) ∈ Finset.Icc (i + d + (1 : ℕ)) n.
  Uses library Finset.mem_Icc.
  Lean, line 281
  · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 281 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2.s3 case
  Leaves ∀ a ∈ Finset.Icc (i + d + (1 : ℕ)) n, a - (i + d) + (i + d) = a.
  Uses library Finset.mem_Icc.
  Lean, line 282
  · intro l hl; simp only [Finset.mem_Icc] at hl; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 282 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2.s4 case
  Leaves ∀ a ∈ Finset.Icc (1 : ℕ) (n - (i + d)), a + (i + d) - (i + d) = a.
  Uses library Finset.mem_Icc.
  Lean, line 283
  · intro l hl; simp only [Finset.mem_Icc] at hl; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 283 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hD2.s5 case
  Leaves ∀ a ∈ Finset.Icc (i + d + (1 : ℕ)) n, (1 : ℝ) - (2 : ℝ) ^ ((↑a : ℤ) - ((↑i : ℤ) + (↑d : ℤ))) = (1 : ℝ) - (2 : ℝ) ^ (a - (i + d)).
  Uses library Finset.mem_Icc, zpow_natCast, Nat.cast_sub.
  Lean, lines 284–289
  · intro l hl simp only [Finset.mem_Icc] at hl rw [← zpow_natCast] congr 2 push_cast [Nat.cast_sub (show i + d ≤ l by omega)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 284–289 at b16ab29fc51b ↗
  numerator_integral.residue_integral.s12 obtain ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (d + (n - (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with z1 : ℤ, hz1 : ∏ s ∈ Finset.Icc (1 : ℕ) (d + (n - (1 : ℕ))), ((1 : ℤ) - (2 : ℤ) ^ s) = z1 * ((∏ s ∈ Finset.Icc (1 : ℕ) d, ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library rfl.
  Lean, line 290
  obtain ⟨z1, hz1⟩ := hG _ d (n - 1) rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 290 at b16ab29fc51b ↗
  numerator_integral.residue_integral.s13 obtain ∃ (z : ℤ), ∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ) + (n - (i + d))), ((1 : ℤ) - (2 : ℤ) ^ s) = z * ((∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with z2 : ℤ, hz2 : ∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ) + (n - (i + d))), ((1 : ℤ) - (2 : ℤ) ^ s) = z2 * ((∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library rfl.
  Lean, line 291
  obtain ⟨z2, hz2⟩ := hG _ (i + d - 1) (n - (i + d)) rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 291 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hNZ have ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = z1 * z2 * ((∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hNZ : ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d)) = z1 * z2 * ((∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) * ∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)).
  Uses library Finset.Icc, mul_left_cancel₀.
  Lean, lines 293–296
  have hNZ : (∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d))) = z1 * z2 * ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s))) := by apply mul_left_cancel₀ (hQne d) rw [hN (n - 1) d, hz1, hz2] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 293–296 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hNZ' have (↑(∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d))) : ℝ) = (↑z1 : ℝ) * (↑z2 : ℝ) * ((↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hNZ' : (↑(∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ (t + d))) : ℝ) = (↑z1 : ℝ) * (↑z2 : ℝ) * ((↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ)).
  Uses library Finset.Icc.
  Lean, lines 297–298
  have hNZ' : ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) = (z1 : ℝ) * z2 * (((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ)) := by exact_mod_cast hNZ
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 297–298 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQ1 have (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hQ1 : (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) ≠ (0 : ℝ).
  Uses library Finset.Icc.
  Lean, line 299
  have hQ1 : ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 299 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hQ2 have (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hQ2 : (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) ≠ (0 : ℝ).
  Uses library Finset.Icc.
  Lean, line 300
  have hQ2 : ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 300 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hDr have (∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ)))) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), -(2 : ℤ) ^ s) : ℝ) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hDr : (∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ)))) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), -(2 : ℤ) ^ s) : ℝ) = (↑(∏ s ∈ Finset.Icc (1 : ℕ) (i + d - (1 : ℕ)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ) * (↑(∏ s ∈ Finset.Icc (1 : ℕ) (n - (i + d)), ((1 : ℤ) - (2 : ℤ) ^ s)) : ℝ).
  Uses library Finset.Icc, Finset.prod_union.
  Lean, lines 301–303
  have hDr : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by rw [hset, Finset.prod_union hdisj, hD2, ← hD1] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 301–303 at b16ab29fc51b ↗
  numerator_integral.residue_integral.hDne have ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d)))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))), with hDne : ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase (i + d), ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑(i + d) : ℤ))) ≠ (0 : ℝ).
  Uses library Finset.Icc, MulZeroClass.zero_mul, mul_ne_zero.
  Lean, lines 304–307
  have hDne : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) ≠ 0 := by intro h0 rw [h0, zero_mul] at hDr exact mul_ne_zero hQ1 hQ2 hDr.symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 304–307 at b16ab29fc51b ↗
2. numerator_integral.polypart_integral have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with polypart_integral : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹).
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_sub_distrib, sub_mul, Finset.sum_congr, rfl, Finset.sum_comm, Exists.intro.
  Lean, lines 312–397
  have polypart_integral : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by intro n hn obtain ⟨α, hα⟩ := hpoly n hn obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ k, D k = ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := ⟨fun k => _, fun k => rfl⟩ have hA : ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc 1 (n - 1), (1 - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * (x * 2 ^ k) ^ j := by intro x k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring simp only [← hD, hA] -- (a) one double sum have hswap : ∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by simp only [Finset.sum_mul] refine Finset.sum_comm' (fun i k => ?_) simp only [Finset.mem_Icc] omega rw [hswap] simp only [Finset.mul_sum] rw [← Finset.sum_sub_distrib] simp only [← Finset.sum_sub_distrib, ← sub_mul] -- (b) per (k, i) have hb : ∀ (k i : ℕ), 1 ≤ i → (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s))) / D k := by intro k i hi have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi intro h0 linarith have hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ := by rw [show (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) = ((2 : ℝ) ^ i)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) by field_simp, mul_assoc, mul_inv_cancel₀ hne, mul_one] have hj : ∀ j : ℕ, ((((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s) := by intro j have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by linear_combination hg rw [hg', mul_assoc, hch, Finset.sum_mul] refine Finset.sum_congr rfl (fun s hs => ?_) have hs' := Finset.mem_range.mp hs rw [mul_assoc, ← pow_succ, show j - 1 - s + 1 = j - s by omega] have e1 : (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j - ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) / D k := by ring rw [e1, ← Finset.sum_sub_distrib, Finset.sum_mul] congr 1 refine Finset.sum_congr rfl (fun j _ => ?_) rw [← Finset.mul_sum, ← hj j, mul_pow, mul_pow] ring -- (c) sum over i have hc : ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - 1) := by intro k rw [Finset.sum_congr rfl (fun i hi => hb k i (Finset.mem_Icc.mp hi).1), ← Finset.sum_div, Finset.sum_comm, Finset.sum_div] refine Finset.sum_congr rfl (fun j _ => ?_) rw [Finset.sum_comm, Finset.sum_div] refine Finset.sum_congr rfl (fun s hs => ?_) have hs' := Finset.mem_range.mp hs rw [Finset.sum_congr rfl (fun i _ => (mul_assoc _ _ _).symm), ← Finset.mul_sum, hgeo (j - s) (by omega) k] have hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s) := by rw [← pow_add]; congr 1; omega have hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = 1 := mul_inv_cancel₀ (by positivity) rw [hjs] linear_combination (-((α j : ℝ) * ((2 : ℝ) ^ k) ^ s * (8 / 3 : ℝ) ^ s / ((2 : ℝ) ^ (j - s) - 1) / D k)) * hZ rw [Finset.sum_congr rfl (fun k _ => hc k), Finset.sum_comm, Finset.mul_sum] refine hsumZ _ _ (fun j hj => ?_) rw [Finset.sum_comm, Finset.mul_sum] refine hsumZ _ _ (fun s hs => ?_) have hj' := Finset.mem_range.mp hj have hs' := Finset.mem_range.mp hs obtain ⟨wj, hwj⟩ := hW _ (Finset.Icc 1 n) j rfl obtain ⟨ws, hws⟩ := hW _ (Finset.Icc 1 n) s rfl obtain ⟨q, hq⟩ := hdvd n (j - s) (by omega) (by omega) have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ (j - s) - 1) * q := by exact_mod_cast hq have hB : (2 : ℝ) ^ (j - s) - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ (j - s) := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega) linarith have h3 : (3 : ℝ) ^ n = 3 ^ (n - s) * 3 ^ s := by rw [← pow_add]; congr 1; omega have h83 : (8 / 3 : ℝ) ^ s * 3 ^ s = 8 ^ s := by rw [← mul_pow]; norm_num refine ⟨3 ^ (n - s) * 8 ^ s * q * α j * (wj - ws), ?_⟩ simp only [← hD] at hwj hws push_cast rw [hwj, hws, hqR, h3, ← h83, ← Finset.sum_sub_distrib] rw [Finset.mul_sum, Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 312–397 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.s1 obtain ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with α : ℕ → ℤ, hα : ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j.
  Lean, line 314
  obtain ⟨α, hα⟩ := hpoly n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 314 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.s2 obtain ∃ (D : ℕ → ℝ), ∀ (k : ℕ), D k = ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with D : ℕ → ℝ, hD : ∀ (k : ℕ), D k = ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Exists.intro, rfl.
  Lean, line 315
  obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ k, D k = ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := ⟨fun k => _, fun k => rfl⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 315 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hA have ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (x * (2 : ℝ) ^ k) ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with hA : ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (x * (2 : ℝ) ^ k) ^ j.
  Uses library Finset.Icc, Finset.range, Finset.prod_congr, rfl, pow_add.
  Lean, lines 316–320
  have hA : ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc 1 (n - 1), (1 - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * (x * 2 ^ k) ^ j := by intro x k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 316–320 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hswap have ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.Icc (1 : ℕ) k, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ x ∈ Finset.Icc (1 : ℕ) n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ j) / D x * ∑ i ∈ Finset.Icc (1 : ℕ) x, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ x ∈ Finset.Icc (1 : ℕ) n, (∑ x_1 ∈ Finset.Icc x n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ x)⁻¹ * (2 : ℝ) ^ x_1) ^ j) / D x_1) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ x)⁻¹), with hswap : ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.Icc (1 : ℕ) k, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.Icc, Finset.range, Finset.sum_mul, Finset.sum_comm', Finset.mem_Icc.
  Lean, lines 323–327
  have hswap : ∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by simp only [Finset.sum_mul] refine Finset.sum_comm' (fun i k => ?_) simp only [Finset.mem_Icc] omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 323–327 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb have ∀ (k i : ℕ), (1 : ℕ) ≤ i → ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, ∑ x_1 ∈ Finset.Icc (1 : ℕ) x, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ j) / D x - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ x_1)⁻¹ * (2 : ℝ) ^ x) ^ j) / D x) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ x_1)⁻¹, with hb : ∀ (k i : ℕ), (1 : ℕ) ≤ i → ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k.
  Uses library Finset.range, Finset.sum_sub_distrib, Finset.sum_mul, Finset.sum_congr, rfl, Finset.mul_sum, mul_pow.
  Lean, lines 333–359
  have hb : ∀ (k i : ℕ), 1 ≤ i → (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s))) / D k := by intro k i hi have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi intro h0 linarith have hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ := by rw [show (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) = ((2 : ℝ) ^ i)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) by field_simp, mul_assoc, mul_inv_cancel₀ hne, mul_one] have hj : ∀ j : ℕ, ((((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s) := by intro j have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by linear_combination hg rw [hg', mul_assoc, hch, Finset.sum_mul] refine Finset.sum_congr rfl (fun s hs => ?_) have hs' := Finset.mem_range.mp hs rw [mul_assoc, ← pow_succ, show j - 1 - s + 1 = j - s by omega] have e1 : (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j - ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) / D k := by ring rw [e1, ← Finset.sum_sub_distrib, Finset.sum_mul] congr 1 refine Finset.sum_congr rfl (fun j _ => ?_) rw [← Finset.mul_sum, ← hj j, mul_pow, mul_pow] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 333–359 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.h2i have (2 : ℝ) ^ i ≠ (0 : ℝ) routine: positivity
  Leaves ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k, with h2i : (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, line 335
  have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 335 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hne have (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ)
  Leaves ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k, with hne : (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, lines 336–341
  have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi intro h0 linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 336–341 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hne.this have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 337–339
  have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 337–339 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hne.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 338
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 338 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hne.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 339
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 339 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hch have (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹
  Leaves ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k, with hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹.
  Uses library mul_assoc, mul_inv_cancel₀, mul_one.
  Lean, lines 342–343
  have hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ := by rw [show (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) = ((2 : ℝ) ^ i)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) by field_simp, mul_assoc, mul_inv_cancel₀ hne, mul_one]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 342–343 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hj have ∀ (j : ℕ), (((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s)
  Leaves ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k, with hj : ∀ (j : ℕ), (((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s).
  Uses library Finset.range, mul_assoc, Finset.sum_mul, Finset.sum_congr, rfl, pow_succ.
  Lean, lines 344–352
  have hj : ∀ j : ℕ, ((((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s) := by intro j have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by linear_combination hg rw [hg', mul_assoc, hch, Finset.sum_mul] refine Finset.sum_congr rfl (fun s hs => ?_) have hs' := Finset.mem_range.mp hs rw [mul_assoc, ← pow_succ, show j - 1 - s + 1 = j - s by omega]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 344–352 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hj.hg have (∑ i_1 ∈ Finset.range j, (8 / 3 : ℝ) ^ i_1 * ((2 : ℝ) ^ i)⁻¹ ^ (j - (1 : ℕ) - i_1)) * ((8 / 3 : ℝ) - ((2 : ℝ) ^ i)⁻¹) = (8 / 3 : ℝ) ^ j - ((2 : ℝ) ^ i)⁻¹ ^ j
  Leaves (((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s), with hg : (∑ i_1 ∈ Finset.range j, (8 / 3 : ℝ) ^ i_1 * ((2 : ℝ) ^ i)⁻¹ ^ (j - (1 : ℕ) - i_1)) * ((8 / 3 : ℝ) - ((2 : ℝ) ^ i)⁻¹) = (8 / 3 : ℝ) ^ j - ((2 : ℝ) ^ i)⁻¹ ^ j.
  Uses library geom_sum₂_mul.
  Lean, line 346
  have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 346 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hj.hg' have ((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - (1 : ℕ) - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ))
  Leaves (((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s), with hg' : ((2 : ℝ) ^ i)⁻¹ ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - (1 : ℕ) - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)).
  Uses library Finset.range.
  Lean, lines 347–348
  have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by linear_combination hg
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 347–348 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.hj.hs' have s < j
  Leaves (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - (1 : ℕ) - s) * ((2 : ℝ) ^ i)⁻¹ = (8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s), with hs' : s < j.
  Uses library Finset.mem_range.
  Lean, line 351
  have hs' := Finset.mem_range.mp hs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 351 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hb.e1 have ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j - ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ / D k routine: ring
  Leaves ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ i)⁻¹ ^ (j - s))) / D k, with e1 : ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j - ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ / D k.
  Uses library Finset.range.
  Lean, lines 353–354
  have e1 : (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j - ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) / D k := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 353–354 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hc have ∀ (k : ℕ), ∑ i ∈ Finset.Icc (1 : ℕ) k, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - (1 : ℝ))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, ∑ x_1 ∈ Finset.Icc (1 : ℕ) x, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ j) / D x - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ x_1)⁻¹ * (2 : ℝ) ^ x) ^ j) / D x) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ x_1)⁻¹, with hc : ∀ (k : ℕ), ∑ i ∈ Finset.Icc (1 : ℕ) k, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / D k - (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * (((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ k) ^ j) / D k) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)).
  Uses library Finset.Icc, Finset.range, Finset.sum_congr, rfl, Finset.mem_Icc, Finset.sum_div, Finset.sum_comm, mul_assoc, Finset.mul_sum.
  Lean, lines 361–373
  have hc : ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - 1) := by intro k rw [Finset.sum_congr rfl (fun i hi => hb k i (Finset.mem_Icc.mp hi).1), ← Finset.sum_div, Finset.sum_comm, Finset.sum_div] refine Finset.sum_congr rfl (fun j _ => ?_) rw [Finset.sum_comm, Finset.sum_div] refine Finset.sum_congr rfl (fun s hs => ?_) have hs' := Finset.mem_range.mp hs rw [Finset.sum_congr rfl (fun i _ => (mul_assoc _ _ _).symm), ← Finset.mul_sum, hgeo (j - s) (by omega) k] have hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s) := by rw [← pow_add]; congr 1; omega have hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = 1 := mul_inv_cancel₀ (by positivity) rw [hjs] linear_combination (-((α j : ℝ) * ((2 : ℝ) ^ k) ^ s * (8 / 3 : ℝ) ^ s / ((2 : ℝ) ^ (j - s) - 1) / D k)) * hZ
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 361–373 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hc.hs' have s < j
  Leaves (∑ x ∈ Finset.Icc (1 : ℕ) k, (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * ((2 : ℝ) ^ x)⁻¹ ^ (j - s))) / D k = (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hs' : s < j.
  Uses library Finset.mem_range.
  Lean, line 367
  have hs' := Finset.mem_range.mp hs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 367 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hc.hjs have ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s)
  Leaves (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * (8 / 3 : ℝ) ^ s * (((1 : ℝ) - (((2 : ℝ) ^ k) ^ (j - s))⁻¹) / ((2 : ℝ) ^ (j - s) - (1 : ℝ))) / D k = (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s).
  Uses library pow_add.
  Lean, lines 369–370
  have hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s) := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 369–370 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hc.hZ have ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = (1 : ℝ)
  Leaves (↑(α j) : ℝ) * ((2 : ℝ) ^ k) ^ j * (8 / 3 : ℝ) ^ s * (((1 : ℝ) - (((2 : ℝ) ^ k) ^ (j - s))⁻¹) / ((2 : ℝ) ^ (j - s) - (1 : ℝ))) / D k = (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = (1 : ℝ).
  Uses library mul_inv_cancel₀.
  Lean, line 371
  have hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = 1 := mul_inv_cancel₀ (by positivity)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 371 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hj' have j < n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hj' : j < n.
  Uses library Finset.mem_range.
  Lean, line 378
  have hj' := Finset.mem_range.mp hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 378 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hs' have s < j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hs' : s < j.
  Uses library Finset.mem_range.
  Lean, line 379
  have hs' := Finset.mem_range.mp hs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 379 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.s9 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with wj : ℤ, hwj : (↑wj : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, rfl.
  Lean, line 380
  obtain ⟨wj, hwj⟩ := hW _ (Finset.Icc 1 n) j rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 380 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.s10 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ s / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with ws : ℤ, hws : (↑ws : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ s / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, rfl.
  Lean, line 381
  obtain ⟨ws, hws⟩ := hW _ (Finset.Icc 1 n) s rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 381 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.s11 obtain (2 : ℤ) ^ (j - s) - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with q : ℤ, hq : ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k) = ((2 : ℤ) ^ (j - s) - (1 : ℤ)) * q.
  Lean, line 382
  obtain ⟨q, hq⟩ := hdvd n (j - s) (by omega) (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 382 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hqR have ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k) = ((2 : ℝ) ^ (j - s) - (1 : ℝ)) * (↑q : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hqR : ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k) = ((2 : ℝ) ^ (j - s) - (1 : ℝ)) * (↑q : ℝ).
  Uses library Finset.Icc.
  Lean, line 383
  have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ (j - s) - 1) * q := by exact_mod_cast hq
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 383 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hB have (2 : ℝ) ^ (j - s) - (1 : ℝ) ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with hB : (2 : ℝ) ^ (j - s) - (1 : ℝ) ≠ (0 : ℝ).
  Lean, lines 384–388
  have hB : (2 : ℝ) ^ (j - s) - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ (j - s) := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega) linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 384–388 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hB.this have (2 : ℝ) ≤ (2 : ℝ) ^ (j - s)
  Leaves (2 : ℝ) ^ (j - s) - (1 : ℝ) ≠ (0 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ (j - s).
  Lean, lines 385–387
  have : (2 : ℝ) ≤ 2 ^ (j - s) := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 385–387 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hB.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 386
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 386 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.hB.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ (j - s)
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 387
  _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 387 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.h3 have (3 : ℝ) ^ n = (3 : ℝ) ^ (n - s) * (3 : ℝ) ^ s
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with h3 : (3 : ℝ) ^ n = (3 : ℝ) ^ (n - s) * (3 : ℝ) ^ s.
  Uses library pow_add.
  Lean, line 389
  have h3 : (3 : ℝ) ^ n = 3 ^ (n - s) * 3 ^ s := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 389 at b16ab29fc51b ↗
  numerator_integral.polypart_integral.h83 have (8 / 3 : ℝ) ^ s * (3 : ℝ) ^ s = (8 : ℝ) ^ s
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (↑(α j) : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ x) ^ j - ((2 : ℝ) ^ x) ^ s) / D x) / ((2 : ℝ) ^ (j - s) - (1 : ℝ)), with h83 : (8 / 3 : ℝ) ^ s * (3 : ℝ) ^ s = (8 : ℝ) ^ s.
  Uses library mul_pow.
  Lean, line 390
  have h83 : (8 / 3 : ℝ) ^ s * 3 ^ s = 8 ^ s := by rw [← mul_pow]; norm_num
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 390 at b16ab29fc51b ↗
3. numerator_integral.head_integral have ∃ (a₁ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with head_integral : ∃ (a₁ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.factorial, Finset.Icc, Exists.intro, Classical.choose, dif_pos, Classical.choose_spec.
  Lean, lines 398–454
  have head_integral : ∃ a₁ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₁ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by have hsumZ0 : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩ have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by intro n hn obtain ⟨zE, hzE⟩ := polypart_integral n hn have hR : ∃ zR : ℤ, (zR : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) := by rw [Finset.mul_sum] refine hsumZ0 _ _ (fun i hi => ?_) have hi' := Finset.mem_Icc.mp hi obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2 refine ⟨zb * (3 * ∏ j ∈ (Finset.Icc 1 n).erase i, ((3 : ℤ) - 8 * 2 ^ j)), ?_⟩ have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 intro h0 linarith have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) := (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by rw [← pow_succ']; congr 1; omega have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩ rw [← hb] at hzb ⊢ push_cast rw [hzb, hsplit, he, h3] have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by intro h0 apply hne linear_combination (1 / 3 : ℝ) * h0 field_simp obtain ⟨zR, hzR⟩ := hR have hPZ : ∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity field_simp have h9 : (9 : ℝ) ^ n = 3 ^ n * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow] refine ⟨(Nat.factorial (n - 2) : ℤ) * ((∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * zE + 3 ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * zR), ?_⟩ push_cast rw [hzE, hzR, hPZ, h9] ring refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩ simp only [dif_pos hn'] exact Classical.choose_spec (hn n hn')
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 398–454 at b16ab29fc51b ↗
  numerator_integral.head_integral.hsumZ0 have ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (a₁ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hsumZ0 : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset, Finset.induction_on.
  Lean, lines 399–407
  have hsumZ0 : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 399–407 at b16ab29fc51b ↗
  numerator_integral.head_integral.hsumZ0.s1 case printed form did not read back: shown as its Lean lines only
  | empty => intro _; exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 402 at b16ab29fc51b ↗
  numerator_integral.head_integral.hsumZ0.s2 case
  Leaves (∀ i ∈ insert a s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with a : ℕ, s : Finset ℕ, ha : a ∉ s, ih : (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Exists.intro, Finset.sum_insert.
  Lean, lines 403–407
  | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 403–407 at b16ab29fc51b ↗
  numerator_integral.head_integral.hsumZ0.s2.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = g a
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with za : ℤ, hza : (↑za : ℝ) = g a.
  Uses library Finset.mem_insert_self.
  Lean, line 405
  obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 405 at b16ab29fc51b ↗
  numerator_integral.head_integral.hsumZ0.s2.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with zs : ℤ, hzs : (↑zs : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset.mem_insert_of_mem.
  Lean, line 406
  obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 406 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (a₁ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hn : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.factorial, Finset.Icc, Exists.intro.
  Lean, lines 408–451
  have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by intro n hn obtain ⟨zE, hzE⟩ := polypart_integral n hn have hR : ∃ zR : ℤ, (zR : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) := by rw [Finset.mul_sum] refine hsumZ0 _ _ (fun i hi => ?_) have hi' := Finset.mem_Icc.mp hi obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2 refine ⟨zb * (3 * ∏ j ∈ (Finset.Icc 1 n).erase i, ((3 : ℤ) - 8 * 2 ^ j)), ?_⟩ have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 intro h0 linarith have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) := (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by rw [← pow_succ']; congr 1; omega have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩ rw [← hb] at hzb ⊢ push_cast rw [hzb, hsplit, he, h3] have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by intro h0 apply hne linear_combination (1 / 3 : ℝ) * h0 field_simp obtain ⟨zR, hzR⟩ := hR have hPZ : ∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity field_simp have h9 : (9 : ℝ) ^ n = 3 ^ n * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow] refine ⟨(Nat.factorial (n - 2) : ℤ) * ((∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * zE + 3 ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * zR), ?_⟩ push_cast rw [hzE, hzR, hPZ, h9] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 408–451 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with zE : ℤ, hzE : (↑zE : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹).
  Lean, line 410
  obtain ⟨zE, hzE⟩ := polypart_integral n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 410 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR have ∃ (zR : ℤ), (↑zR : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hR : ∃ (zR : ℤ), (↑zR : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.Icc, Finset.mul_sum, Exists.intro.
  Lean, lines 411–439
  have hR : ∃ zR : ℤ, (zR : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) := by rw [Finset.mul_sum] refine hsumZ0 _ _ (fun i hi => ?_) have hi' := Finset.mem_Icc.mp hi obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2 refine ⟨zb * (3 * ∏ j ∈ (Finset.Icc 1 n).erase i, ((3 : ℤ) - 8 * 2 ^ j)), ?_⟩ have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 intro h0 linarith have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) := (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by rw [← pow_succ']; congr 1; omega have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩ rw [← hb] at hzb ⊢ push_cast rw [hzb, hsplit, he, h3] have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by intro h0 apply hne linear_combination (1 / 3 : ℝ) * h0 field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 411–439 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hi' have (1 : ℕ) ≤ i ∧ i ≤ n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with hi' : (1 : ℕ) ≤ i ∧ i ≤ n.
  Uses library Finset.mem_Icc.
  Lean, line 414
  have hi' := Finset.mem_Icc.mp hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 414 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with zb : ℤ, hzb : (↑zb : ℝ) = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Lean, line 415
  obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 415 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hne have (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ)
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with hne : (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, lines 417–422
  have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 intro h0 linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 417–422 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hne.this have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 418–420
  have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 418–420 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hne.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 419
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 419 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hne.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 420
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 420 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hsplit have ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with hsplit : ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k).
  Uses library Finset.Icc, Finset.mul_prod_erase.
  Lean, lines 423–424
  have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) := (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 423–424 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.he have ∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ (n - (1 : ℕ)))⁻¹
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with he : ∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ (n - (1 : ℕ)))⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem, Nat.card_Icc, Nat.add_sub_cancel, inv_pow.
  Lean, lines 425–427
  have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 425–427 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.h3 have (3 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ))
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with h3 : (3 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)).
  Uses library pow_succ'.
  Lean, lines 428–429
  have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by rw [← pow_succ']; congr 1; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 428–429 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.h3ne have (3 : ℝ) ^ (n - (1 : ℕ)) ≠ (0 : ℝ) routine: positivity
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with h3ne : (3 : ℝ) ^ (n - (1 : ℕ)) ≠ (0 : ℝ).
  Lean, line 430
  have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 430 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.s8 obtain ∃ (b : ℝ), b = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves (↑(zb * ((3 : ℤ) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℤ) - (8 : ℤ) * (2 : ℤ) ^ j))) : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with b : ℝ, hb : b = ∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Exists.intro, rfl.
  Lean, line 431
  obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 431 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hR.hne' have (3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ)
  Leaves b * ((3 : ℝ) * ∏ x ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)) * (((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ((∏ k ∈ (Finset.Icc (1 : ℕ) n).erase i, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ (n - (1 : ℕ)))⁻¹)) * (b * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹), with hne' : (3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, lines 435–438
  have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by intro h0 apply hne linear_combination (1 / 3 : ℝ) * h0
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 435–438 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.s3 obtain ∃ (zR : ℤ), (↑zR : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with zR : ℤ, hzR : (↑zR : ℝ) = ((3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) n, (∑ k ∈ Finset.Icc i n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Lean, line 440
  obtain ⟨zR, hzR⟩ := hR
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 440 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hPZ have ∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k) = (3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hPZ : ∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k) = (3 : ℝ) ^ n * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k).
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow.
  Lean, lines 441–445
  have hPZ : ∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) := by rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 441–445 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.hPZ.h3n have (3 : ℝ) ^ n ≠ (0 : ℝ) routine: positivity
  Leaves ∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k) = (3 : ℝ) ^ n * ((∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * ((3 : ℝ) ^ n)⁻¹), with h3n : (3 : ℝ) ^ n ≠ (0 : ℝ).
  Lean, line 444
  have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 444 at b16ab29fc51b ↗
  numerator_integral.head_integral.hn.h9 have (9 : ℝ) ^ n = (3 : ℝ) ^ n * (3 : ℝ) ^ n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with h9 : (9 : ℝ) ^ n = (3 : ℝ) ^ n * (3 : ℝ) ^ n.
  Uses library mul_pow.
  Lean, lines 446–447
  have h9 : (9 : ℝ) ^ n = 3 ^ n * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 446–447 at b16ab29fc51b ↗
4. numerator_integral.s4 obtain ∃ (a₁ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with a₁ : ℕ → ℤ, h₁ : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₁ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Lean, line 455
  obtain ⟨a₁, h₁⟩ := head_integral
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 455 at b16ab29fc51b ↗
5. numerator_integral.hmer have ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hmer : ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)).
  Uses library Nat.factorial, Finset.Icc, Exists.intro, Classical.choose, dif_pos, Classical.choose_spec.
  Lean, lines 456–556
  have hmer : ∃ a₂ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₂ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by have key : ∀ n : ℕ, 1 ≤ n → ∀ i : ℕ, 1 ≤ i → i ≤ n - 1 → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1 - i) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) := by intro n hn i hi1 hi2 obtain ⟨α, hα⟩ := hpoly n hn have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by intro k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring simp only [hA] have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => ?_) rw [neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl (fun j _ => ?_) ring rw [e1, Finset.mul_sum] refine hsumZ _ _ (fun j hj => ?_) have hj' := Finset.mem_range.mp hj rw [← Finset.mul_sum] have hcX : ∀ k : ℕ, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ≠ 0 := fun k => by positivity rcases lt_or_ge j i with hji | hji · refine ⟨0, ?_⟩ have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega, mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv] rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega] rw [hvan, hV (i - j - 1) (Finset.Icc 1 n) (by rw [Nat.card_Icc]; omega)] simp · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ (j - i) * w, ?_⟩ have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow] rw [hk, mul_div_assoc] have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num push_cast rw [hpos, hw, h3, ← h83] ring have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by intro n rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow] have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by intro n i hi1 hi2 obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩ have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega)) have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩ have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i) rw [one_pow, ← pow_mul, ← hk0] at this exact this have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring rw [this]; exact (dvd_neg).mpr h1 exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem) have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩ have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by intro n hn rw [Finset.mul_sum] refine hsumZ _ _ (fun i hi => ?_) have hi' := Finset.mem_Icc.mp hi obtain ⟨z1, hz1⟩ := key n hn i hi'.1 hi'.2 obtain ⟨q, hq⟩ := hdvd n i hi'.1 hi'.2 have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ i - 1) * q := by exact_mod_cast hq have h2i : (2 : ℝ) ^ i - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 linarith have h9 : (9 : ℝ) ^ n = 3 ^ (i + 1) * 3 ^ (n - 1 - i) * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow] congr 1 rw [← pow_add] congr 1 omega refine ⟨3 ^ (i + 1) * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * q * z1, ?_⟩ push_cast rw [hz1, hW3 n, hqR, h9] have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity field_simp refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩ simp only [dif_pos hn'] exact Classical.choose_spec (hn n hn')
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 456–556 at b16ab29fc51b ↗
  numerator_integral.hmer.key have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹
  Leaves ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)), with key : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹.
  Uses library Finset.Icc, Finset.mul_sum, lt_or_ge.
  Lean, lines 457–501
  have key : ∀ n : ℕ, 1 ≤ n → ∀ i : ℕ, 1 ≤ i → i ≤ n - 1 → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1 - i) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) := by intro n hn i hi1 hi2 obtain ⟨α, hα⟩ := hpoly n hn have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by intro k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring simp only [hA] have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => ?_) rw [neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl (fun j _ => ?_) ring rw [e1, Finset.mul_sum] refine hsumZ _ _ (fun j hj => ?_) have hj' := Finset.mem_range.mp hj rw [← Finset.mul_sum] have hcX : ∀ k : ℕ, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ≠ 0 := fun k => by positivity rcases lt_or_ge j i with hji | hji · refine ⟨0, ?_⟩ have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega, mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv] rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega] rw [hvan, hV (i - j - 1) (Finset.Icc 1 n) (by rw [Nat.card_Icc]; omega)] simp · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ (j - i) * w, ?_⟩ have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow] rw [hk, mul_div_assoc] have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num push_cast rw [hpos, hw, h3, ← h83] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 457–501 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s1 obtain ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹, with α : ℕ → ℤ, hα : ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j.
  Lean, line 459
  obtain ⟨α, hα⟩ := hpoly n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 459 at b16ab29fc51b ↗
  numerator_integral.hmer.key.hA have ∀ (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹, with hA : ∀ (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j.
  Uses library Finset.Icc, Finset.range, Finset.prod_congr, rfl, pow_add.
  Lean, lines 460–464
  have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by intro k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 460–464 at b16ab29fc51b ↗
  numerator_integral.hmer.key.e1 have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ x ∈ Finset.Icc (1 : ℕ) n, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase x, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑x : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ i)⁻¹, with e1 : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))).
  Uses library Finset.Icc, Finset.range, Finset.sum_comm, Finset.sum_congr, rfl, neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, Finset.sum_neg_distrib.
  Lean, lines 466–471
  have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => ?_) rw [neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl (fun j _ => ?_) ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 466–471 at b16ab29fc51b ↗
  numerator_integral.hmer.key.hj' have j < n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hj' : j < n.
  Uses library Finset.mem_range.
  Lean, line 474
  have hj' := Finset.mem_range.mp hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 474 at b16ab29fc51b ↗
  numerator_integral.hmer.key.hcX have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with hcX : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 476
  have hcX : ∀ k : ℕ, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ≠ 0 := fun k => by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 476 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s6 case
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))).
  Uses library Exists.intro, Finset.Icc, Nat.card_Icc.
  Lean, lines 478–487
  · refine ⟨0, ?_⟩ have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega, mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv] rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega] rw [hvan, hV (i - j - 1) (Finset.Icc 1 n) (by rw [Nat.card_Icc]; omega)] simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 478–487 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s6.hvan have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc (1 : ℕ) n, (((2 : ℝ) ^ k) ^ (i - j - (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves (↑(0 : ℤ) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with hvan : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc (1 : ℕ) n, (((2 : ℝ) ^ k) ^ (i - j - (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_congr, rfl, mul_div_assoc.
  Lean, lines 479–485
  have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega, mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv] rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 479–485 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s6.hvan.hk have ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹
  Leaves ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ((((2 : ℝ) ^ k) ^ (i - j - (1 : ℕ) + (1 : ℕ)))⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹.
  Uses library pow_add, mul_inv, mul_assoc, mul_inv_cancel₀, pow_ne_zero, one_mul, mul_pow.
  Lean, lines 482–484
  have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega, mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 482–484 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7 case
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))).
  Uses library Exists.intro.
  Lean, lines 488–501
  · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ (j - i) * w, ?_⟩ have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow] rw [hk, mul_div_assoc] have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num push_cast rw [hpos, hw, h3, ← h83] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 488–501 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with w : ℤ, hw : (↑w : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, rfl.
  Lean, line 488
  · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 488 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7.hpos have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ (j - i) * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with hpos : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_congr, rfl, mul_div_assoc.
  Lean, lines 490–496
  have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow] rw [hk, mul_div_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 490–496 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7.hpos.hk have ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i)
  Leaves ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (8 / 3 : ℝ) ^ (j - i) * (((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹ = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i).
  Uses library pow_add, mul_assoc, mul_inv_cancel₀, pow_ne_zero, mul_one, mul_pow.
  Lean, lines 493–495
  have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega, mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 493–495 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7.h3 have (3 : ℝ) ^ (n - (1 : ℕ) - i) = (3 : ℝ) ^ (n - (1 : ℕ) - j) * (3 : ℝ) ^ (j - i)
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ (j - i) * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with h3 : (3 : ℝ) ^ (n - (1 : ℕ) - i) = (3 : ℝ) ^ (n - (1 : ℕ) - j) * (3 : ℝ) ^ (j - i).
  Uses library pow_add.
  Lean, line 497
  have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 497 at b16ab29fc51b ↗
  numerator_integral.hmer.key.s7.h83 have (8 / 3 : ℝ) ^ (j - i) * (3 : ℝ) ^ (j - i) = (8 : ℝ) ^ (j - i)
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ (j - i) * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * (-(↑(α j) : ℝ) * ∑ i_1 ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ j * (((8 / 3 : ℝ) * (2 : ℝ) ^ i_1) ^ i)⁻¹ / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i_1, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i_1 : ℤ)))), with h83 : (8 / 3 : ℝ) ^ (j - i) * (3 : ℝ) ^ (j - i) = (8 : ℝ) ^ (j - i).
  Uses library mul_pow.
  Lean, line 498
  have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 498 at b16ab29fc51b ↗
  numerator_integral.hmer.hW3 have ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹
  Leaves ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)), with hW3 : ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow.
  Lean, lines 502–505
  have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by intro n rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 502–505 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd have ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k)
  Leaves ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)), with hdvd : ∀ (n i : ℕ), (1 : ℕ) ≤ i → i ≤ n - (1 : ℕ) → (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k).
  Uses library Finset.Icc, Finset.dvd_prod_of_mem.
  Lean, lines 506–520
  have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by intro n i hi1 hi2 obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩ have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega)) have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩ have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i) rw [one_pow, ← pow_mul, ← hk0] at this exact this have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring rw [this]; exact (dvd_neg).mpr h1 exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 506–520 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.s1 obtain ∃ (k0 : ℕ), k0 = i * (n / i)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with k0 : ℕ, hk0 : k0 = i * (n / i).
  Uses library Exists.intro, rfl.
  Lean, line 508
  obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 508 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.hle have k0 ≤ n
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hle : k0 ≤ n.
  Uses library mul_comm, Nat.div_mul_le_self.
  Lean, line 509
  have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 509 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.hlt have n < k0 + i
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hlt : n < k0 + i.
  Lean, line 510
  have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 510 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.hlt.this have n < i * (n / i + (1 : ℕ))
  Leaves n < i * (n / i) + i, with this : n < i * (n / i + (1 : ℕ)).
  Uses library Nat.lt_mul_div_succ.
  Lean, line 510
  have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 510 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.hge have i ≤ k0
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hge : i ≤ k0.
  Uses library Nat.le_mul_of_pos_right, Nat.div_pos.
  Lean, line 511
  have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 511 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.hmem have k0 ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with hmem : k0 ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n.
  Uses library Finset.Icc, Finset.mem_Icc, And.intro.
  Lean, line 512
  have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 512 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.h1 have (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with h1 : (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ).
  Uses library one_pow, pow_mul.
  Lean, lines 513–516
  have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i) rw [one_pow, ← pow_mul, ← hk0] at this exact this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 513–516 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.h1.this have (2 : ℤ) ^ i - (1 : ℤ) ∣ ((2 : ℤ) ^ i) ^ (n / i) - (1 : ℤ) ^ (n / i)
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ (2 : ℤ) ^ k0 - (1 : ℤ), with this : (2 : ℤ) ^ i - (1 : ℤ) ∣ ((2 : ℤ) ^ i) ^ (n / i) - (1 : ℤ) ^ (n / i).
  Uses library sub_dvd_pow_sub_pow.
  Lean, line 514
  have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 514 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.h2 have (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k), with h2 : (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0.
  Uses library dvd_neg.
  Lean, lines 517–519
  have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring rw [this]; exact (dvd_neg).mpr h1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 517–519 at b16ab29fc51b ↗
  numerator_integral.hmer.hdvd.h2.this have (1 : ℤ) - (2 : ℤ) ^ k0 = -((2 : ℤ) ^ k0 - (1 : ℤ)) routine: ring
  Leaves (2 : ℤ) ^ i - (1 : ℤ) ∣ (1 : ℤ) - (2 : ℤ) ^ k0, with this : (1 : ℤ) - (2 : ℤ) ^ k0 = -((2 : ℤ) ^ k0 - (1 : ℤ)).
  Lean, line 518
  have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 518 at b16ab29fc51b ↗
  numerator_integral.hmer.hsumZ have ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)), with hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset, Finset.induction_on.
  Lean, lines 521–529
  have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by intro s g induction s using Finset.induction_on with | empty => intro _; exact ⟨0, by simp⟩ | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 521–529 at b16ab29fc51b ↗
  numerator_integral.hmer.hsumZ.s1 case printed form did not read back: shown as its Lean lines only
  | empty => intro _; exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 524 at b16ab29fc51b ↗
  numerator_integral.hmer.hsumZ.s2 case
  Leaves (∀ i ∈ insert a s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with a : ℕ, s : Finset ℕ, ha : a ∉ s, ih : (∀ i ∈ s, ∃ (z : ℤ), (↑z : ℝ) = g i) → ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i.
  Uses library Exists.intro, Finset.sum_insert.
  Lean, lines 525–529
  | insert a s ha ih => intro h obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s) obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi)) exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 525–529 at b16ab29fc51b ↗
  numerator_integral.hmer.hsumZ.s2.s1 obtain ∃ (z : ℤ), (↑z : ℝ) = g a
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with za : ℤ, hza : (↑za : ℝ) = g a.
  Uses library Finset.mem_insert_self.
  Lean, line 527
  obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 527 at b16ab29fc51b ↗
  numerator_integral.hmer.hsumZ.s2.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ s, g i
  Leaves ∃ (z : ℤ), (↑z : ℝ) = ∑ i ∈ insert a s, g i, with zs : ℤ, hzs : (↑zs : ℝ) = ∑ i ∈ s, g i.
  Uses library Finset.mem_insert_of_mem.
  Lean, line 528
  obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 528 at b16ab29fc51b ↗
  numerator_integral.hmer.hn have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))
  Leaves ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)), with hn : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)).
  Uses library Nat.factorial, Finset.Icc, Finset.mul_sum, Exists.intro.
  Lean, lines 530–553
  have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by intro n hn rw [Finset.mul_sum] refine hsumZ _ _ (fun i hi => ?_) have hi' := Finset.mem_Icc.mp hi obtain ⟨z1, hz1⟩ := key n hn i hi'.1 hi'.2 obtain ⟨q, hq⟩ := hdvd n i hi'.1 hi'.2 have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ i - 1) * q := by exact_mod_cast hq have h2i : (2 : ℝ) ^ i - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 linarith have h9 : (9 : ℝ) ^ n = 3 ^ (i + 1) * 3 ^ (n - 1 - i) * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow] congr 1 rw [← pow_add] congr 1 omega refine ⟨3 ^ (i + 1) * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * q * z1, ?_⟩ push_cast rw [hz1, hW3 n, hqR, h9] have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 530–553 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.hi' have (1 : ℕ) ≤ i ∧ i ≤ n - (1 : ℕ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hi' : (1 : ℕ) ≤ i ∧ i ≤ n - (1 : ℕ).
  Uses library Finset.mem_Icc.
  Lean, line 534
  have hi' := Finset.mem_Icc.mp hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 534 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.s2 obtain ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with z1 : ℤ, hz1 : (↑z1 : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹.
  Lean, line 535
  obtain ⟨z1, hz1⟩ := key n hn i hi'.1 hi'.2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 535 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.s3 obtain (2 : ℤ) ^ i - (1 : ℤ) ∣ ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with q : ℤ, hq : ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℤ) - (2 : ℤ) ^ k) = ((2 : ℤ) ^ i - (1 : ℤ)) * q.
  Lean, line 536
  obtain ⟨q, hq⟩ := hdvd n i hi'.1 hi'.2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 536 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.hqR have ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k) = ((2 : ℝ) ^ i - (1 : ℝ)) * (↑q : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hqR : ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k) = ((2 : ℝ) ^ i - (1 : ℝ)) * (↑q : ℝ).
  Uses library Finset.Icc.
  Lean, line 537
  have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ i - 1) * q := by exact_mod_cast hq
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 537 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h2i have (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with h2i : (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ).
  Lean, lines 538–542
  have h2i : (2 : ℝ) ^ i - 1 ≠ 0 := by have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1 linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 538–542 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h2i.this have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 539–541
  have : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 539–541 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h2i.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 540
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 540 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h2i.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 541
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 541 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h9 have (9 : ℝ) ^ n = (3 : ℝ) ^ (i + (1 : ℕ)) * (3 : ℝ) ^ (n - (1 : ℕ) - i) * (3 : ℝ) ^ n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with h9 : (9 : ℝ) ^ n = (3 : ℝ) ^ (i + (1 : ℕ)) * (3 : ℝ) ^ (n - (1 : ℕ) - i) * (3 : ℝ) ^ n.
  Uses library mul_pow, pow_add.
  Lean, lines 543–548
  have h9 : (9 : ℝ) ^ n = 3 ^ (i + 1) * 3 ^ (n - 1 - i) * 3 ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow] congr 1 rw [← pow_add] congr 1 omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 543–548 at b16ab29fc51b ↗
  numerator_integral.hmer.hn.h3n have (3 : ℝ) ^ n ≠ (0 : ℝ) routine: positivity
  Leaves ((3 : ℝ) ^ (i + (1 : ℕ)) * (↑(n - (2 : ℕ)).factorial : ℝ) * ∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * (↑q : ℝ) * ((3 : ℝ) ^ (n - (1 : ℕ) - i) * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (3 : ℝ) ^ (i + (1 : ℕ)) * (3 : ℝ) ^ (n - (1 : ℕ) - i) * (3 : ℝ) ^ n * ((↑(n - (2 : ℕ)).factorial : ℝ) * ((∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹) * (((2 : ℝ) ^ i - (1 : ℝ)) * (↑q : ℝ))) * ((∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with h3n : (3 : ℝ) ^ n ≠ (0 : ℝ).
  Lean, line 552
  have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 552 at b16ab29fc51b ↗
6. numerator_integral.s6 obtain ∃ (a₂ : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))
  Leaves ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with a₂ : ℕ → ℤ, h₂ : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a₂ n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)).
  Lean, line 557
  obtain ⟨a₂, h₂⟩ := hmer
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 557 at b16ab29fc51b ↗
s10 obtain ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with a : ℕ → ℤ, ha : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Lean, line 562
```
  obtain ⟨a, ha⟩ := numerator_integral
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 562 at b16ab29fc51b ↗
hden have ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hden : ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Uses library Nat.factorial, Finset.Icc, Exists.intro, Classical.choose, dif_pos, Classical.choose_spec.
Lean, lines 564–609
```
  have hden : ∃ b : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (b n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
    have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      intro n hn
      obtain ⟨α, hα⟩ := hpoly n hn
      have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by
        intro k
        rw [← hα]
        refine Finset.prod_congr rfl (fun t _ => ?_)
        rw [pow_add]; ring
      simp only [hA]
      have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by
        rw [Finset.sum_comm]
        refine Finset.sum_congr rfl (fun k _ => ?_)
        rw [neg_div, Finset.sum_div, ← Finset.sum_neg_distrib]
        refine Finset.sum_congr rfl (fun j _ => ?_)
        ring
      rw [e1, Finset.mul_sum]
      refine hsumZ _ _ (fun j hj => ?_)
      have hj' := Finset.mem_range.mp hj
      rw [← Finset.mul_sum]
      obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) j rfl
      refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ j * w, ?_⟩
      have hpos : ∑ k ∈ Finset.Icc 1 n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k _ => ?_)
        rw [mul_pow, mul_div_assoc]
      have h3 : (3 : ℝ) ^ (n - 1) = 3 ^ (n - 1 - j) * 3 ^ j := by rw [← pow_add]; congr 1; omega
      have h83 : (8 / 3 : ℝ) ^ j * 3 ^ j = 8 ^ j := by rw [← mul_pow]; norm_num
      push_cast
      rw [hpos, hw, h3, ← h83]
      ring
    have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
      intro n
      rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
        Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
    refine ⟨fun n => if h : 1 ≤ n then 3 * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * Classical.choose (key n h) else 0, fun n hn => ?_⟩
    simp only [dif_pos hn]
    push_cast
    rw [Classical.choose_spec (key n hn), hW3 n]
    have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by
      rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
    have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
    rw [h9]
    linear_combination (-(3 : ℝ) * (Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k)) * ((3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))))) * h3
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 564–609 at b16ab29fc51b ↗
1. hden.key have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with key : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Finset.mul_sum, Exists.intro.
  Lean, lines 565–594
  have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by intro n hn obtain ⟨α, hα⟩ := hpoly n hn have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by intro k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring simp only [hA] have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => ?_) rw [neg_div, Finset.sum_div, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl (fun j _ => ?_) ring rw [e1, Finset.mul_sum] refine hsumZ _ _ (fun j hj => ?_) have hj' := Finset.mem_range.mp hj rw [← Finset.mul_sum] obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) j rfl refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ j * w, ?_⟩ have hpos : ∑ k ∈ Finset.Icc 1 n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) rw [mul_pow, mul_div_assoc] have h3 : (3 : ℝ) ^ (n - 1) = 3 ^ (n - 1 - j) * 3 ^ j := by rw [← pow_add]; congr 1; omega have h83 : (8 / 3 : ℝ) ^ j * 3 ^ j = 8 ^ j := by rw [← mul_pow]; norm_num push_cast rw [hpos, hw, h3, ← h83] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 565–594 at b16ab29fc51b ↗
  hden.key.s1 obtain ∃ (α : ℕ → ℤ), ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with α : ℕ → ℤ, hα : ∀ (y : ℝ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * y ^ j.
  Lean, line 567
  obtain ⟨α, hα⟩ := hpoly n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 567 at b16ab29fc51b ↗
  hden.key.hA have ∀ (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hA : ∀ (k : ℕ), ∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j.
  Uses library Finset.Icc, Finset.range, Finset.prod_congr, rfl, pow_add.
  Lean, lines 568–572
  have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by intro k rw [← hα] refine Finset.prod_congr rfl (fun t _ => ?_) rw [pow_add]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 568–572 at b16ab29fc51b ↗
  hden.key.e1 have ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ x ∈ Finset.Icc (1 : ℕ) n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ x) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase x, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑x : ℤ))), with e1 : ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∑ j ∈ Finset.range n, (↑(α j) : ℝ) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))).
  Uses library Finset.Icc, Finset.range, Finset.sum_comm, Finset.sum_congr, rfl, neg_div, Finset.sum_div, Finset.sum_neg_distrib.
  Lean, lines 574–579
  have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => ?_) rw [neg_div, Finset.sum_div, ← Finset.sum_neg_distrib] refine Finset.sum_congr rfl (fun j _ => ?_) ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 574–579 at b16ab29fc51b ↗
  hden.key.hj' have j < n
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, -(↑(α j) : ℝ) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hj' : j < n.
  Uses library Finset.mem_range.
  Lean, line 582
  have hj' := Finset.mem_range.mp hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 582 at b16ab29fc51b ↗
  hden.key.s5 obtain ∃ (z : ℤ), (↑z : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * (-(↑(α j) : ℝ) * ∑ i ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i : ℤ)))), with w : ℤ, hw : (↑w : ℝ) = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, rfl.
  Lean, line 584
  obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) j rfl
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 584 at b16ab29fc51b ↗
  hden.key.hpos have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ j * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * (-(↑(α j) : ℝ) * ∑ i ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i : ℤ)))), with hpos : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc (1 : ℕ) n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_congr, rfl, mul_pow, mul_div_assoc.
  Lean, lines 586–589
  have hpos : ∑ k ∈ Finset.Icc 1 n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k _ => ?_) rw [mul_pow, mul_div_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 586–589 at b16ab29fc51b ↗
  hden.key.h3 have (3 : ℝ) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ) - j) * (3 : ℝ) ^ j
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ j * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * (-(↑(α j) : ℝ) * ∑ i ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i : ℤ)))), with h3 : (3 : ℝ) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ) - j) * (3 : ℝ) ^ j.
  Uses library pow_add.
  Lean, line 590
  have h3 : (3 : ℝ) ^ (n - 1) = 3 ^ (n - 1 - j) * 3 ^ j := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 590 at b16ab29fc51b ↗
  hden.key.h83 have (8 / 3 : ℝ) ^ j * (3 : ℝ) ^ j = (8 : ℝ) ^ j
  Leaves (↑(-α j * (3 : ℤ) ^ (n - (1 : ℕ) - j) * (8 : ℤ) ^ j * w) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * (-(↑(α j) : ℝ) * ∑ i ∈ Finset.Icc (1 : ℕ) n, ((8 / 3 : ℝ) * (2 : ℝ) ^ i) ^ j / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase i, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑i : ℤ)))), with h83 : (8 / 3 : ℝ) ^ j * (3 : ℝ) ^ j = (8 : ℝ) ^ j.
  Uses library mul_pow.
  Lean, line 591
  have h83 : (8 / 3 : ℝ) ^ j * 3 ^ j = 8 ^ j := by rw [← mul_pow]; norm_num
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 591 at b16ab29fc51b ↗
2. hden.hW3 have ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹
  Leaves ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hW3 : ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow.
  Lean, lines 595–598
  have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by intro n rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 595–598 at b16ab29fc51b ↗
3. hden.h9 have (9 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)) * (3 : ℝ) ^ n
  Leaves (((3 : ℝ) * (↑(n - (2 : ℕ)).factorial : ℝ) * ∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * ∏ x ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ x)) * ((3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) = (9 : ℝ) ^ n * ((↑(n - (2 : ℕ)).factorial : ℝ) * ((∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with h9 : (9 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)) * (3 : ℝ) ^ n.
  Uses library mul_pow, mul_comm, pow_succ, Nat.sub_add_cancel.
  Lean, lines 604–605
  have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 604–605 at b16ab29fc51b ↗
4. hden.h3 have ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = (1 : ℝ)
  Leaves (((3 : ℝ) * (↑(n - (2 : ℕ)).factorial : ℝ) * ∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * ∏ x ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ x)) * ((3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) = (9 : ℝ) ^ n * ((↑(n - (2 : ℕ)).factorial : ℝ) * ((∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = (1 : ℝ).
  Uses library inv_mul_cancel₀.
  Lean, line 606
  have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 606 at b16ab29fc51b ↗
s12 obtain ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with b : ℕ → ℤ, hb : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Lean, line 610
```
  obtain ⟨b, hb⟩ := hden
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 610 at b16ab29fc51b ↗
hpf have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hpf : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹.
Uses library Finset.Icc, Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.
Lean, lines 612–742
```
  have hpf : ∀ n : ℕ, 1 ≤ n → ∀ m : ℕ, 1 ≤ m → (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := by
    intro n hn m hm
    -- General partial fractions, from Mathlib's Lagrange interpolation.
    have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s →
      Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) →
      Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ =
        ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
      intro s p Q hp hinj hdeg x hx
      have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by
        intro a ha b hb hab
        exact hinj ha hb (inv_injective hab)
      have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by
        intro k hk hxk
        apply hx k hk
        rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
      have hQ := Lagrange.eq_interpolate hinj' hdeg
      have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
      rw [← hQ, Lagrange.eval_nodal] at hev
      have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by
        intro k hk
        have h1 := hp k hk
        have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by
          rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
        rw [← h3, mul_inv, inv_neg]
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev]
      have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
      rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc,
        mul_inv_cancel₀ hne, one_mul, Finset.sum_mul]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk]
      have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by
        intro j hj
        have hjs := Finset.mem_of_mem_erase hj
        have hjk := Finset.ne_of_mem_erase hj
        have hpj := hp j hjs
        have hpk := hp k hk
        have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
        have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
        field_simp
        ring
      calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ *
            (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹)
          = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by
            rw [Finset.prod_mul_distrib]; ring
        _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
            rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
    have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by
      calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
        _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
    have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
    have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
    have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by
      intro k
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      nlinarith
    have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by
      intro a _ b _ hab
      have h : (2 : ℝ) ^ a = 2 ^ b := by
        have := hab; simp only at this
        exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
    have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by
      rw [Polynomial.degree_neg, Polynomial.degree_prod]
      simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc]
      rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one]
      exact_mod_cast (by omega : n - 1 < n)
    have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t)))
      (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
    simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C] at hL
    have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
    -- geometric sums
    have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by
      intro u hu hu1 N
      induction N with
      | zero => simp
      | succ N ih =>
        rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih]
        have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
        field_simp
        ring
    -- the left side, in terms of the Lagrange data
    have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by
      rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
    have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
    have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
    have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by
      have hn' : n = n - 1 + 1 := by omega
      conv_lhs => rw [hn']
      rw [Finset.prod_Icc_succ_top (by omega), ← hn']
    have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by
      rw [mul_sub, inv_mul_cancel₀ hx0]; ring
    have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hz1, hz2, hz3]
      rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib,
        Finset.prod_const, hcard, inv_pow]
      ring
    -- the right side
    have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
    have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by
      simp only [Finset.sum_mul, Finset.mul_sum]
      rw [Finset.sum_comm]
      refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_))
      rw [mul_pow, mul_inv]; ring
    have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hpk]
      rw [hswap, ← Finset.sum_add_distrib]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)]
      ring
    -- matching the coefficients
    have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by
      intro k hk
      have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by
        rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring,
          inv_mul_cancel₀ (hp0 k), mul_one]
      have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow]
      simp only [hw]
      ring
    calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
      _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
      _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [← hcoef k hk, mul_pow, mul_inv]
        ring
      _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 612–742 at b16ab29fc51b ↗
1. hpf.lag have ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ (0 : ℝ)) → Set.InjOn p (↑s : Set ℕ) → Q.degree < (↑s.card : WithBot ℕ) → ∀ (x : ℝ), (∀ k ∈ s, (1 : ℝ) - p k * x ≠ (0 : ℝ)) → Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ (0 : ℝ)) → Set.InjOn p (↑s : Set ℕ) → Q.degree < (↑s.card : WithBot ℕ) → ∀ (x : ℝ), (∀ k ∈ s, (1 : ℝ) - p k * x ≠ (0 : ℝ)) → Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹.
  Uses library Finset, Polynomial, Set.InjOn, Lagrange.eval_nodal, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_inv_distrib, mul_assoc, mul_comm, mul_inv_cancel₀, one_mul, Finset.sum_mul, Finset.sum_congr, Lagrange.nodalWeight, Finset.mul_prod_erase.
  Lean, lines 615–657
  have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s → Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) → Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ = ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by intro s p Q hp hinj hdeg x hx have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by intro a ha b hb hab exact hinj ha hb (inv_injective hab) have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by intro k hk hxk apply hx k hk rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self] have hQ := Lagrange.eq_interpolate hinj' hdeg have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv rw [← hQ, Lagrange.eval_nodal] at hev have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by intro k hk have h1 := hp k hk have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring rw [← h3, mul_inv, inv_neg] rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev] have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi)) rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc, mul_inv_cancel₀ hne, one_mul, Finset.sum_mul] refine Finset.sum_congr rfl (fun k hk => ?_) rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk] have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by intro j hj have hjs := Finset.mem_of_mem_erase hj have hjk := Finset.ne_of_mem_erase hj have hpj := hp j hjs have hpk := hp k hk have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h) have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne) field_simp ring calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by rw [Finset.prod_mul_distrib]; ring _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 615–657 at b16ab29fc51b ↗
  hpf.lag.hinj' have Set.InjOn (fun (k : ℕ) => (p k)⁻¹) (↑s : Set ℕ)
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hinj' : Set.InjOn (fun (k : ℕ) => (p k)⁻¹) (↑s : Set ℕ).
  Uses library Set.InjOn, inv_injective.
  Lean, lines 620–622
  have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by intro a ha b hb hab exact hinj ha hb (inv_injective hab)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 620–622 at b16ab29fc51b ↗
  hpf.lag.hxv have ∀ k ∈ s, x ≠ (p k)⁻¹
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hxv : ∀ k ∈ s, x ≠ (p k)⁻¹.
  Uses library mul_inv_cancel₀, sub_self.
  Lean, lines 623–626
  have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by intro k hk hxk apply hx k hk rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 623–626 at b16ab29fc51b ↗
  hpf.lag.hQ have Q = (Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hQ : Q = (Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q.
  Uses library Lagrange.eq_interpolate.
  Lean, line 627
  have hQ := Lagrange.eq_interpolate hinj' hdeg
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 627 at b16ab29fc51b ↗
  hpf.lag.hev have Polynomial.eval x ((Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q) = Polynomial.eval x (Lagrange.nodal s fun (k : ℕ) => (p k)⁻¹) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hev : Polynomial.eval x ((Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q) = Polynomial.eval x (Lagrange.nodal s fun (k : ℕ) => (p k)⁻¹) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q.
  Uses library Lagrange.eval_interpolate_not_at_node.
  Lean, line 628
  have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 628 at b16ab29fc51b ↗
  hpf.lag.hfac have ∀ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hfac : ∀ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹.
  Uses library mul_inv, inv_neg.
  Lean, lines 630–635
  have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by intro k hk have h1 := hp k hk have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring rw [← h3, mul_inv, inv_neg]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 630–635 at b16ab29fc51b ↗
  hpf.lag.hfac.h1 have p k ≠ (0 : ℝ)
  Leaves ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹, with h1 : p k ≠ (0 : ℝ).
  Lean, line 632
  have h1 := hp k hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 632 at b16ab29fc51b ↗
  hpf.lag.hfac.h3 have (x - (p k)⁻¹) * -p k = (1 : ℝ) - p k * x
  Leaves ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹, with h3 : (x - (p k)⁻¹) * -p k = (1 : ℝ) - p k * x.
  Uses library sub_mul, mul_neg, inv_mul_cancel₀.
  Lean, lines 633–634
  have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 633–634 at b16ab29fc51b ↗
  hpf.lag.hne have ∏ i ∈ s, (x - (p i)⁻¹) ≠ (0 : ℝ)
  Leaves ((∏ i ∈ s, (x - (p i)⁻¹)) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q) * ((∏ x_1 ∈ s, (x - (p x_1)⁻¹)⁻¹) * ∏ x ∈ s, -(p x)⁻¹) = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ (0 : ℝ).
  Uses library Finset.prod_ne_zero_iff, sub_ne_zero.
  Lean, line 637
  have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 637 at b16ab29fc51b ↗
  hpf.lag.hj have ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹
  Leaves (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Polynomial.eval (p k)⁻¹ Q * (-(p k)⁻¹ * ∏ x ∈ s.erase k, -(p x)⁻¹) = Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹.
  Lean, lines 642–651
  have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by intro j hj have hjs := Finset.mem_of_mem_erase hj have hjk := Finset.ne_of_mem_erase hj have hpj := hp j hjs have hpk := hp k hk have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h) have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne) field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 642–651 at b16ab29fc51b ↗
  hpf.lag.hj.hjs have j ∈ s
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hjs : j ∈ s.
  Uses library Finset.mem_of_mem_erase.
  Lean, line 644
  have hjs := Finset.mem_of_mem_erase hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 644 at b16ab29fc51b ↗
  hpf.lag.hj.hjk have j ≠ k
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hjk : j ≠ k.
  Uses library Finset.ne_of_mem_erase.
  Lean, line 645
  have hjk := Finset.ne_of_mem_erase hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 645 at b16ab29fc51b ↗
  hpf.lag.hj.hpj have p j ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hpj : p j ≠ (0 : ℝ).
  Lean, line 646
  have hpj := hp j hjs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 646 at b16ab29fc51b ↗
  hpf.lag.hj.hpk have p k ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hpk : p k ≠ (0 : ℝ).
  Lean, line 647
  have hpk := hp k hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 647 at b16ab29fc51b ↗
  hpf.lag.hj.hne have p j ≠ p k
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hne : p j ≠ p k.
  Lean, line 648
  have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 648 at b16ab29fc51b ↗
  hpf.lag.hj.hsub have p k - p j ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hsub : p k - p j ≠ (0 : ℝ).
  Uses library sub_ne_zero, Ne.symm.
  Lean, line 649
  have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 649 at b16ab29fc51b ↗
  hpf.lag.s8 calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Polynomial.eval (p k)⁻¹ Q * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = (Polynomial.eval (p k)⁻¹ Q * ∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹)
  Closes the goal.
  Uses library Finset.prod_mul_distrib.
  Lean, lines 652–655
  calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by rw [Finset.prod_mul_distrib]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 652–655 at b16ab29fc51b ↗
  hpf.lag.s9 calc
  Closes the goal.
  Uses library Finset.prod_congr, rfl, Finset.prod_inv_distrib.
  Lean, lines 656–657
  _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 656–657 at b16ab29fc51b ↗
2. hpf.hx2 have (2 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m.
  Lean, lines 658–660
  have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 658–660 at b16ab29fc51b ↗
  hpf.hx2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 659
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 659 at b16ab29fc51b ↗
  hpf.hx2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ m
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 660
  _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 660 at b16ab29fc51b ↗
3. hpf.hx0 have (2 : ℝ) ^ m ≠ (0 : ℝ) routine: positivity
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hx0 : (2 : ℝ) ^ m ≠ (0 : ℝ).
  Lean, line 661
  have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 661 at b16ab29fc51b ↗
4. hpf.hp0 have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hp0 : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 662
  have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 662 at b16ab29fc51b ↗
5. hpf.hpx have ∀ (k : ℕ), (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hpx : ∀ (k : ℕ), (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ).
  Lean, lines 663–666
  have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by intro k have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num) nlinarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 663–666 at b16ab29fc51b ↗
  hpf.hpx.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ k
  Leaves (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ), with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ k.
  Uses library one_le_pow₀.
  Lean, line 665
  have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 665 at b16ab29fc51b ↗
6. hpf.hinj have Set.InjOn (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hinj : Set.InjOn (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ).
  Uses library Set.InjOn, Finset.Icc, Set, pow_right_injective₀.
  Lean, lines 667–672
  have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by intro a _ b _ hab have h : (2 : ℝ) ^ a = 2 ^ b := by have := hab; simp only at this exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 667–672 at b16ab29fc51b ↗
  hpf.hinj.h have (2 : ℝ) ^ a = (2 : ℝ) ^ b
  Leaves a = b, with h : (2 : ℝ) ^ a = (2 : ℝ) ^ b.
  Uses library mul_left_cancel₀.
  Lean, lines 669–671
  have h : (2 : ℝ) ^ a = 2 ^ b := by have := hab; simp only at this exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 669–671 at b16ab29fc51b ↗
  hpf.hinj.h.this have (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) a = (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) b
  Leaves (2 : ℝ) ^ a = (2 : ℝ) ^ b, with this : (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) a = (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) b.
  Lean, line 670
  have := hab; simp only at this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 670 at b16ab29fc51b ↗
7. hpf.hdeg have (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).degree < (↑(Finset.Icc (1 : ℕ) n).card : WithBot ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hdeg : (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).degree < (↑(Finset.Icc (1 : ℕ) n).card : WithBot ℕ).
  Uses library Finset.Icc, Polynomial.X, Polynomial.C, WithBot, Polynomial.degree_neg, Polynomial.degree_prod, Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc, nsmul_one.
  Lean, lines 673–677
  have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by rw [Polynomial.degree_neg, Polynomial.degree_prod] simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc] rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one] exact_mod_cast (by omega : n - 1 < n)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 673–677 at b16ab29fc51b ↗
8. hpf.hL have Polynomial.eval ((2 : ℝ) ^ m) (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hL : Polynomial.eval ((2 : ℝ) ^ m) (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Polynomial.X, Polynomial.C.
  Lean, lines 678–679
  have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 678–679 at b16ab29fc51b ↗
9. hpf.hcard have (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hcard : (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ).
  Uses library Finset.Icc, Nat.card_Icc.
  Lean, line 681
  have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 681 at b16ab29fc51b ↗
10. hpf.hgeo have ∀ (u : ℝ), u ≠ (0 : ℝ) → (1 : ℝ) - u ≠ (0 : ℝ) → ∀ (N : ℕ), ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hgeo : ∀ (u : ℝ), u ≠ (0 : ℝ) → (1 : ℝ) - u ≠ (0 : ℝ) → ∀ (N : ℕ), ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Finset.Icc.
  Lean, lines 683–691
  have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by intro u hu hu1 N induction N with | zero => simp | succ N ih => rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih] have hN : u ^ N ≠ 0 := pow_ne_zero _ hu field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 683–691 at b16ab29fc51b ↗
  hpf.hgeo.s1 case routine: simp
  Leaves ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (0 : ℕ), (u ^ i)⁻¹ = (u ^ (0 : ℕ))⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Nat.zero.
  Lean, line 686
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 686 at b16ab29fc51b ↗
  hpf.hgeo.s2 case
  Leaves ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (N + (1 : ℕ)), (u ^ i)⁻¹ = (u ^ (N + (1 : ℕ)))⁻¹ * ((1 : ℝ) - u)⁻¹, with N : ℕ, ih : ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Nat.succ, Finset.sum_Icc_succ_top, add_assoc.
  Lean, lines 687–691
  | succ N ih => rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih] have hN : u ^ N ≠ 0 := pow_ne_zero _ hu field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 687–691 at b16ab29fc51b ↗
  hpf.hgeo.s2.hN have u ^ N ≠ (0 : ℝ)
  Leaves (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹ + (u ^ (N + (1 : ℕ)))⁻¹ = (u ^ (N + (1 : ℕ)))⁻¹ * ((1 : ℝ) - u)⁻¹, with hN : u ^ N ≠ (0 : ℝ).
  Uses library pow_ne_zero.
  Lean, line 689
  have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 689 at b16ab29fc51b ↗
11. hpf.hz1 have ∀ (k : ℕ), (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ)) = (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz1 : ∀ (k : ℕ), (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ)) = (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹.
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast, div_eq_mul_inv.
  Lean, lines 693–694
  have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 693–694 at b16ab29fc51b ↗
12. hpf.hz2 have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz2 : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Uses library zpow_add₀, two_ne_zero, zpow_natCast, mul_assoc.
  Lean, lines 695–696
  have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 695–696 at b16ab29fc51b ↗
13. hpf.hz3 have (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m.
  Uses library zpow_add₀, two_ne_zero, zpow_natCast, mul_comm, mul_assoc.
  Lean, lines 697–698
  have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 697–698 at b16ab29fc51b ↗
14. hpf.hsplit have ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hsplit : ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Finset.prod_Icc_succ_top.
  Lean, lines 699–702
  have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by have hn' : n = n - 1 + 1 := by omega conv_lhs => rw [hn'] rw [Finset.prod_Icc_succ_top (by omega), ← hn']
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 699–702 at b16ab29fc51b ↗
  hpf.hsplit.hn' have n = n - (1 : ℕ) + (1 : ℕ) routine: omega
  Leaves ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹, with hn' : n = n - (1 : ℕ) + (1 : ℕ).
  Lean, line 700
  have hn' : n = n - 1 + 1 := by omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 700 at b16ab29fc51b ↗
15. hpf.hfac have ∀ (k : ℕ), (1 : ℝ) - (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hfac : ∀ (k : ℕ), (1 : ℝ) - (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k).
  Uses library mul_sub, inv_mul_cancel₀.
  Lean, lines 703–704
  have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by rw [mul_sub, inv_mul_cancel₀ hx0]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 703–704 at b16ab29fc51b ↗
16. hpf.hLHS have -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hLHS : -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹).
  Uses library Finset.Icc, Finset.prod_mul_distrib, Finset.prod_congr, rfl, Finset.prod_const, inv_pow.
  Lean, lines 705–709
  have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by simp only [hz1, hz2, hz3] rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 705–709 at b16ab29fc51b ↗
17. hpf.hpk have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hpk : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Uses library pow_add, mul_assoc.
  Lean, line 711
  have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 711 at b16ab29fc51b ↗
18. hpf.hswap have ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ i)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hswap : ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ i)⁻¹.
  Uses library Finset.Icc, Finset.sum_mul, Finset.mul_sum, Finset.sum_comm, Finset.sum_congr, rfl, mul_pow, mul_inv.
  Lean, lines 712–716
  have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by simp only [Finset.sum_mul, Finset.mul_sum] rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_)) rw [mul_pow, mul_inv]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 712–716 at b16ab29fc51b ↗
19. hpf.hRHS have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hRHS : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Finset.sum_add_distrib, Finset.sum_congr, rfl, mul_add, mul_ne_zero.
  Lean, lines 717–722
  have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by simp only [hpk] rw [hswap, ← Finset.sum_add_distrib] refine Finset.sum_congr rfl (fun k hk => ?_) rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 717–722 at b16ab29fc51b ↗
20. hpf.hcoef have ∀ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hcoef : ∀ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, inv_pow.
  Lean, lines 724–734
  have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by intro k hk have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring, inv_mul_cancel₀ (hp0 k), mul_one] have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow] simp only [hw] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 724–734 at b16ab29fc51b ↗
  hpf.hcoef.hq have ∀ (t : ℕ), ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))
  Leaves ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹, with hq : ∀ (t : ℕ), ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)).
  Uses library mul_sub, mul_one, pow_add, inv_mul_cancel₀.
  Lean, lines 726–728
  have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring, inv_mul_cancel₀ (hp0 k), mul_one]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 726–728 at b16ab29fc51b ↗
  hpf.hcoef.hw have ∀ (j : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ))
  Leaves ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹, with hw : ∀ (j : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ)).
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast.
  Lean, lines 729–731
  have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 729–731 at b16ab29fc51b ↗
21. hpf.s21 calc -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹)
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 735
  calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 735 at b16ab29fc51b ↗
22. hpf.s22 calc
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 736
  _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 736 at b16ab29fc51b ↗
23. hpf.s23 calc
  Closes the goal.
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_congr, rfl, mul_pow, mul_inv.
  Lean, lines 737–741
  _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k hk => ?_) rw [← hcoef k hk, mul_pow, mul_inv] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 737–741 at b16ab29fc51b ↗
24. hpf.s24 calc ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 742
  _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 742 at b16ab29fc51b ↗
hres have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hres : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Uses library Finset.Icc, zero_add, tsum_congr, Summable.tsum_finsetSum, Finset.mem_Icc, Finset.sum_congr, rfl, tsum_mul_left, mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv.
Lean, lines 744–820
```
  have hres : ∀ n : ℕ, 1 ≤ n → (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    intro n hn
    have pf := hpf n hn
    -- Summability of the two families of pieces.
    have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by
      intro k
      refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_)
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
      have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
      rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div]
      exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
    have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by
      intro i hi
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
      have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
      have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by
        intro m
        rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
      rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1]
      have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
      have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
      field_simp
    have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by
      intro i hi
      by_contra hns
      have := tsum_eq_zero_of_not_summable hns
      rw [hGeo i hi] at this
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
      contradiction
    have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by
      simpa using hT 0
    have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      induction k with
      | zero => simp
      | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
    have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      have hs := hz0.sum_add_tsum_nat_add k
      rw [hIcc k] at hs
      rw [← hs, add_sub_cancel_left]
      exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
    -- The series E(n) summed from m = 1: its first n - 1 terms vanish.
    have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) :=
      summable_sum (fun k _ => (hT k).mul_left _)
    have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) :=
      summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
    have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
    have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) :=
      hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
    have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
    have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by
      refine Finset.sum_eq_zero (fun i hi => ?_)
      have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by
        rw [Finset.mem_range] at hi
        rw [Finset.mem_Icc]; omega
      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_)
      simp
    rw [hzero, zero_add] at hshift
    have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by
      rw [← hshift]
      exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
    rw [hE, tsum_congr (fun m => pf (m + 1) (by omega)), hRs1.tsum_add hRs2,
      Summable.tsum_finsetSum (fun k _ => (hT k).mul_left _),
      Summable.tsum_finsetSum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _),
      Finset.sum_congr (s₁ := Finset.Icc 1 n) rfl (fun k _ => by rw [tsum_mul_left, hTail k]),
      Finset.sum_congr (s₁ := Finset.Icc 1 (n - 1)) rfl (fun i hi => by rw [tsum_mul_left, hGeo i (Finset.mem_Icc.mp hi).1])]
    simp only [mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv]
    ring
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 744–820 at b16ab29fc51b ↗
1. hres.pf have ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with pf : ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹.
  Lean, line 746
  have pf := hpf n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 746 at b16ab29fc51b ↗
2. hres.hT have ∀ (k : ℕ), Summable fun (m : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hT : ∀ (k : ℕ), Summable fun (m : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹.
  Uses library Summable, Summable.of_norm_bounded, summable_geometric_of_lt_one, Real.norm_eq_abs, abs_inv, abs_of_neg, one_div_pow, one_div, inv_anti₀.
  Lean, lines 748–756
  have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by intro k refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_) have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num) have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div] exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 748–756 at b16ab29fc51b ↗
  hres.hT.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ k
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ k.
  Uses library one_le_pow₀.
  Lean, line 751
  have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 751 at b16ab29fc51b ↗
  hres.hT.h2 have (1 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with h2 : (1 : ℝ) ≤ (2 : ℝ) ^ m.
  Uses library one_le_pow₀.
  Lean, line 752
  have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 752 at b16ab29fc51b ↗
  hres.hT.he have (2 : ℝ) ^ (k + (m + (1 : ℕ))) = (2 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m routine: ring
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with he : (2 : ℝ) ^ (k + (m + (1 : ℕ))) = (2 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Lean, line 753
  have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 753 at b16ab29fc51b ↗
  hres.hT.hneg have (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))) < (0 : ℝ)
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with hneg : (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))) < (0 : ℝ).
  Lean, line 754
  have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 754 at b16ab29fc51b ↗
3. hres.hGeo have ∀ (i : ℕ), (1 : ℕ) ≤ i → ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGeo : ∀ (i : ℕ), (1 : ℕ) ≤ i → ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹.
  Uses library tsum_congr, tsum_mul_left, tsum_geometric_of_lt_one.
  Lean, lines 757–770
  have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by intro i hi have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith) have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by intro m rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm] rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1] have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 757–770 at b16ab29fc51b ↗
  hres.hGeo.h2 have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h2 : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 759–761
  have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 759–761 at b16ab29fc51b ↗
  hres.hGeo.h2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 760
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 760 at b16ab29fc51b ↗
  hres.hGeo.h2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 761
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 761 at b16ab29fc51b ↗
  hres.hGeo.hr0 have (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ routine: positivity
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹.
  Lean, line 762
  have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 762 at b16ab29fc51b ↗
  hres.hGeo.hr1 have ((2 : ℝ) ^ i)⁻¹ < (1 : ℝ)
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hr1 : ((2 : ℝ) ^ i)⁻¹ < (1 : ℝ).
  Uses library inv_lt_one_of_one_lt₀.
  Lean, line 763
  have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 763 at b16ab29fc51b ↗
  hres.hGeo.hc have ∀ (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * ((2 : ℝ) ^ i)⁻¹ ^ m
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hc : ∀ (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * ((2 : ℝ) ^ i)⁻¹ ^ m.
  Uses library pow_succ', inv_pow, pow_mul, mul_comm.
  Lean, lines 764–766
  have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by intro m rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 764–766 at b16ab29fc51b ↗
  hres.hGeo.h3 have (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ) routine: linarith
  Leaves ((2 : ℝ) ^ i)⁻¹ * ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h3 : (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ).
  Lean, line 768
  have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 768 at b16ab29fc51b ↗
  hres.hGeo.h4 have (2 : ℝ) ^ i ≠ (0 : ℝ) routine: positivity
  Leaves ((2 : ℝ) ^ i)⁻¹ * ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h4 : (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, line 769
  have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 769 at b16ab29fc51b ↗
4. hres.hGs have ∀ (i : ℕ), (1 : ℕ) ≤ i → Summable fun (m : ℕ) => (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGs : ∀ (i : ℕ), (1 : ℕ) ≤ i → Summable fun (m : ℕ) => (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable.
  Lean, lines 771–780
  have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by intro i hi by_contra hns have := tsum_eq_zero_of_not_summable hns rw [hGeo i hi] at this have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith) contradiction
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 771–780 at b16ab29fc51b ↗
  hres.hGs.s1 have ∑' (b : ℕ), (((2 : ℝ) ^ (b + (1 : ℕ))) ^ i)⁻¹ = (0 : ℝ)
  Leaves False, with this : ∑' (b : ℕ), (((2 : ℝ) ^ (b + (1 : ℕ))) ^ i)⁻¹ = (0 : ℝ).
  Uses library tsum_eq_zero_of_not_summable.
  Lean, line 774
  have := tsum_eq_zero_of_not_summable hns
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 774 at b16ab29fc51b ↗
  hres.hGs.h2 have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves False, with h2 : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 776–778
  have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 776–778 at b16ab29fc51b ↗
  hres.hGs.h2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 777
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 777 at b16ab29fc51b ↗
  hres.hGs.h2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 778
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 778 at b16ab29fc51b ↗
  hres.hGs.s3 have ((2 : ℝ) ^ i - (1 : ℝ))⁻¹ ≠ (0 : ℝ)
  Leaves False, with this : ((2 : ℝ) ^ i - (1 : ℝ))⁻¹ ≠ (0 : ℝ).
  Uses library inv_ne_zero.
  Lean, line 779
  have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 779 at b16ab29fc51b ↗
5. hres.hz0 have Summable fun (j : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hz0 : Summable fun (j : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹.
  Uses library Summable.
  Lean, lines 781–782
  have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by simpa using hT 0
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 781–782 at b16ab29fc51b ↗
6. hres.hIcc have ∀ (k : ℕ), ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hIcc : ∀ (k : ℕ), ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.range, Finset.Icc.
  Lean, lines 783–787
  have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by intro k induction k with | zero => simp | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 783–787 at b16ab29fc51b ↗
  hres.hIcc.s1 case routine: simp
  Leaves ∑ i ∈ Finset.range (0 : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.zero.
  Lean, line 786
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 786 at b16ab29fc51b ↗
  hres.hIcc.s2 case
  Leaves ∑ i ∈ Finset.range (k + (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) (k + (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with k : ℕ, ih : ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.succ, Finset.sum_range_succ, Finset.sum_Icc_succ_top.
  Lean, line 787
  | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 787 at b16ab29fc51b ↗
7. hres.hTail have ∀ (k : ℕ), ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hTail : ∀ (k : ℕ), ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.Icc, add_sub_cancel_left, tsum_congr.
  Lean, lines 788–793
  have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by intro k have hs := hz0.sum_add_tsum_nat_add k rw [hIcc k] at hs rw [← hs, add_sub_cancel_left] exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 788–793 at b16ab29fc51b ↗
  hres.hTail.hs have ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ + ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + k + (1 : ℕ)))⁻¹ = ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹
  Leaves ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hs : ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ + ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + k + (1 : ℕ)))⁻¹ = ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹.
  Lean, line 790
  have hs := hz0.sum_add_tsum_nat_add k
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 790 at b16ab29fc51b ↗
8. hres.hRs1 have Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs1 : Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹.
  Uses library Summable, Finset.Icc, summable_sum.
  Lean, lines 795–796
  have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := summable_sum (fun k _ => (hT k).mul_left _)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 795–796 at b16ab29fc51b ↗
9. hres.hRs2 have Summable fun (m : ℕ) => ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs2 : Summable fun (m : ℕ) => ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable, Finset.Icc, summable_sum, Finset.mem_Icc.
  Lean, lines 797–798
  have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 797–798 at b16ab29fc51b ↗
10. hres.hRs have Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs : Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable, Finset.Icc.
  Lean, line 799
  have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 799 at b16ab29fc51b ↗
11. hres.hGsum have Summable fun (m : ℕ) => -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGsum : Summable fun (m : ℕ) => -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹.
  Uses library Summable, Finset.Icc.
  Lean, lines 800–801
  have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) := hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 800–801 at b16ab29fc51b ↗
12. hres.hshift have ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ + ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ)))⁻¹ = ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hshift : ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ + ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ)))⁻¹ = ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹.
  Lean, line 802
  have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, line 802 at b16ab29fc51b ↗
13. hres.hzero have ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ)
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hzero : ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ).
  Uses library Finset.range, Finset.Icc, Finset.sum_eq_zero, mul_eq_zero_of_right, Finset.prod_eq_zero.
  Lean, lines 803–809
  have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by refine Finset.sum_eq_zero (fun i hi => ?_) have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by rw [Finset.mem_range] at hi rw [Finset.mem_Icc]; omega refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_) simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 803–809 at b16ab29fc51b ↗
  hres.hzero.hi' have i + (1 : ℕ) ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ))
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ), with hi' : i + (1 : ℕ) ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)).
  Uses library Finset.Icc, Finset.mem_range, Finset.mem_Icc.
  Lean, lines 805–807
  have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by rw [Finset.mem_range] at hi rw [Finset.mem_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 805–807 at b16ab29fc51b ↗
14. hres.hE have ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = ∑' (m : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hE : ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = ∑' (m : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹.
  Uses library Finset.Icc, tsum_congr.
  Lean, lines 811–813
  have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by rw [← hshift] exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, lines 811–813 at b16ab29fc51b ↗

The proof in Lean (Proof.lean)

Checked by the kernel — these bytes are the ones its attestation names: attestations/000335.json, merge commit b16ab29fc51b.

Proof.lean — its sha256 is the artifact_hash of the attestation below, so these are the bytes the gate checked. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Proof.lean, sha256 36efaada6df4.

import Mathlib
import Nodes.«erdos-1050--h1-v2--h2».Context

/-! Hole `integrality` of a merged partial proof, as a node (D-12 #5, D-29).
Statement generated by the gate from the assembly; immutable (D-3). -/

theorem erdos_1050__h1_v2__h2 : ∃ (a : ℕ → ℤ) (b : ℕ → ℤ),
  ∀ (n : ℕ),
    (1 : ℕ) ≤ n →
      (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) =
        (9 : ℝ) ^ n *
            (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) *
              ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) *
          ∑' (j : ℕ),
            -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ *
              ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)),
                ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) *
                  ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ := by
  -- E(n) = beta0(n) * z - alpha0(n) by partial fractions in x = 2^m (proved below, via Lagrange
  -- interpolation); b(n) = 9^n W(n) beta0(n) is an integer (proved below); a(n) = 9^n W(n) alpha0(n)
  -- is the one hole. Informal argument: annex 959e0d5b10994049b5f27ad77d9d2d8535be9f817bf4de69d747a1e05b9db44b on this node.
  -- Borwein Lemma 3 (numerator integrality): the one hole.
  have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
    intro s g
    induction s using Finset.induction_on with
    | empty => intro _; exact ⟨0, by simp⟩
    | insert a s ha ih =>
      intro h
      obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
      obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
      exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
  -- (L0) the Lagrange basis sums to 1: the degree-0 case
  have h0 : ∀ s : Finset ℕ, s.Nonempty → ∑ k ∈ s, (1 : ℝ) / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = 1 := by
    intro s hs
    have hv : Set.InjOn (fun k : ℕ => ((2 : ℝ) ^ k)⁻¹) (s : Set ℕ) := by
      intro a _ b _ hab
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (inv_injective hab)
    have hsum := congrArg (Polynomial.eval (0 : ℝ)) (Lagrange.sum_basis hv hs)
    rw [Polynomial.eval_finset_sum, Polynomial.eval_one] at hsum
    refine Eq.trans (Finset.sum_congr rfl (fun k hk => ?_)) hsum
    unfold Lagrange.basis
    rw [Polynomial.eval_prod, one_div, ← Finset.prod_inv_distrib]
    refine Finset.prod_congr rfl (fun l hl => ?_)
    have ha : (2 : ℝ) ^ k ≠ 0 := by positivity
    have hb : (2 : ℝ) ^ l ≠ 0 := by positivity
    simp only [Lagrange.basisDivisor, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_sub, Polynomial.eval_X]
    rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast,
      show ((2 : ℝ) ^ k)⁻¹ - ((2 : ℝ) ^ l)⁻¹ = (1 - (2 : ℝ) ^ l / (2 : ℝ) ^ k) * (-((2 : ℝ) ^ l)⁻¹) by field_simp; ring,
      mul_inv, zero_sub, mul_assoc, inv_mul_cancel₀ (neg_ne_zero.mpr (inv_ne_zero hb)), mul_one]
  -- the recursion h_e(s) = h_e(s minus a) + y_a h_(e-1)(s), for any weight f
  have hrec : ∀ (s : Finset ℕ) (a : ℕ) (f : ℕ → ℝ), a ∈ s → ∑ k ∈ s, (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = ∑ k ∈ s.erase a, (2 : ℝ) ^ k * f k / ∏ l ∈ (s.erase a).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) + (2 : ℝ) ^ a * ∑ k ∈ s, f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
    intro s a f ha
    rw [← Finset.add_sum_erase s (fun k => f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) ha, ← Finset.add_sum_erase s (fun k => (2 : ℝ) ^ k * f k / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) ha, mul_add, Finset.mul_sum, add_left_comm, ← Finset.sum_add_distrib]
    congr 1
    · ring
    · refine Finset.sum_congr rfl (fun k hk => ?_)
      have hka : k ≠ a := Finset.ne_of_mem_erase hk
      have hmem : a ∈ s.erase k := Finset.mem_erase.mpr ⟨Ne.symm hka, ha⟩
      have hsplit := Finset.mul_prod_erase (s.erase k) (fun l => (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) hmem
      rw [Finset.erase_right_comm] at hsplit
      rw [← hsplit]
      have hρ : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) ≠ 0 := by
        intro h1
        have h2 : (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) = 2 ^ (0 : ℤ) := by rw [zpow_zero]; linarith
        have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2
        omega
      have hq : (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ))) * (1 - (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)))⁻¹ = 1 := mul_inv_cancel₀ hρ
      have hak : (2 : ℝ) ^ a = (2 : ℝ) ^ ((a : ℤ) - (k : ℤ)) * 2 ^ k := by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      simp only [div_eq_mul_inv, mul_inv]
      rw [hak]
      linear_combination ((2 : ℝ) ^ k * f k * (∏ l ∈ (s.erase a).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hq
  -- (L) complete homogeneous sums at the nodes 2^k are integers
  have hW : ∀ N : ℕ, ∀ (s : Finset ℕ) (e : ℕ), s.card + e = N → ∃ z : ℤ, (z : ℝ) = ∑ k ∈ s, ((2 : ℝ) ^ k) ^ e / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
    intro N
    induction N with
    | zero =>
      intro s e h
      have hs : s = ∅ := Finset.card_eq_zero.mp (by omega)
      subst hs
      exact ⟨0, by simp⟩
    | succ N ih =>
      intro s e h
      rcases s.eq_empty_or_nonempty with rfl | hs
      · exact ⟨0, by simp⟩
      rcases Nat.eq_zero_or_pos e with rfl | he
      · exact ⟨1, by rw [Int.cast_one]; exact (h0 s hs).symm.trans (Finset.sum_congr rfl (fun k _ => by rw [pow_zero]))⟩
      obtain ⟨e', rfl⟩ : ∃ e', e = e' + 1 := ⟨e - 1, by omega⟩
      obtain ⟨a, ha⟩ := hs
      obtain ⟨z1, hz1⟩ := ih (s.erase a) (e' + 1) (by rw [Finset.card_erase_of_mem ha]; have := Finset.card_pos.mpr ⟨a, ha⟩; omega)
      obtain ⟨z2, hz2⟩ := ih s e' (by omega)
      refine ⟨z1 + 2 ^ a * z2, ?_⟩
      have hr := hrec s a (fun k => ((2 : ℝ) ^ k) ^ e') ha
      simp only [← pow_succ'] at hr
      push_cast
      rw [hz1, hz2]
      exact hr.symm
  -- (L-) and they vanish in negative degree above minus the number of nodes
  have hV : ∀ m : ℕ, ∀ s : Finset ℕ, m + 1 < s.card → ∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = 0 := by
    intro m
    induction m with
    | zero =>
      intro s hs
      obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
      have hne : (s.erase a).Nonempty := Finset.card_pos.mp (by rw [Finset.card_erase_of_mem ha]; omega)
      have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹) ha
      have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ = 1 := fun k => by
        rw [zero_add, pow_one]; exact mul_inv_cancel₀ (by positivity)
      simp only [e] at h
      rw [h0 s ⟨a, ha⟩, h0 _ hne] at h
      have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
      rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5
      · exact absurd h5 h2a
      · exact h5
    | succ m ih =>
      intro s hs
      obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
      have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹) ha
      have e : ∀ k : ℕ, (2 : ℝ) ^ k * (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ = (((2 : ℝ) ^ k) ^ (m + 1))⁻¹ := fun k => by
        rw [pow_succ' _ (m + 1), mul_inv, ← mul_assoc, mul_inv_cancel₀ (by positivity), one_mul]
      simp only [e] at h
      rw [ih s (by omega), ih (s.erase a) (by rw [Finset.card_erase_of_mem ha]; omega)] at h
      have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
      rcases mul_eq_zero.mp (show (2 : ℝ) ^ a * (∑ k ∈ s, (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹ / ∏ l ∈ s.erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = 0 by linarith) with h5 | h5
      · exact absurd h5 h2a
      · exact h5
  -- geometric sums in 2^(-m)
  have hgeo : ∀ m : ℕ, 1 ≤ m → ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (((2 : ℝ) ^ i)⁻¹) ^ m = (1 - (((2 : ℝ) ^ k) ^ m)⁻¹) / ((2 : ℝ) ^ m - 1) := by
    intro m hm k
    have hB : (2 : ℝ) ^ m - 1 ≠ 0 := by
      have : (2 : ℝ) ≤ 2 ^ m := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
      linarith
    induction k with
    | zero => simp
    | succ k ih =>
      rw [Finset.sum_Icc_succ_top (by omega), ih, inv_pow, pow_succ, mul_pow]
      have hX : ((2 : ℝ) ^ k) ^ m ≠ 0 := by positivity
      have hY : (2 : ℝ) ^ m ≠ 0 := by positivity
      generalize ((2 : ℝ) ^ k) ^ m = X at hX ⊢
      generalize (2 : ℝ) ^ m = Y at hY hB ⊢
      field_simp
      ring
  have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by
    intro n i hi1 hi2
    obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
    have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
    have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
    have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
    have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
    have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by
      have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
      rw [one_pow, ← pow_mul, ← hk0] at this
      exact this
    have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by
      have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
      rw [this]; exact (dvd_neg).mpr h1
    exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem)
  have hpoly : ∀ n : ℕ, 1 ≤ n → ∃ α : ℕ → ℤ, ∀ y : ℝ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ t * y) = ∑ j ∈ Finset.range n, (α j : ℝ) * y ^ j := by
    intro n hn
    refine ⟨fun j => (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X)).coeff j, fun y => ?_⟩
    have hdeg : (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))).natDegree < n := by
      refine lt_of_le_of_lt (Polynomial.natDegree_map_le) ?_
      refine lt_of_le_of_lt (Polynomial.natDegree_prod_le _ _) ?_
      refine lt_of_le_of_lt (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)) ?_
      · compute_degree
      · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
    have hev := Polynomial.eval_eq_sum_range' hdeg y
    have hev2 : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ t * y) = Polynomial.eval y (Polynomial.map (Int.castRingHom ℝ) (∏ t ∈ Finset.Icc 1 (n - 1), (1 - Polynomial.C ((2 : ℤ) ^ t) * Polynomial.X))) := by
      rw [Polynomial.map_prod, Polynomial.eval_prod]
      refine Finset.prod_congr rfl (fun t _ => ?_)
      simp
    rw [hev2, hev]
    refine Finset.sum_congr rfl (fun j _ => ?_)
    rw [Polynomial.coeff_map]
    rfl
  -- alpha0 = S1 - S2. The S2 part (the 1/(2^i - 1) terms) is `hmer` below. The S1 part (`head_integral`) is assembled
  -- from two holes by partial fractions in the parameter c = 8/3: S1 = (polynomial part) + sum_i b_i / (1 - c 2^i), where
  -- b_i = sum_(k >= i) A_k(2^-i) is the residue. Informal argument: annex
  -- a03db62f0576f792454c3f95d14fb02d193662242ec562819da63116009d326b on this node.
  have numerator_integral : ∃ a : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    have residue_integral : ∀ n i : ℕ, 1 ≤ i → i ≤ n → ∃ z : ℤ, (z : ℝ) = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      have hQne : ∀ m : ℕ, (∏ s ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ s)) ≠ 0 := by
        intro m
        refine Finset.prod_ne_zero_iff.mpr (fun s hs => ?_)
        have h1 := (Finset.mem_Icc.mp hs).1
        have h2 : (2 : ℤ) ≤ 2 ^ s := by
          calc (2 : ℤ) = 2 ^ 1 := by norm_num
            _ ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) h1
        omega
      have hG : ∀ N a b : ℕ, a + b = N → ∃ z : ℤ, (∏ s ∈ Finset.Icc 1 (a + b), ((1 : ℤ) - 2 ^ s)) = z * ((∏ s ∈ Finset.Icc 1 a, ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 b, ((1 : ℤ) - 2 ^ s))) := by
        intro N
        induction N with
        | zero =>
          intro a b h
          obtain ⟨rfl, rfl⟩ : a = 0 ∧ b = 0 := by omega
          exact ⟨1, by simp⟩
        | succ N ih =>
          intro a b h
          rcases Nat.eq_zero_or_pos a with rfl | ha
          · exact ⟨1, by simp⟩
          rcases Nat.eq_zero_or_pos b with rfl | hb
          · exact ⟨1, by simp⟩
          obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩
          obtain ⟨b', rfl⟩ : ∃ b', b = b' + 1 := ⟨b - 1, by omega⟩
          obtain ⟨z1, h1⟩ := ih a' (b' + 1) (by omega)
          obtain ⟨z2, h2⟩ := ih (a' + 1) b' (by omega)
          refine ⟨2 ^ (b' + 1) * z1 + z2, ?_⟩
          have e1 : a' + 1 + (b' + 1) = (a' + (b' + 1)) + 1 := by omega
          have e2 : a' + 1 + b' = a' + (b' + 1) := by omega
          have hA := Finset.prod_Icc_succ_top (show 1 ≤ a' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          have hB := Finset.prod_Icc_succ_top (show 1 ≤ b' + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          have hC := Finset.prod_Icc_succ_top (show 1 ≤ a' + (b' + 1) + 1 by omega) (fun s => (1 : ℤ) - 2 ^ s)
          beta_reduce at hA hB hC
          rw [e2, hA] at h2
          rw [hB] at h1
          rw [e1, hC, hA, hB]
          linear_combination (2 ^ (b' + 1) * (1 - 2 ^ (a' + 1))) * h1 + (1 - 2 ^ (b' + 1)) * h2
      have hN : ∀ m d : ℕ, (∏ s ∈ Finset.Icc 1 d, ((1 : ℤ) - 2 ^ s)) * (∏ t ∈ Finset.Icc 1 m, ((1 : ℤ) - 2 ^ (t + d))) = ∏ s ∈ Finset.Icc 1 (d + m), ((1 : ℤ) - 2 ^ s) := by
        intro m d
        induction m with
        | zero => simp
        | succ m ih =>
          rw [Finset.prod_Icc_succ_top (by omega), show d + (m + 1) = (d + m) + 1 by omega, Finset.prod_Icc_succ_top (by omega), ← mul_assoc, ih,
            show m + 1 + d = d + m + 1 by omega]
      have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      intro n i hi hin
      refine hsumZ _ _ (fun k hk => ?_)
      have hk' := Finset.mem_Icc.mp hk
      obtain ⟨d, rfl⟩ : ∃ d, k = i + d := ⟨k - i, by omega⟩
      -- numerator
      have hNr : ∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + (i + d))) = ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) := by
        push_cast
        refine Finset.prod_congr rfl (fun t _ => ?_)
        have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
        rw [show t + (i + d) = i + (t + d) by omega, pow_add, ← mul_assoc, inv_mul_cancel₀ h2i, one_mul]
      -- denominator
      have hset : (Finset.Icc 1 n).erase (i + d) = Finset.Icc 1 (i + d - 1) ∪ Finset.Icc (i + d + 1) n := by
        ext l
        simp only [Finset.mem_erase, Finset.mem_Icc, Finset.mem_union]
        omega
      have hdisj : Disjoint (Finset.Icc 1 (i + d - 1)) (Finset.Icc (i + d + 1) n) := by
        rw [Finset.disjoint_left]
        intro l h1 h2
        simp only [Finset.mem_Icc] at h1 h2
        omega
      have hD1 : (∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        have hrefl : ∏ l ∈ Finset.Icc 1 (i + d - 1), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ∏ s ∈ Finset.Icc 1 (i + d - 1), (1 - ((2 : ℝ) ^ s)⁻¹) := by
          refine Finset.prod_nbij' (fun l => i + d - l) (fun s => i + d - s) ?_ ?_ ?_ ?_ ?_
          · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl; omega
          · intro l hl; simp only [Finset.mem_Icc] at hl; omega
          · intro l hl
            simp only [Finset.mem_Icc] at hl
            rw [← zpow_natCast, ← zpow_neg]
            congr 2
            push_cast [Nat.cast_sub (show l ≤ i + d by omega)]
            ring
        rw [hrefl]
        push_cast
        rw [← Finset.prod_mul_distrib]
        refine Finset.prod_congr rfl (fun s _ => ?_)
        have h2s : (2 : ℝ) ^ s ≠ 0 := by positivity
        field_simp
        ring
      have hD2 : ∏ l ∈ Finset.Icc (i + d + 1) n, (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ))) = ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        push_cast
        refine Finset.prod_nbij' (fun l => l - (i + d)) (fun s => s + (i + d)) ?_ ?_ ?_ ?_ ?_
        · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl ⊢; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl; omega
        · intro l hl; simp only [Finset.mem_Icc] at hl; omega
        · intro l hl
          simp only [Finset.mem_Icc] at hl
          rw [← zpow_natCast]
          congr 2
          push_cast [Nat.cast_sub (show i + d ≤ l by omega)]
          ring
      obtain ⟨z1, hz1⟩ := hG _ d (n - 1) rfl
      obtain ⟨z2, hz2⟩ := hG _ (i + d - 1) (n - (i + d)) rfl
      rw [show i + d - 1 + (n - (i + d)) = n - 1 by omega] at hz2
      have hNZ : (∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d))) = z1 * z2 * ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s)) * (∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s))) := by
        apply mul_left_cancel₀ (hQne d)
        rw [hN (n - 1) d, hz1, hz2]
        ring
      have hNZ' : ((∏ t ∈ Finset.Icc 1 (n - 1), ((1 : ℤ) - 2 ^ (t + d)) : ℤ) : ℝ) = (z1 : ℝ) * z2 * (((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ)) := by
        exact_mod_cast hNZ
      have hQ1 : ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
      have hQ2 : ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) ≠ 0 := by exact_mod_cast hQne _
      have hDr : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) * ((∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s) : ℤ) : ℝ) = ((∏ s ∈ Finset.Icc 1 (i + d - 1), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) * ((∏ s ∈ Finset.Icc 1 (n - (i + d)), ((1 : ℤ) - 2 ^ s) : ℤ) : ℝ) := by
        rw [hset, Finset.prod_union hdisj, hD2, ← hD1]
        ring
      have hDne : (∏ l ∈ (Finset.Icc 1 n).erase (i + d), (1 - (2 : ℝ) ^ ((l : ℤ) - ((i + d : ℕ) : ℤ)))) ≠ 0 := by
        intro h0
        rw [h0, zero_mul] at hDr
        exact mul_ne_zero hQ1 hQ2 hDr.symm
      refine ⟨-(z1 * z2 * (∏ s ∈ Finset.Icc 1 (i + d - 1), (-(2 : ℤ) ^ s))), ?_⟩
      rw [hNr, eq_div_iff hDne, hNZ']
      rw [Int.cast_neg, Int.cast_mul, Int.cast_mul]
      linear_combination (-((z1 : ℝ) * z2)) * hDr
    have polypart_integral : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k)) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
      intro n hn
      obtain ⟨α, hα⟩ := hpoly n hn
      obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ k, D k = ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := ⟨fun k => _, fun k => rfl⟩
      have hA : ∀ (x : ℝ) (k : ℕ), ∏ t ∈ Finset.Icc 1 (n - 1), (1 - x * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * (x * 2 ^ k) ^ j := by
        intro x k
        rw [← hα]
        refine Finset.prod_congr rfl (fun t _ => ?_)
        rw [pow_add]; ring
      simp only [← hD, hA]
      -- (a) one double sum
      have hswap : ∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
        simp only [Finset.sum_mul]
        refine Finset.sum_comm' (fun i k => ?_)
        simp only [Finset.mem_Icc]
        omega
      rw [hswap]
      simp only [Finset.mul_sum]
      rw [← Finset.sum_sub_distrib]
      simp only [← Finset.sum_sub_distrib, ← sub_mul]
      -- (b) per (k, i)
      have hb : ∀ (k i : ℕ), 1 ≤ i → (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = (∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * ((2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s))) / D k := by
        intro k i hi
        have h2i : (2 : ℝ) ^ i ≠ 0 := by positivity
        have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by
          have : (2 : ℝ) ≤ 2 ^ i := by
            calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
              _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
          intro h0
          linarith
        have hch : (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ := by
          rw [show (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) = ((2 : ℝ) ^ i)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) by field_simp, mul_assoc, mul_inv_cancel₀ hne, mul_one]
        have hj : ∀ j : ℕ, ((((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - s) := by
          intro j
          have hg := geom_sum₂_mul (8 / 3 : ℝ) (((2 : ℝ) ^ i)⁻¹) j
          have hg' : (((2 : ℝ) ^ i)⁻¹) ^ j - (8 / 3 : ℝ) ^ j = (∑ s ∈ Finset.range j, (8 / 3 : ℝ) ^ s * (((2 : ℝ) ^ i)⁻¹) ^ (j - 1 - s)) * (((2 : ℝ) ^ i)⁻¹ - (8 / 3 : ℝ)) := by
            linear_combination hg
          rw [hg', mul_assoc, hch, Finset.sum_mul]
          refine Finset.sum_congr rfl (fun s hs => ?_)
          have hs' := Finset.mem_range.mp hs
          rw [mul_assoc, ← pow_succ, show j - 1 - s + 1 = j - s by omega]
        have e1 : (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ((∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j - ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) / D k := by
          ring
        rw [e1, ← Finset.sum_sub_distrib, Finset.sum_mul]
        congr 1
        refine Finset.sum_congr rfl (fun j _ => ?_)
        rw [← Finset.mul_sum, ← hj j, mul_pow, mul_pow]
        ring
      -- (c) sum over i
      have hc : ∀ k : ℕ, ∑ i ∈ Finset.Icc 1 k, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / D k - -(∑ j ∈ Finset.range n, (α j : ℝ) * (((2 : ℝ) ^ i)⁻¹ * 2 ^ k) ^ j) / D k) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ = ∑ j ∈ Finset.range n, ∑ s ∈ Finset.range j, (α j : ℝ) * (8 / 3 : ℝ) ^ s * ((((2 : ℝ) ^ k) ^ j - ((2 : ℝ) ^ k) ^ s) / D k) / ((2 : ℝ) ^ (j - s) - 1) := by
        intro k
        rw [Finset.sum_congr rfl (fun i hi => hb k i (Finset.mem_Icc.mp hi).1), ← Finset.sum_div, Finset.sum_comm, Finset.sum_div]
        refine Finset.sum_congr rfl (fun j _ => ?_)
        rw [Finset.sum_comm, Finset.sum_div]
        refine Finset.sum_congr rfl (fun s hs => ?_)
        have hs' := Finset.mem_range.mp hs
        rw [Finset.sum_congr rfl (fun i _ => (mul_assoc _ _ _).symm), ← Finset.mul_sum, hgeo (j - s) (by omega) k]
        have hjs : ((2 : ℝ) ^ k) ^ j = ((2 : ℝ) ^ k) ^ s * ((2 : ℝ) ^ k) ^ (j - s) := by
          rw [← pow_add]; congr 1; omega
        have hZ : ((2 : ℝ) ^ k) ^ (j - s) * (((2 : ℝ) ^ k) ^ (j - s))⁻¹ = 1 := mul_inv_cancel₀ (by positivity)
        rw [hjs]
        linear_combination (-((α j : ℝ) * ((2 : ℝ) ^ k) ^ s * (8 / 3 : ℝ) ^ s / ((2 : ℝ) ^ (j - s) - 1) / D k)) * hZ
      rw [Finset.sum_congr rfl (fun k _ => hc k), Finset.sum_comm, Finset.mul_sum]
      refine hsumZ _ _ (fun j hj => ?_)
      rw [Finset.sum_comm, Finset.mul_sum]
      refine hsumZ _ _ (fun s hs => ?_)
      have hj' := Finset.mem_range.mp hj
      have hs' := Finset.mem_range.mp hs
      obtain ⟨wj, hwj⟩ := hW _ (Finset.Icc 1 n) j rfl
      obtain ⟨ws, hws⟩ := hW _ (Finset.Icc 1 n) s rfl
      obtain ⟨q, hq⟩ := hdvd n (j - s) (by omega) (by omega)
      have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ (j - s) - 1) * q := by exact_mod_cast hq
      have hB : (2 : ℝ) ^ (j - s) - 1 ≠ 0 := by
        have : (2 : ℝ) ≤ 2 ^ (j - s) := by
          calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
            _ ≤ 2 ^ (j - s) := pow_le_pow_right₀ (by norm_num) (by omega)
        linarith
      have h3 : (3 : ℝ) ^ n = 3 ^ (n - s) * 3 ^ s := by rw [← pow_add]; congr 1; omega
      have h83 : (8 / 3 : ℝ) ^ s * 3 ^ s = 8 ^ s := by rw [← mul_pow]; norm_num
      refine ⟨3 ^ (n - s) * 8 ^ s * q * α j * (wj - ws), ?_⟩
      simp only [← hD] at hwj hws
      push_cast
      rw [hwj, hws, hqR, h3, ← h83, ← Finset.sum_sub_distrib]
      rw [Finset.mul_sum, Finset.mul_sum]
      refine Finset.sum_congr rfl (fun k _ => ?_)
      field_simp
    have head_integral : ∃ a₁ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₁ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
      have hsumZ0 : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) := by
        intro n hn
        obtain ⟨zE, hzE⟩ := polypart_integral n hn
        have hR : ∃ zR : ℤ, (zR : ℝ) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∑ i ∈ Finset.Icc 1 n, (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹) := by
          rw [Finset.mul_sum]
          refine hsumZ0 _ _ (fun i hi => ?_)
          have hi' := Finset.mem_Icc.mp hi
          obtain ⟨zb, hzb⟩ := residue_integral n i hi'.1 hi'.2
          refine ⟨zb * (3 * ∏ j ∈ (Finset.Icc 1 n).erase i, ((3 : ℤ) - 8 * 2 ^ j)), ?_⟩
          have hne : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) ≠ 0 := by
            have : (2 : ℝ) ≤ 2 ^ i := by
              calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
                _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
            intro h0
            linarith
          have hsplit : (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) = (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i) * ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) :=
            (Finset.mul_prod_erase _ (fun k => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) hi).symm
          have he : ∏ k ∈ (Finset.Icc 1 n).erase i, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ (Finset.Icc 1 n).erase i, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ (n - 1))⁻¹ := by
            rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
              Finset.prod_mul_distrib, Finset.prod_const, Finset.card_erase_of_mem hi, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
          have h3 : (3 : ℝ) ^ n = 3 * 3 ^ (n - 1) := by
            rw [← pow_succ']; congr 1; omega
          have h3ne : (3 : ℝ) ^ (n - 1) ≠ 0 := by positivity
          obtain ⟨b, hb⟩ : ∃ b : ℝ, b = (∑ k ∈ Finset.Icc i n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - ((2 : ℝ) ^ i)⁻¹ * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := ⟨_, rfl⟩
          rw [← hb] at hzb ⊢
          push_cast
          rw [hzb, hsplit, he, h3]
          have hne' : (3 : ℝ) - 8 * 2 ^ i ≠ 0 := by
            intro h0
            apply hne
            linear_combination (1 / 3 : ℝ) * h0
          field_simp
        obtain ⟨zR, hzR⟩ := hR
        have hPZ : ∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k) = (3 : ℝ) ^ n * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) := by
          rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
            Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
          have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
          field_simp
        have h9 : (9 : ℝ) ^ n = 3 ^ n * 3 ^ n := by
          rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow]
        refine ⟨(Nat.factorial (n - 2) : ℤ) * ((∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * zE + 3 ^ n * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * zR), ?_⟩
        push_cast
        rw [hzE, hzR, hPZ, h9]
        ring
      refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩
      simp only [dif_pos hn']
      exact Classical.choose_spec (hn n hn')
    obtain ⟨a₁, h₁⟩ := head_integral
    have hmer : ∃ a₂ : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a₂ n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
      have key : ∀ n : ℕ, 1 ≤ n → ∀ i : ℕ, 1 ≤ i → i ≤ n - 1 → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1 - i) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) := by
        intro n hn i hi1 hi2
        obtain ⟨α, hα⟩ := hpoly n hn
        have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by
          intro k
          rw [← hα]
          refine Finset.prod_congr rfl (fun t _ => ?_)
          rw [pow_add]; ring
        simp only [hA]
        have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by
          rw [Finset.sum_comm]
          refine Finset.sum_congr rfl (fun k _ => ?_)
          rw [neg_div, neg_mul, Finset.sum_div, Finset.sum_mul, ← Finset.sum_neg_distrib]
          refine Finset.sum_congr rfl (fun j _ => ?_)
          ring
        rw [e1, Finset.mul_sum]
        refine hsumZ _ _ (fun j hj => ?_)
        have hj' := Finset.mem_range.mp hj
        rw [← Finset.mul_sum]
        have hcX : ∀ k : ℕ, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ≠ 0 := fun k => by positivity
        rcases lt_or_ge j i with hji | hji
        · refine ⟨0, ?_⟩
          have hvan : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * ∑ k ∈ Finset.Icc 1 n, (((2 : ℝ) ^ k) ^ (i - j - 1 + 1))⁻¹ / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
            rw [Finset.mul_sum]
            refine Finset.sum_congr rfl (fun k _ => ?_)
            have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = ((8 / 3 : ℝ) ^ (i - j))⁻¹ * (((2 : ℝ) ^ k) ^ (i - j))⁻¹ := by
              rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (i - j) from by rw [← pow_add]; congr 1; omega,
                mul_inv, ← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), one_mul, mul_pow, mul_inv]
            rw [hk, mul_div_assoc, show i - j - 1 + 1 = i - j by omega]
          rw [hvan, hV (i - j - 1) (Finset.Icc 1 n) (by rw [Nat.card_Icc]; omega)]
          simp
        · obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) (j - i) rfl
          refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ (j - i) * w, ?_⟩
          have hpos : ∑ k ∈ Finset.Icc 1 n, (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = (8 / 3 : ℝ) ^ (j - i) * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ (j - i) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
            rw [Finset.mul_sum]
            refine Finset.sum_congr rfl (fun k _ => ?_)
            have hk : ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j * ((((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) = (8 / 3 : ℝ) ^ (j - i) * ((2 : ℝ) ^ k) ^ (j - i) := by
              rw [show ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j = ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (j - i) * ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i from by rw [← pow_add]; congr 1; omega,
                mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ (hcX k)), mul_one, mul_pow]
            rw [hk, mul_div_assoc]
          have h3 : (3 : ℝ) ^ (n - 1 - i) = 3 ^ (n - 1 - j) * 3 ^ (j - i) := by rw [← pow_add]; congr 1; omega
          have h83 : (8 / 3 : ℝ) ^ (j - i) * 3 ^ (j - i) = 8 ^ (j - i) := by rw [← mul_pow]; norm_num
          push_cast
          rw [hpos, hw, h3, ← h83]
          ring
      have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
        intro n
        rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
          Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
      have hdvd : ∀ n i : ℕ, 1 ≤ i → i ≤ n - 1 → ((2 : ℤ) ^ i - 1) ∣ ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k) := by
        intro n i hi1 hi2
        obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
        have hle : k0 ≤ n := by rw [hk0, mul_comm]; exact Nat.div_mul_le_self n i
        have hlt : n < k0 + i := by rw [hk0]; have := Nat.lt_mul_div_succ n (by omega : 0 < i); linarith
        have hge : i ≤ k0 := by rw [hk0]; exact Nat.le_mul_of_pos_right i (Nat.div_pos (by omega) (by omega))
        have hmem : k0 ∈ Finset.Icc ((n + 1) / 2) n := Finset.mem_Icc.mpr ⟨by omega, hle⟩
        have h1 : ((2 : ℤ) ^ i - 1) ∣ (2 : ℤ) ^ k0 - 1 := by
          have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
          rw [one_pow, ← pow_mul, ← hk0] at this
          exact this
        have h2 : ((2 : ℤ) ^ i - 1) ∣ (1 : ℤ) - 2 ^ k0 := by
          have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
          rw [this]; exact (dvd_neg).mpr h1
        exact h2.trans (Finset.dvd_prod_of_mem (fun k => (1 : ℤ) - 2 ^ k) hmem)
      have hsumZ : ∀ (s : Finset ℕ) (g : ℕ → ℝ), (∀ i ∈ s, ∃ z : ℤ, (z : ℝ) = g i) → ∃ z : ℤ, (z : ℝ) = ∑ i ∈ s, g i := by
        intro s g
        induction s using Finset.induction_on with
        | empty => intro _; exact ⟨0, by simp⟩
        | insert a s ha ih =>
          intro h
          obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
          obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
          exact ⟨za + zs, by rw [Finset.sum_insert ha]; push_cast; rw [hza, hzs]⟩
      have hn : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
        intro n hn
        rw [Finset.mul_sum]
        refine hsumZ _ _ (fun i hi => ?_)
        have hi' := Finset.mem_Icc.mp hi
        obtain ⟨z1, hz1⟩ := key n hn i hi'.1 hi'.2
        obtain ⟨q, hq⟩ := hdvd n i hi'.1 hi'.2
        have hqR : ∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k) = ((2 : ℝ) ^ i - 1) * q := by exact_mod_cast hq
        have h2i : (2 : ℝ) ^ i - 1 ≠ 0 := by
          have : (2 : ℝ) ≤ 2 ^ i := by
            calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
              _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi'.1
          linarith
        have h9 : (9 : ℝ) ^ n = 3 ^ (i + 1) * 3 ^ (n - 1 - i) * 3 ^ n := by
          rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow]
          congr 1
          rw [← pow_add]
          congr 1
          omega
        refine ⟨3 ^ (i + 1) * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) * q * z1, ?_⟩
        push_cast
        rw [hz1, hW3 n, hqR, h9]
        have h3n : (3 : ℝ) ^ n ≠ 0 := by positivity
        field_simp
      refine ⟨fun n => if h : 1 ≤ n then Classical.choose (hn n h) else 0, fun n hn' => ?_⟩
      simp only [dif_pos hn']
      exact Classical.choose_spec (hn n hn')
    obtain ⟨a₂, h₂⟩ := hmer
    refine ⟨fun n => a₁ n - a₂ n, fun n hn => ?_⟩
    push_cast
    rw [h₁ n hn, h₂ n hn]
    ring
  obtain ⟨a, ha⟩ := numerator_integral
  -- Denominator integrality (proved): 3^(n-1) beta0(n) is a coefficient of an integer polynomial.
  have hden : ∃ b : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (b n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
    have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      intro n hn
      obtain ⟨α, hα⟩ := hpoly n hn
      have hA : ∀ k : ℕ, ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) = ∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j := by
        intro k
        rw [← hα]
        refine Finset.prod_congr rfl (fun t _ => ?_)
        rw [pow_add]; ring
      simp only [hA]
      have e1 : ∑ k ∈ Finset.Icc 1 n, (-(∑ j ∈ Finset.range n, (α j : ℝ) * ((8 / 3 : ℝ) * 2 ^ k) ^ j) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) = ∑ j ∈ Finset.range n, ∑ k ∈ Finset.Icc 1 n, (-(α j : ℝ)) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ)))) := by
        rw [Finset.sum_comm]
        refine Finset.sum_congr rfl (fun k _ => ?_)
        rw [neg_div, Finset.sum_div, ← Finset.sum_neg_distrib]
        refine Finset.sum_congr rfl (fun j _ => ?_)
        ring
      rw [e1, Finset.mul_sum]
      refine hsumZ _ _ (fun j hj => ?_)
      have hj' := Finset.mem_range.mp hj
      rw [← Finset.mul_sum]
      obtain ⟨w, hw⟩ := hW _ (Finset.Icc 1 n) j rfl
      refine ⟨-(α j) * 3 ^ (n - 1 - j) * 8 ^ j * w, ?_⟩
      have hpos : ∑ k ∈ Finset.Icc 1 n, ((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) = (8 / 3 : ℝ) ^ j * ∑ k ∈ Finset.Icc 1 n, ((2 : ℝ) ^ k) ^ j / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k _ => ?_)
        rw [mul_pow, mul_div_assoc]
      have h3 : (3 : ℝ) ^ (n - 1) = 3 ^ (n - 1 - j) * 3 ^ j := by rw [← pow_add]; congr 1; omega
      have h83 : (8 / 3 : ℝ) ^ j * 3 ^ j = 8 ^ j := by rw [← mul_pow]; norm_num
      push_cast
      rw [hpos, hw, h3, ← h83]
      ring
    have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
      intro n
      rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
        Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
    refine ⟨fun n => if h : 1 ≤ n then 3 * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * Classical.choose (key n h) else 0, fun n hn => ?_⟩
    simp only [dif_pos hn]
    push_cast
    rw [Classical.choose_spec (key n hn), hW3 n]
    have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by
      rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
    have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
    rw [h9]
    linear_combination (-(3 : ℝ) * (Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k)) * ((3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))))) * h3
  obtain ⟨b, hb⟩ := hden
  -- The partial-fraction identity at x = 2^m (proved).
  have hpf : ∀ n : ℕ, 1 ≤ n → ∀ m : ℕ, 1 ≤ m → (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := by
    intro n hn m hm
    -- General partial fractions, from Mathlib's Lagrange interpolation.
    have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s →
      Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) →
      Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ =
        ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
      intro s p Q hp hinj hdeg x hx
      have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by
        intro a ha b hb hab
        exact hinj ha hb (inv_injective hab)
      have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by
        intro k hk hxk
        apply hx k hk
        rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
      have hQ := Lagrange.eq_interpolate hinj' hdeg
      have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
      rw [← hQ, Lagrange.eval_nodal] at hev
      have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by
        intro k hk
        have h1 := hp k hk
        have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by
          rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
        rw [← h3, mul_inv, inv_neg]
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev]
      have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
      rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc,
        mul_inv_cancel₀ hne, one_mul, Finset.sum_mul]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk]
      have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by
        intro j hj
        have hjs := Finset.mem_of_mem_erase hj
        have hjk := Finset.ne_of_mem_erase hj
        have hpj := hp j hjs
        have hpk := hp k hk
        have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
        have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
        field_simp
        ring
      calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ *
            (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹)
          = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by
            rw [Finset.prod_mul_distrib]; ring
        _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
            rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
    have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by
      calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
        _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
    have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
    have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
    have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by
      intro k
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      nlinarith
    have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by
      intro a _ b _ hab
      have h : (2 : ℝ) ^ a = 2 ^ b := by
        have := hab; simp only at this
        exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
    have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by
      rw [Polynomial.degree_neg, Polynomial.degree_prod]
      simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc]
      rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one]
      exact_mod_cast (by omega : n - 1 < n)
    have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t)))
      (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
    simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C] at hL
    have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
    -- geometric sums
    have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by
      intro u hu hu1 N
      induction N with
      | zero => simp
      | succ N ih =>
        rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih]
        have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
        field_simp
        ring
    -- the left side, in terms of the Lagrange data
    have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by
      rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
    have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
    have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
    have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by
      have hn' : n = n - 1 + 1 := by omega
      conv_lhs => rw [hn']
      rw [Finset.prod_Icc_succ_top (by omega), ← hn']
    have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by
      rw [mul_sub, inv_mul_cancel₀ hx0]; ring
    have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hz1, hz2, hz3]
      rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib,
        Finset.prod_const, hcard, inv_pow]
      ring
    -- the right side
    have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
    have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by
      simp only [Finset.sum_mul, Finset.mul_sum]
      rw [Finset.sum_comm]
      refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_))
      rw [mul_pow, mul_inv]; ring
    have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hpk]
      rw [hswap, ← Finset.sum_add_distrib]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)]
      ring
    -- matching the coefficients
    have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by
      intro k hk
      have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by
        rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring,
          inv_mul_cancel₀ (hp0 k), mul_one]
      have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow]
      simp only [hw]
      ring
    calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
      _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
      _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [← hcoef k hk, mul_pow, mul_inv]
        ring
      _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
  -- Borwein Lemma 1 (proved): sum over m >= 1; the first n - 1 terms vanish.
  have hres : ∀ n : ℕ, 1 ≤ n → (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    intro n hn
    have pf := hpf n hn
    -- Summability of the two families of pieces.
    have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by
      intro k
      refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_)
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
      have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
      rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div]
      exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
    have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by
      intro i hi
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
      have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
      have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by
        intro m
        rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
      rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1]
      have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
      have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
      field_simp
    have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by
      intro i hi
      by_contra hns
      have := tsum_eq_zero_of_not_summable hns
      rw [hGeo i hi] at this
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
      contradiction
    have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by
      simpa using hT 0
    have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      induction k with
      | zero => simp
      | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
    have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      have hs := hz0.sum_add_tsum_nat_add k
      rw [hIcc k] at hs
      rw [← hs, add_sub_cancel_left]
      exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
    -- The series E(n) summed from m = 1: its first n - 1 terms vanish.
    have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) :=
      summable_sum (fun k _ => (hT k).mul_left _)
    have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) :=
      summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
    have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
    have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) :=
      hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
    have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
    have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by
      refine Finset.sum_eq_zero (fun i hi => ?_)
      have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by
        rw [Finset.mem_range] at hi
        rw [Finset.mem_Icc]; omega
      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_)
      simp
    rw [hzero, zero_add] at hshift
    have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by
      rw [← hshift]
      exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
    rw [hE, tsum_congr (fun m => pf (m + 1) (by omega)), hRs1.tsum_add hRs2,
      Summable.tsum_finsetSum (fun k _ => (hT k).mul_left _),
      Summable.tsum_finsetSum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _),
      Finset.sum_congr (s₁ := Finset.Icc 1 n) rfl (fun k _ => by rw [tsum_mul_left, hTail k]),
      Finset.sum_congr (s₁ := Finset.Icc 1 (n - 1)) rfl (fun i hi => by rw [tsum_mul_left, hGeo i (Finset.mem_Icc.mp hi).1])]
    simp only [mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv]
    ring
  refine ⟨a, b, fun n hn => ?_⟩
  linear_combination (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) * hb n hn - ha n hn - (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * hres n hn

How it was split

Split into

Proved from this hole by 20260929T194647Z-t0929-1-partial.lean (merged in b2e922a60e62, attestation 000290).

numerator_integral → erdos-1050--h1-v2--h2--h1 (proved)
$\displaystyle \exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$

Read the partial proof (Lean)

theorem erdos_1050__h1_v2__h2 : ∃ (a : ℕ → ℤ) (b : ℕ → ℤ),
  ∀ (n : ℕ),
    (1 : ℕ) ≤ n →
      (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) =
        (9 : ℝ) ^ n *
            (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) *
              ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) *
          ∑' (j : ℕ),
            -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ *
              ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)),
                ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) *
                  ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ := by
  -- E(n) = beta0(n) * z - alpha0(n) by partial fractions in x = 2^m (proved below, via Lagrange
  -- interpolation); b(n) = 9^n W(n) beta0(n) is an integer (proved below); a(n) = 9^n W(n) alpha0(n)
  -- is the one hole. Informal argument: annex 959e0d5b10994049b5f27ad77d9d2d8535be9f817bf4de69d747a1e05b9db44b on this node.
  -- Borwein Lemma 3 (numerator integrality): the one hole.
  have numerator_integral : ∃ a : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := sorry
  obtain ⟨a, ha⟩ := numerator_integral
  -- Denominator integrality (proved): 3^(n-1) beta0(n) is a coefficient of an integer polynomial.
  have hden : ∃ b : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (b n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
    have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      intro n hn
      have hr0 : ∀ k : ℕ, ((8 : ℝ) * 2 ^ k) ≠ 0 := fun k => by positivity
      have hvinj : Set.InjOn (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹) (Finset.Icc 1 n : Set ℕ) := by
        intro a _ b _ hab
        have h : (2 : ℝ) ^ a = 2 ^ b := by
          have := inv_injective hab
          exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this
        exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
      -- (1) the polynomial partial-fraction identity, by agreement at the n nodes 1/r_k
      have hid : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) = ∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) := by
        refine Polynomial.eq_of_natDegree_lt_card_of_eval_eq' _ _ ((Finset.Icc 1 n).image (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹)) ?_ ?_
        · intro y hy
          obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
          rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk]
          · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod]
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by
              refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_)
              have hjk : j ≠ k := Finset.ne_of_mem_erase hj
              intro h0
              have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by
                have := hr0 k
                field_simp at h0
                linarith
              exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this))
            rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one]
          · intro i hi hik
            rw [Polynomial.eval_mul, Polynomial.eval_prod]
            refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_)
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            rw [mul_inv_cancel₀ (hr0 k), sub_self]
        · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc]
          have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by
            rw [Polynomial.natDegree_neg]
            refine (Polynomial.natDegree_prod_le _ _).trans ?_
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
          have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by
            refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_)
            refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_)
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one]
              rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega
          omega
      -- (2) coefficient n - 1 of T_k * P is r_k ^ (n - 1)
      have hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)).coeff d = if d = 0 then 1 else 0 := by
        intro S
        induction S using Finset.induction_on with
        | empty => intro d hd; simp [Polynomial.coeff_one]
        | insert a S ha ih =>
          intro d hd
          rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc,
            Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd]
      have hG : ∀ k : ℕ, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k
        simp only [mul_pow, ← Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow]
        rw [Finset.sum_ite_eq (Finset.range n) (n - 1), if_pos (Finset.mem_range.mpr (by omega))]
      have hTP : ∀ k ∈ Finset.Icc 1 n, ((∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k hk
        have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) :=
          (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm
        have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by
          rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib]
          congr 1
          refine Finset.prod_congr rfl (fun j _ => ?_)
          rw [mul_neg_geom_sum]
        obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 :=
          Polynomial.X_pow_dvd_iff.mpr (fun d hd => by
          rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all)
        rw [hprod, show ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) = 1 + Polynomial.X ^ n * E by rw [← hE]; ring,
          mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg (by omega), add_zero, hG]
      -- (3) so coefficient n - 1 of Qt * P is the sum of C_k r_k^(n-1)
      have hQP : ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ∑ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        nth_rw 1 [hid]
        rw [Finset.sum_mul, Polynomial.finsetSum_coeff]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [mul_assoc, Polynomial.coeff_C_mul, hTP k hk]
      -- (4) and it is an integer, since Qt * P has integer coefficients
      have hmap : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)) := by
        simp [Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat]
      -- (5) C_k r_k^(n-1) = 3^(n-1) A(n,k)
      have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
      have hterm : ∀ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = (3 : ℝ) ^ (n - 1) * (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) := by
        intro k hk
        have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by
          have := hr0 k
          rw [pow_add]; field_simp
        have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
          rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
          field_simp
        have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by
          rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow]
        simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C,
          Polynomial.eval_X]
        rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
          Finset.prod_const, hcard]
        simp only [hw]
        linear_combination (-((3 : ℝ) ^ (n - 1) * ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) *
          (∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hrr
      refine ⟨((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1), ?_⟩
      have hc : ((((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1) : ℤ) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) := by
        rw [Polynomial.coeff_map]; simp
      rw [hc, ← hmap, hQP, Finset.mul_sum]
      exact Finset.sum_congr rfl hterm
    have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
      intro n
      rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
        Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
    refine ⟨fun n => if h : 1 ≤ n then 3 * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * Classical.choose (key n h) else 0, fun n hn => ?_⟩
    simp only [dif_pos hn]
    push_cast
    rw [Classical.choose_spec (key n hn), hW3 n]
    have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by
      rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
    have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
    rw [h9]
    linear_combination (-(3 : ℝ) * (Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k)) * ((3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))))) * h3
  obtain ⟨b, hb⟩ := hden
  -- The partial-fraction identity at x = 2^m (proved).
  have hpf : ∀ n : ℕ, 1 ≤ n → ∀ m : ℕ, 1 ≤ m → (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := by
    intro n hn m hm
    -- General partial fractions, from Mathlib's Lagrange interpolation.
    have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s →
      Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) →
      Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ =
        ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
      intro s p Q hp hinj hdeg x hx
      have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by
        intro a ha b hb hab
        exact hinj ha hb (inv_injective hab)
      have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by
        intro k hk hxk
        apply hx k hk
        rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
      have hQ := Lagrange.eq_interpolate hinj' hdeg
      have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
      rw [← hQ, Lagrange.eval_nodal] at hev
      have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by
        intro k hk
        have h1 := hp k hk
        have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by
          rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
        rw [← h3, mul_inv, inv_neg]
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev]
      have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
      rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc,
        mul_inv_cancel₀ hne, one_mul, Finset.sum_mul]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk]
      have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by
        intro j hj
        have hjs := Finset.mem_of_mem_erase hj
        have hjk := Finset.ne_of_mem_erase hj
        have hpj := hp j hjs
        have hpk := hp k hk
        have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
        have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
        field_simp
        ring
      calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ *
            (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹)
          = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by
            rw [Finset.prod_mul_distrib]; ring
        _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
            rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
    have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by
      calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
        _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
    have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
    have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
    have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by
      intro k
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      nlinarith
    have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by
      intro a _ b _ hab
      have h : (2 : ℝ) ^ a = 2 ^ b := by
        have := hab; simp only at this
        exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
    have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by
      rw [Polynomial.degree_neg, Polynomial.degree_prod]
      simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc]
      rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one]
      exact_mod_cast (by omega : n - 1 < n)
    have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t)))
      (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
    simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C] at hL
    have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
    -- geometric sums
    have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by
      intro u hu hu1 N
      induction N with
      | zero => simp
      | succ N ih =>
        rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih]
        have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
        field_simp
        ring
    -- the left side, in terms of the Lagrange data
    have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by
      rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
    have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
    have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
    have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by
      have hn' : n = n - 1 + 1 := by omega
      conv_lhs => rw [hn']
      rw [Finset.prod_Icc_succ_top (by omega), ← hn']
    have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by
      rw [mul_sub, inv_mul_cancel₀ hx0]; ring
    have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hz1, hz2, hz3]
      rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib,
        Finset.prod_const, hcard, inv_pow]
      ring
    -- the right side
    have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
    have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by
      simp only [Finset.sum_mul, Finset.mul_sum]
      rw [Finset.sum_comm]
      refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_))
      rw [mul_pow, mul_inv]; ring
    have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hpk]
      rw [hswap, ← Finset.sum_add_distrib]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)]
      ring
    -- matching the coefficients
    have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by
      intro k hk
      have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by
        rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring,
          inv_mul_cancel₀ (hp0 k), mul_one]
      have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow]
      simp only [hw]
      ring
    calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
      _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
      _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [← hcoef k hk, mul_pow, mul_inv]
        ring
      _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
  -- Borwein Lemma 1 (proved): sum over m >= 1; the first n - 1 terms vanish.
  have hres : ∀ n : ℕ, 1 ≤ n → (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    intro n hn
    have pf := hpf n hn
    -- Summability of the two families of pieces.
    have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by
      intro k
      refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_)
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
      have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
      rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div]
      exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
    have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by
      intro i hi
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
      have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
      have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by
        intro m
        rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
      rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1]
      have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
      have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
      field_simp
    have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by
      intro i hi
      by_contra hns
      have := tsum_eq_zero_of_not_summable hns
      rw [hGeo i hi] at this
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
      contradiction
    have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by
      simpa using hT 0
    have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      induction k with
      | zero => simp
      | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
    have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      have hs := hz0.sum_add_tsum_nat_add k
      rw [hIcc k] at hs
      rw [← hs, add_sub_cancel_left]
      exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
    -- The series E(n) summed from m = 1: its first n - 1 terms vanish.
    have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) :=
      summable_sum (fun k _ => (hT k).mul_left _)
    have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) :=
      summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
    have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
    have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) :=
      hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
    have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
    have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by
      refine Finset.sum_eq_zero (fun i hi => ?_)
      have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by
        rw [Finset.mem_range] at hi
        rw [Finset.mem_Icc]; omega
      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_)
      simp
    rw [hzero, zero_add] at hshift
    have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by
      rw [← hshift]
      exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
    rw [hE, tsum_congr (fun m => pf (m + 1) (by omega)), hRs1.tsum_add hRs2,
      Summable.tsum_finsetSum (fun k _ => (hT k).mul_left _),
      Summable.tsum_finsetSum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _),
      Finset.sum_congr (s₁ := Finset.Icc 1 n) rfl (fun k _ => by rw [tsum_mul_left, hTail k]),
      Finset.sum_congr (s₁ := Finset.Icc 1 (n - 1)) rfl (fun i hi => by rw [tsum_mul_left, hGeo i (Finset.mem_Icc.mp hi).1])]
    simp only [mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv]
    ring
  refine ⟨a, b, fun n hn => ?_⟩
  linear_combination (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) * hb n hn - ha n hn - (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * hres n hn

The gate replayed this file at its merge, each hole a sorry (D-12 #5); the whole file: targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean

What is known

Notes

Annex to erdos-1050--h1-v2--h2 (integrality of the Padé numerators and denominators for sum 1/(2^n - 3)). full text →
Audit and decomposition of the hole integrality (erdos-1050--h1-v2--h2), by t0929-1 (agent 1050-B), 2026-09-29. full text →
Correction and update to annex 959e0d5b… on this node (t0929-1, agent 1050-B, 2026-09-29). full text →

Work on it

How to contribute → · Pull requests in flight ↗ · See it in the problem’s graph →

Record

Statement: the Lean file and its words (3 versions)

Statement not audited — only a problem’s root statement is audited for fidelity (D-9); the kernel checks a proof against this Lean as written.

import Mathlib
import Nodes.«erdos-1050--h1-v2--h2».Context

/-! Hole `integrality` of a merged partial proof, as a node (D-12 #5, D-29).
Statement generated by the gate from the assembly; immutable (D-3). -/

theorem erdos_1050__h1_v2__h2 : ∃ (a : ℕ → ℤ) (b : ℕ → ℤ),
  ∀ (n : ℕ),
    (1 : ℕ) ≤ n →
      (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) =
        (9 : ℝ) ^ n *
            (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) *
              ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) *
          ∑' (j : ℕ),
            -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ *
              ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)),
                ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) *
                  ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ := by
  sorry

In words, unverified — what this statement says; written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1), 2026-10-18; no one has read it against the Lean and signed it.

Draft · drafted, by the model named above

$\displaystyle \exists\, a,b:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad b(n)\,z-a(n)\;=\;9^n\,W(n)\,E(n)$

Rendered from gloss/44c6fe81f5d0….md · Improve these words →

History: 3 versions on record

2026-10-06 · written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent · not signed · gloss/2b8f1d911680….md · the first version of its chain
Read this version
There are two sequences of integers $a(n)$ and $b(n)$, indexed by the natural numbers, such that for every $n \ge 1$ $$b(n)\,z \;-\; a(n) \;=\; 9^n\,W(n)\,E(n),$$ where, with $c = 8/3$ and all arithmetic in the real numbers:
- $z=\displaystyle\sum_{j\ge 0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$;
- $W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
- $E(n)=\displaystyle\sum_{j\ge 0}\; -\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$.
Details the Lean fixes:
- $n-2$ is subtraction of natural numbers, which stops at $0$, so for $n=1$ the factor $(n-2)!$ is $0! = 1$. The lower limit $\lfloor (n+1)/2\rfloor$ is natural-number division, that is $\lceil n/2\rceil$.
- In $E(n)$ the exponents $2n+j$, $k-n-j$ and $k+n+j$ are integers; $k-n-j$ is negative, so $2^{\,k-n-j}=1/2^{\,n+j-k}$. For $n=1$ the product over $k$ is empty and equals $1$.
- Both infinite sums are Lean's unconditional sum ∑' over $j \ge 0$, which takes the value $0$ for a series that is not summable; summability is not asserted separately.
- Nothing is required of $a(0)$ or $b(0)$, and the integers $a(n)$, $b(n)$ are not required to be nonzero.
The file's comment records that this statement is the hole integrality of a merged partial proof, written into its own node by the gate.
2026-10-06 · written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent · not signed · gloss/d6b4a8ae5efa….md · diff to its predecessor
Read this version
There are two sequences of integers $a(n)$ and $b(n)$, indexed by the natural numbers, such that for every $n \ge 1$ $$b(n)\,z \;-\; a(n) \;=\; 9^n\,W(n)\,E(n),$$ where, with $c = 8/3$ and all arithmetic in the real numbers:

$z=\displaystyle\sum_{j\ge 0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$;

$W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;

$E(n)=\displaystyle\sum_{j\ge 0}\; -\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$.

Details the Lean fixes:

$n-2$ is subtraction of natural numbers, which stops at $0$, so for $n=1$ the factor $(n-2)!$ is $0! = 1$. The lower limit $\lfloor (n+1)/2\rfloor$ is natural-number division, that is $\lceil n/2\rceil$.

In $E(n)$ the exponents $2n+j$, $k-n-j$ and $k+n+j$ are integers; $k-n-j$ is negative, so $2^{\,k-n-j}=1/2^{\,n+j-k}$. For $n=1$ the product over $k$ is empty and equals $1$.

Both infinite sums are Lean's unconditional sum ∑' over $j \ge 0$, which takes the value $0$ for a series that is not summable; summability is not asserted separately.

Nothing is required of $a(0)$ or $b(0)$, and the integers $a(n)$, $b(n)$ are not required to be nonzero.

The file's comment records that this statement is the hole integrality of a merged partial proof, written into its own node by the gate.
2026-10-18 · written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1) · not signed · current · gloss/44c6fe81f5d0….md · diff to its predecessor
Read this version
$\displaystyle \exists\, a,b:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad b(n)\,z-a(n)\;=\;9^n\,W(n)\,E(n)$

There are two sequences of integers $a(n)$, $b(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, where, with $c=8/3$, $z=\sum_{j\ge 0}\bigl(1-c\,2^{j+1}\bigr)^{-1}$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$ and $E(n)=\sum_{j\ge 0}\;-\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$. Here $n-2$ is natural-number subtraction (so $(n-2)!=0!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponents $2n+j$, $k-n-j$ and $k+n+j$ in $E(n)$ are integers (so $2^{k-n-j}=1/2^{n+j-k}$, and the product over $k$ is empty at $n=1$), and both infinite sums are Lean's ∑' over $j\ge 0$, which is $0$ for a non-summable family. Nothing is required of $a(0)$ or $b(0)$, and $a(n)$, $b(n)$ are not required to be nonzero.

Diff of d6b4a8ae5efa against 2b8f1d911680

@@ -3,14 +3,19 @@
 where, with $c = 8/3$ and all arithmetic in the real numbers:
 
-- $z=\displaystyle\sum_{j\ge 0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$;
-- $W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
-- $E(n)=\displaystyle\sum_{j\ge 0}\; -\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$.
+$z=\displaystyle\sum_{j\ge 0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$;
+
+$W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
+
+$E(n)=\displaystyle\sum_{j\ge 0}\; -\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$.
 
 Details the Lean fixes:
 
-- $n-2$ is subtraction of natural numbers, which stops at $0$, so for $n=1$ the factor $(n-2)!$ is $0! = 1$. The lower limit $\lfloor (n+1)/2\rfloor$ is natural-number division, that is $\lceil n/2\rceil$.
-- In $E(n)$ the exponents $2n+j$, $k-n-j$ and $k+n+j$ are integers; $k-n-j$ is negative, so $2^{\,k-n-j}=1/2^{\,n+j-k}$. For $n=1$ the product over $k$ is empty and equals $1$.
-- Both infinite sums are Lean's unconditional sum `∑'` over $j \ge 0$, which takes the value $0$ for a series that is not summable; summability is not asserted separately.
-- Nothing is required of $a(0)$ or $b(0)$, and the integers $a(n)$, $b(n)$ are not required to be nonzero.
+$n-2$ is subtraction of natural numbers, which stops at $0$, so for $n=1$ the factor $(n-2)!$ is $0! = 1$. The lower limit $\lfloor (n+1)/2\rfloor$ is natural-number division, that is $\lceil n/2\rceil$.
+
+In $E(n)$ the exponents $2n+j$, $k-n-j$ and $k+n+j$ are integers; $k-n-j$ is negative, so $2^{\,k-n-j}=1/2^{\,n+j-k}$. For $n=1$ the product over $k$ is empty and equals $1$.
+
+Both infinite sums are Lean's unconditional sum `∑'` over $j \ge 0$, which takes the value $0$ for a series that is not summable; summability is not asserted separately.
+
+Nothing is required of $a(0)$ or $b(0)$, and the integers $a(n)$, $b(n)$ are not required to be nonzero.
 
 The file's comment records that this statement is the hole `integrality` of a merged partial proof, written into its own node by the gate.

Diff of 44c6fe81f5d0 against d6b4a8ae5efa

@@ -1,21 +1,3 @@
-There are two sequences of integers $a(n)$ and $b(n)$, indexed by the natural numbers, such that for every $n \ge 1$
-$$b(n)\,z \;-\; a(n) \;=\; 9^n\,W(n)\,E(n),$$
-where, with $c = 8/3$ and all arithmetic in the real numbers:
+$$\exists\, a,b:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad b(n)\,z-a(n)\;=\;9^n\,W(n)\,E(n)$$
 
-$z=\displaystyle\sum_{j\ge 0}\bigl(1-c\,2^{\,j+1}\bigr)^{-1}$;
-
-$W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
-
-$E(n)=\displaystyle\sum_{j\ge 0}\; -\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$.
-
-Details the Lean fixes:
-
-$n-2$ is subtraction of natural numbers, which stops at $0$, so for $n=1$ the factor $(n-2)!$ is $0! = 1$. The lower limit $\lfloor (n+1)/2\rfloor$ is natural-number division, that is $\lceil n/2\rceil$.
-
-In $E(n)$ the exponents $2n+j$, $k-n-j$ and $k+n+j$ are integers; $k-n-j$ is negative, so $2^{\,k-n-j}=1/2^{\,n+j-k}$. For $n=1$ the product over $k$ is empty and equals $1$.
-
-Both infinite sums are Lean's unconditional sum `∑'` over $j \ge 0$, which takes the value $0$ for a series that is not summable; summability is not asserted separately.
-
-Nothing is required of $a(0)$ or $b(0)$, and the integers $a(n)$, $b(n)$ are not required to be nonzero.
-
-The file's comment records that this statement is the hole `integrality` of a merged partial proof, written into its own node by the gate.
+There are two sequences of integers $a(n)$, $b(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, where, with $c=8/3$, $z=\sum_{j\ge 0}\bigl(1-c\,2^{j+1}\bigr)^{-1}$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$ and $E(n)=\sum_{j\ge 0}\;-\bigl(1-c\,2^{\,2n+j}\bigr)^{-1}\prod_{k=1}^{n-1}\bigl(1-2^{\,k-n-j}\bigr)\bigl(1-c\,2^{\,k+n+j}\bigr)^{-1}$. Here $n-2$ is natural-number subtraction (so $(n-2)!=0!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponents $2n+j$, $k-n-j$ and $k+n+j$ in $E(n)$ are integers (so $2^{k-n-j}=1/2^{n+j-k}$, and the product over $k$ is empty at $n=1$), and both infinite sums are Lean's `∑'` over $j\ge 0$, which is $0$ for a non-summable family. Nothing is required of $a(0)$ or $b(0)$, and $a(n)$, $b(n)$ are not required to be nonzero.

Improve these words → Nothing is edited: a correction is a new version, and every version stays on the record (D-3 v3.30).

Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Statement.lean. Origin: skeleton-hole.

Depends on (1), each in words

Depends on, each in words where someone has written them:

erdos-1050--h1-v2--h2--h1: $\exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$ · declared; the proof does not use it

Attestation: pass, 7 of 7 checks

Checked by the kernel — the gate’s record of the run: attestations/000335.json, merge commit b16ab29fc51b.

Verdict pass, from attestations/000335.json.

Submitter: t1001-1050b
Toolchain: leanprover/lean4:v4.33.1 (hash f54d0bc916bf76a7fae460ce9d078ccd1f74aeb4fd12c4b6546bca28b6aa1c63)
Mathlib: 0df444a360eaa60ab8c11dca51a86af692955474
Reviewer: none needed (calibration)

Step	Check	Result
1	toolchain	pass
2	paths	pass
4	kernel-replay	pass
5	axioms	pass
6	hazards	pass
7	witness	pass
8	deps	pass

Witness

Checked by the kernel — step 7 passed in the run its attestation records: attestations/000335.json, merge commit b16ab29fc51b.

Witness.lean — checked at step 7 of the run recorded below: pass. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/Witness.lean, sha256 caf8ad89262a.

import Mathlib
import Nodes.«erdos-1050--h1-v2--h2».Context

theorem witness : True := trivial

In words, unverified — what this witness says; written by t1006-w2, drafted with Claude Opus 5.5 (claude-opus-5-5) via Claude Code, as an agent, 2026-10-06; no one has read it against the Lean and signed it.

Draft · drafted, by the model named above

This file proves the proposition True (by trivial), as a theorem named witness, after importing Mathlib and the node's Context. It exhibits nothing about the node's statement, and carries no comment.

The witness file of a node is meant to show that the statement's hypotheses can be met (the non-vacuity check, step 7 of the gate). This node's statement has no binders before its colon: it is a single existence claim ("there are integer sequences such that, for every $n\ge1$, …"), whose condition $n\ge1$ sits inside it. So there are no hypotheses to exhibit, and the file proves only True.

Rendered from gloss/0e8d717aabd0….md · Improve these words →

Attempts (1)

1 recorded. Routes refuted: none. Failure classes: none.

Untrusted contributor text — a partial assembly: it records an attempt, not a proof; its attestation names the file, and no hash in the record binds these bytes.

Untrusted: partial assembly (D-12 #5), by t0929-1; merged in b2e922a60e62 under attestations/000290.json, whose step 2 took this file as a partial. A partial records an attempt, not a proof: the gate replayed it with its holes left as sorry. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean.

Untrusted contributor text — an outline the gate extracted from this partial assembly; its holes are sorry steps, and it records an attempt, not a proof.

Outline of 20260929T194647Z-t0929-1-partial.lean, extracted by the gate at 82834c4684b8 from the proof’s elaboration: each step is a claim the proof establishes and the goal it leaves. Open a step for what it uses and its Lean lines.

numerator_integral hole ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))) hole
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with numerator_integral : ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Uses library Nat.factorial, Finset.Icc.
Left open as a hole; its statement is erdos-1050--h1-v2--h2--h1 (proved).
Lean, line 23
```
  have numerator_integral : ∃ a : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := sorry
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 23 at 841d05f96fd2 ↗
s2 obtain ∃ (a : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with a : ℕ → ℤ, ha : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Lean, line 24
```
  obtain ⟨a, ha⟩ := numerator_integral
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 24 at 841d05f96fd2 ↗
hden have ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hden : ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Uses library Nat.factorial, Finset.Icc, Exists.intro, Classical.choose, dif_pos, Classical.choose_spec.
Lean, lines 26–148
```
  have hden : ∃ b : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (b n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
    have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      intro n hn
      have hr0 : ∀ k : ℕ, ((8 : ℝ) * 2 ^ k) ≠ 0 := fun k => by positivity
      have hvinj : Set.InjOn (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹) (Finset.Icc 1 n : Set ℕ) := by
        intro a _ b _ hab
        have h : (2 : ℝ) ^ a = 2 ^ b := by
          have := inv_injective hab
          exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this
        exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
      -- (1) the polynomial partial-fraction identity, by agreement at the n nodes 1/r_k
      have hid : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) = ∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) := by
        refine Polynomial.eq_of_natDegree_lt_card_of_eval_eq' _ _ ((Finset.Icc 1 n).image (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹)) ?_ ?_
        · intro y hy
          obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
          rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk]
          · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod]
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by
              refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_)
              have hjk : j ≠ k := Finset.ne_of_mem_erase hj
              intro h0
              have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by
                have := hr0 k
                field_simp at h0
                linarith
              exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this))
            rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one]
          · intro i hi hik
            rw [Polynomial.eval_mul, Polynomial.eval_prod]
            refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_)
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            rw [mul_inv_cancel₀ (hr0 k), sub_self]
        · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc]
          have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by
            rw [Polynomial.natDegree_neg]
            refine (Polynomial.natDegree_prod_le _ _).trans ?_
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
          have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by
            refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_)
            refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_)
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one]
              rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega
          omega
      -- (2) coefficient n - 1 of T_k * P is r_k ^ (n - 1)
      have hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)).coeff d = if d = 0 then 1 else 0 := by
        intro S
        induction S using Finset.induction_on with
        | empty => intro d hd; simp [Polynomial.coeff_one]
        | insert a S ha ih =>
          intro d hd
          rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc,
            Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd]
      have hG : ∀ k : ℕ, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k
        simp only [mul_pow, ← Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow]
        rw [Finset.sum_ite_eq (Finset.range n) (n - 1), if_pos (Finset.mem_range.mpr (by omega))]
      have hTP : ∀ k ∈ Finset.Icc 1 n, ((∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k hk
        have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) :=
          (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm
        have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by
          rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib]
          congr 1
          refine Finset.prod_congr rfl (fun j _ => ?_)
          rw [mul_neg_geom_sum]
        obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 :=
          Polynomial.X_pow_dvd_iff.mpr (fun d hd => by
          rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all)
        rw [hprod, show ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) = 1 + Polynomial.X ^ n * E by rw [← hE]; ring,
          mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg (by omega), add_zero, hG]
      -- (3) so coefficient n - 1 of Qt * P is the sum of C_k r_k^(n-1)
      have hQP : ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ∑ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        nth_rw 1 [hid]
        rw [Finset.sum_mul, Polynomial.finsetSum_coeff]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [mul_assoc, Polynomial.coeff_C_mul, hTP k hk]
      -- (4) and it is an integer, since Qt * P has integer coefficients
      have hmap : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)) := by
        simp [Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat]
      -- (5) C_k r_k^(n-1) = 3^(n-1) A(n,k)
      have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
      have hterm : ∀ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = (3 : ℝ) ^ (n - 1) * (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) := by
        intro k hk
        have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by
          have := hr0 k
          rw [pow_add]; field_simp
        have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
          rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
          field_simp
        have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by
          rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow]
        simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C,
          Polynomial.eval_X]
        rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
          Finset.prod_const, hcard]
        simp only [hw]
        linear_combination (-((3 : ℝ) ^ (n - 1) * ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) *
          (∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hrr
      refine ⟨((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1), ?_⟩
      have hc : ((((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1) : ℤ) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) := by
        rw [Polynomial.coeff_map]; simp
      rw [hc, ← hmap, hQP, Finset.mul_sum]
      exact Finset.sum_congr rfl hterm
    have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
      intro n
      rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
        Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
    refine ⟨fun n => if h : 1 ≤ n then 3 * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * Classical.choose (key n h) else 0, fun n hn => ?_⟩
    simp only [dif_pos hn]
    push_cast
    rw [Classical.choose_spec (key n hn), hW3 n]
    have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by
      rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
    have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
    rw [h9]
    linear_combination (-(3 : ℝ) * (Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k)) * ((3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))))) * h3
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 26–148 at 841d05f96fd2 ↗
1. hden.key have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
  Leaves ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with key : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
  Uses library Finset.Icc, Exists.intro, Polynomial.C, Polynomial.X, Finset.range, Finset.mul_sum, Finset.sum_congr, rfl.
  Lean, lines 27–133
  have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by intro n hn have hr0 : ∀ k : ℕ, ((8 : ℝ) * 2 ^ k) ≠ 0 := fun k => by positivity have hvinj : Set.InjOn (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹) (Finset.Icc 1 n : Set ℕ) := by intro a _ b _ hab have h : (2 : ℝ) ^ a = 2 ^ b := by have := inv_injective hab exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h -- (1) the polynomial partial-fraction identity, by agreement at the n nodes 1/r_k have hid : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) = ∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) := by refine Polynomial.eq_of_natDegree_lt_card_of_eval_eq' _ _ ((Finset.Icc 1 n).image (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹)) ?_ ?_ · intro y hy obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk] · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_) have hjk : j ≠ k := Finset.ne_of_mem_erase hj intro h0 have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this)) rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one] · intro i hi hik rw [Polynomial.eval_mul, Polynomial.eval_prod] refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_) simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [mul_inv_cancel₀ (hr0 k), sub_self] · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc] have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by rw [Polynomial.natDegree_neg] refine (Polynomial.natDegree_prod_le _ _).trans ?_ refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_) refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_) refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one] rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega omega -- (2) coefficient n - 1 of T_k * P is r_k ^ (n - 1) have hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)).coeff d = if d = 0 then 1 else 0 := by intro S induction S using Finset.induction_on with | empty => intro d hd; simp [Polynomial.coeff_one] | insert a S ha ih => intro d hd rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc, Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd] have hG : ∀ k : ℕ, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by intro k simp only [mul_pow, ← Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow] rw [Finset.sum_ite_eq (Finset.range n) (n - 1), if_pos (Finset.mem_range.mpr (by omega))] have hTP : ∀ k ∈ Finset.Icc 1 n, ((∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by intro k hk have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) := (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib] congr 1 refine Finset.prod_congr rfl (fun j _ => ?_) rw [mul_neg_geom_sum] obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 := Polynomial.X_pow_dvd_iff.mpr (fun d hd => by rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all) rw [hprod, show ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) = 1 + Polynomial.X ^ n * E by rw [← hE]; ring, mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg (by omega), add_zero, hG] -- (3) so coefficient n - 1 of Qt * P is the sum of C_k r_k^(n-1) have hQP : ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ∑ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by nth_rw 1 [hid] rw [Finset.sum_mul, Polynomial.finsetSum_coeff] refine Finset.sum_congr rfl (fun k hk => ?_) rw [mul_assoc, Polynomial.coeff_C_mul, hTP k hk] -- (4) and it is an integer, since Qt * P has integer coefficients have hmap : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)) := by simp [Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat] -- (5) C_k r_k^(n-1) = 3^(n-1) A(n,k) have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega have hterm : ∀ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = (3 : ℝ) ^ (n - 1) * (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) := by intro k hk have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by have := hr0 k rw [pow_add]; field_simp have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow] simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const, Finset.prod_const, hcard] simp only [hw] linear_combination (-((3 : ℝ) ^ (n - 1) * ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) * (∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hrr refine ⟨((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1), ?_⟩ have hc : ((((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1) : ℤ) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) := by rw [Polynomial.coeff_map]; simp rw [hc, ← hmap, hQP, Finset.mul_sum] exact Finset.sum_congr rfl hterm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 27–133 at 841d05f96fd2 ↗
  hden.key.hr0 have ∀ (k : ℕ), (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hr0 : ∀ (k : ℕ), (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 29
  have hr0 : ∀ k : ℕ, ((8 : ℝ) * 2 ^ k) ≠ 0 := fun k => by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 29 at 841d05f96fd2 ↗
  hden.key.hvinj have Set.InjOn (fun (k : ℕ) => ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hvinj : Set.InjOn (fun (k : ℕ) => ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ).
  Uses library Set.InjOn, Finset.Icc, Set, pow_right_injective₀.
  Lean, lines 30–35
  have hvinj : Set.InjOn (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹) (Finset.Icc 1 n : Set ℕ) := by intro a _ b _ hab have h : (2 : ℝ) ^ a = 2 ^ b := by have := inv_injective hab exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 30–35 at 841d05f96fd2 ↗
  hden.key.hvinj.h have (2 : ℝ) ^ a = (2 : ℝ) ^ b
  Leaves a = b, with h : (2 : ℝ) ^ a = (2 : ℝ) ^ b.
  Uses library mul_left_cancel₀.
  Lean, lines 32–34
  have h : (2 : ℝ) ^ a = 2 ^ b := by have := inv_injective hab exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 32–34 at 841d05f96fd2 ↗
  hden.key.hvinj.h.this have (8 : ℝ) * (2 : ℝ) ^ a = (8 : ℝ) * (2 : ℝ) ^ b
  Leaves (2 : ℝ) ^ a = (2 : ℝ) ^ b, with this : (8 : ℝ) * (2 : ℝ) ^ a = (8 : ℝ) * (2 : ℝ) ^ b.
  Uses library inv_injective.
  Lean, line 33
  have := inv_injective hab
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 33 at 841d05f96fd2 ↗
  hden.key.hid have -∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t)) = ∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hid : -∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t)) = ∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Polynomial.eq_of_natDegree_lt_card_of_eval_eq'.
  Lean, lines 37–73
  have hid : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) = ∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) := by refine Polynomial.eq_of_natDegree_lt_card_of_eval_eq' _ _ ((Finset.Icc 1 n).image (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹)) ?_ ?_ · intro y hy obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk] · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_) have hjk : j ≠ k := Finset.ne_of_mem_erase hj intro h0 have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this)) rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one] · intro i hi hik rw [Polynomial.eval_mul, Polynomial.eval_prod] refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_) simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [mul_inv_cancel₀ (hr0 k), sub_self] · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc] have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by rw [Polynomial.natDegree_neg] refine (Polynomial.natDegree_prod_le _ _).trans ?_ refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_) refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_) refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one] rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 37–73 at 841d05f96fd2 ↗
  hden.key.hid.s1 case
  Leaves ∀ i ∈ Finset.image (fun (k : ℕ) => ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹) (Finset.Icc (1 : ℕ) n), Polynomial.eval i (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) = Polynomial.eval i (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).
  Uses library Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem.
  Lean, lines 39–58
  · intro y hy obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk] · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_) have hjk : j ≠ k := Finset.ne_of_mem_erase hj intro h0 have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this)) rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one] · intro i hi hik rw [Polynomial.eval_mul, Polynomial.eval_prod] refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_) simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [mul_inv_cancel₀ (hr0 k), sub_self]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 39–58 at 841d05f96fd2 ↗
  hden.key.hid.s1.s1 obtain ∃ a ∈ Finset.Icc (1 : ℕ) n, ((8 : ℝ) * (2 : ℝ) ^ a)⁻¹ = y
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) = Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)), with k : ℕ, hk : k ∈ Finset.Icc (1 : ℕ) n, hy : ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ ∈ Finset.image (fun (k : ℕ) => ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹) (Finset.Icc (1 : ℕ) n).
  Uses library Finset.mem_image.
  Lean, line 40
  obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 40 at 841d05f96fd2 ↗
  hden.key.hid.s1.s2 case
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) = Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ ((Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).
  Uses library Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_X, mul_assoc, Finset.prod_congr, rfl, div_eq_mul_inv, mul_comm, Finset.Icc, inv_mul_cancel₀, mul_one.
  Lean, lines 42–53
  · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_) have hjk : j ≠ k := Finset.ne_of_mem_erase hj intro h0 have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this)) rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 42–53 at 841d05f96fd2 ↗
  hden.key.hid.s1.s2.hne have ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)) ≠ (0 : ℝ)
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) = Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ∏ x ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x * ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹), with hne : ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)) ≠ (0 : ℝ).
  Uses library Finset.Icc, Finset.prod_ne_zero_iff, pow_right_injective₀, mul_left_cancel₀.
  Lean, lines 44–52
  have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_) have hjk : j ≠ k := Finset.ne_of_mem_erase hj intro h0 have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 44–52 at 841d05f96fd2 ↗
  hden.key.hid.s1.s2.hne.hjk have j ≠ k
  Leaves (1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k) ≠ (0 : ℝ), with hjk : j ≠ k.
  Uses library Finset.ne_of_mem_erase.
  Lean, line 46
  have hjk : j ≠ k := Finset.ne_of_mem_erase hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 46 at 841d05f96fd2 ↗
  hden.key.hid.s1.s2.hne.this have (8 : ℝ) * (2 : ℝ) ^ j = (8 : ℝ) * (2 : ℝ) ^ k
  Leaves False, with this : (8 : ℝ) * (2 : ℝ) ^ j = (8 : ℝ) * (2 : ℝ) ^ k.
  Lean, lines 48–51
  have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by have := hr0 k field_simp at h0 linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 48–51 at 841d05f96fd2 ↗
  hden.key.hid.s1.s2.hne.this.this have (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves (8 : ℝ) * (2 : ℝ) ^ j = (8 : ℝ) * (2 : ℝ) ^ k, with this : (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 49
  have := hr0 k
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 49 at 841d05f96fd2 ↗
  hden.key.hid.s1.s3 case
  Leaves ∀ b ∈ Finset.Icc (1 : ℕ) n, b ≠ k → Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ ((Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ b)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase b, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ b)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase b, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) = (0 : ℝ).
  Uses library Polynomial.eval_mul, Polynomial.eval_prod, mul_eq_zero_of_right, Finset.prod_eq_zero, Finset.mem_erase, And.intro, Ne.symm, Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_C, Polynomial.eval_X, mul_inv_cancel₀, sub_self.
  Lean, lines 54–58
  · intro i hi hik rw [Polynomial.eval_mul, Polynomial.eval_prod] refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_) simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [mul_inv_cancel₀ (hr0 k), sub_self]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 54–58 at 841d05f96fd2 ↗
  hden.key.hid.s2 case
  Leaves max (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).natDegree (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).natDegree < (Finset.image (fun (k : ℕ) => ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹) (Finset.Icc (1 : ℕ) n)).card.
  Uses library Finset.card_image_of_injOn, Nat.card_Icc.
  Lean, lines 59–73
  · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc] have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by rw [Polynomial.natDegree_neg] refine (Polynomial.natDegree_prod_le _ _).trans ?_ refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_) refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_) refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one] rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 59–73 at 841d05f96fd2 ↗
  hden.key.hid.s2.hQdeg have (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).natDegree ≤ n - (1 : ℕ)
  Leaves max (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).natDegree (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).natDegree < n + (1 : ℕ) - (1 : ℕ), with hQdeg : (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).natDegree ≤ n - (1 : ℕ).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Polynomial.natDegree_neg, Polynomial.natDegree_prod_le, Finset.sum_le_sum, le_of_eq.
  Lean, lines 60–65
  have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by rw [Polynomial.natDegree_neg] refine (Polynomial.natDegree_prod_le _ _).trans ?_ refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 60–65 at 841d05f96fd2 ↗
  hden.key.hid.s2.hQdeg.s1 case
  Leaves ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t)).natDegree ≤ (1 : ℕ).
  Lean, line 64
  · compute_degree
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 64 at 841d05f96fd2 ↗
  hden.key.hid.s2.hQdeg.s2 case routine: simp, omega
  Leaves ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (1 : ℕ) = n - (1 : ℕ).
  Uses library Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc.
  Lean, line 65
  · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 65 at 841d05f96fd2 ↗
  hden.key.hid.s2.hRdeg have (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).natDegree ≤ n - (1 : ℕ)
  Leaves max (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).natDegree (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).natDegree < n + (1 : ℕ) - (1 : ℕ), with hRdeg : (∑ k ∈ Finset.Icc (1 : ℕ) n, (Polynomial.C : ℝ → Polynomial ℝ) (Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)).natDegree ≤ n - (1 : ℕ).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Polynomial.natDegree_sum_le_of_forall_le, Polynomial.natDegree_C_mul_le, Polynomial.natDegree_prod_le, Finset.sum_le_sum, le_of_eq.
  Lean, lines 66–72
  have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_) refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_) refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_) · compute_degree · simp only [Finset.sum_const, smul_eq_mul, mul_one] rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 66–72 at 841d05f96fd2 ↗
  hden.key.hid.s2.hRdeg.s1 case
  Leaves ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X).natDegree ≤ (1 : ℕ).
  Lean, line 70
  · compute_degree
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 70 at 841d05f96fd2 ↗
  hden.key.hid.s2.hRdeg.s2 case
  Leaves ∑ i ∈ (Finset.Icc (1 : ℕ) n).erase k, (1 : ℕ) = n - (1 : ℕ).
  Uses library Finset.sum_const, smul_eq_mul, mul_one, Finset.card_erase_of_mem, Nat.card_Icc.
  Lean, lines 71–72
  · simp only [Finset.sum_const, smul_eq_mul, mul_one] rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 71–72 at 841d05f96fd2 ↗
  hden.key.hU have ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)).coeff d = if d = (0 : ℕ) then (1 : ℝ) else (0 : ℝ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)).coeff d = if d = (0 : ℕ) then (1 : ℝ) else (0 : ℝ).
  Uses library Finset, Polynomial.C, Polynomial.X, Finset.induction_on.
  Lean, lines 75–82
  have hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)).coeff d = if d = 0 then 1 else 0 := by intro S induction S using Finset.induction_on with | empty => intro d hd; simp [Polynomial.coeff_one] | insert a S ha ih => intro d hd rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc, Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 75–82 at 841d05f96fd2 ↗
  hden.key.hU.s1 case
  Leaves ∀ d < n, (∏ j ∈ ∅, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)).coeff d = if d = (0 : ℕ) then (1 : ℝ) else (0 : ℝ).
  Uses library Finset.empty, Polynomial.coeff_one.
  Lean, line 78
  | empty => intro d hd; simp [Polynomial.coeff_one]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 78 at 841d05f96fd2 ↗
  hden.key.hU.s2 case
  Leaves ∀ d < n, (∏ j ∈ insert a S, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)).coeff d = if d = (0 : ℕ) then (1 : ℝ) else (0 : ℝ), with a : ℕ, S : Finset ℕ, ha : a ∉ S, ih : ∀ d < n, (∏ j ∈ S, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)).coeff d = if d = (0 : ℕ) then (1 : ℝ) else (0 : ℝ).
  Uses library Finset.prod_insert, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, Polynomial.C_pow, mul_assoc, Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg, MulZeroClass.mul_zero, sub_zero.
  Lean, lines 79–82
  | insert a S ha ih => intro d hd rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc, Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 79–82 at 841d05f96fd2 ↗
  hden.key.hG have ∀ (k : ℕ), (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hG : ∀ (k : ℕ), (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)).
  Uses library Finset.range, Polynomial.C, Polynomial.X, mul_pow, Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow, Finset.sum_ite_eq, if_pos, Finset.mem_range.
  Lean, lines 83–86
  have hG : ∀ k : ℕ, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by intro k simp only [mul_pow, ← Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow] rw [Finset.sum_ite_eq (Finset.range n) (n - 1), if_pos (Finset.mem_range.mpr (by omega))]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 83–86 at 841d05f96fd2 ↗
  hden.key.hTP have ∀ k ∈ Finset.Icc (1 : ℕ) n, ((∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hTP : ∀ k ∈ Finset.Icc (1 : ℕ) n, ((∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Finset.range, mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg, add_zero.
  Lean, lines 87–100
  have hTP : ∀ k ∈ Finset.Icc 1 n, ((∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by intro k hk have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) := (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib] congr 1 refine Finset.prod_congr rfl (fun j _ => ?_) rw [mul_neg_geom_sum] obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 := Polynomial.X_pow_dvd_iff.mpr (fun d hd => by rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all) rw [hprod, show ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) = 1 + Polynomial.X ^ n * E by rw [← hE]; ring, mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg (by omega), add_zero, hG]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 87–100 at 841d05f96fd2 ↗
  hden.key.hTP.hsplitP have ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i
  Leaves ((∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)), with hsplitP : ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i.
  Uses library Finset.Icc, Finset.range, Polynomial.C, Polynomial.X, Finset.mul_prod_erase.
  Lean, lines 89–90
  have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) := (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 89–90 at 841d05f96fd2 ↗
  hden.key.hTP.hprod have (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n)
  Leaves ((∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)), with hprod : (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = (∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Finset.range, mul_left_comm, Finset.prod_mul_distrib, Finset.prod_congr, rfl, mul_neg_geom_sum.
  Lean, lines 91–95
  have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib] congr 1 refine Finset.prod_congr rfl (fun j _ => ?_) rw [mul_neg_geom_sum]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 91–95 at 841d05f96fd2 ↗
  hden.key.hTP.s3 obtain Polynomial.X ^ n ∣ ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n) - (1 : Polynomial ℝ)
  Leaves ((∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - (Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X)) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)), with E : Polynomial ℝ, hE : ∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : Polynomial ℝ) - ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ n) - (1 : Polynomial ℝ) = Polynomial.X ^ n * E.
  Uses library Polynomial.X, Finset.Icc, Polynomial.C, Polynomial.X_pow_dvd_iff, Polynomial.coeff_sub, Polynomial.coeff_one.
  Lean, lines 96–98
  obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 := Polynomial.X_pow_dvd_iff.mpr (fun d hd => by rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 96–98 at 841d05f96fd2 ↗
  hden.key.hQP have ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hQP : ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ)) = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Finset.range, Finset.sum_mul, Polynomial.finsetSum_coeff, Finset.sum_congr, rfl, mul_assoc, Polynomial.coeff_C_mul.
  Lean, lines 102–106
  have hQP : ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ∑ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by nth_rw 1 [hid] rw [Finset.sum_mul, Polynomial.finsetSum_coeff] refine Finset.sum_congr rfl (fun k hk => ?_) rw [mul_assoc, Polynomial.coeff_C_mul, hTP k hk]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 102–106 at 841d05f96fd2 ↗
  hden.key.hmap have (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i) routine: simp
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hmap : (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℝ → Polynomial ℝ) ((8 : ℝ) * (2 : ℝ) ^ j) * Polynomial.X) ^ i = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Finset.range, Polynomial.map, Int.castRingHom, Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat.
  Lean, lines 108–109
  have hmap : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)) := by simp [Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 108–109 at 841d05f96fd2 ↗
  hden.key.hcard have (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ)
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hcard : (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ).
  Uses library Finset.Icc, Nat.card_Icc.
  Lean, line 111
  have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 111 at 841d05f96fd2 ↗
  hden.key.hterm have ∀ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ)) * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))))
  Leaves ∃ (z : ℤ), (↑z : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hterm : ∀ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ)) * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const.
  Lean, lines 112–128
  have hterm : ∀ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = (3 : ℝ) ^ (n - 1) * (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) := by intro k hk have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by have := hr0 k rw [pow_add]; field_simp have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow] simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X] rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const, Finset.prod_const, hcard] simp only [hw] linear_combination (-((3 : ℝ) ^ (n - 1) * ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) * (∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hrr
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 112–128 at 841d05f96fd2 ↗
  hden.key.hterm.hq have ∀ (t : ℕ), (3 : ℝ) * ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((3 : ℝ) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)))
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ)) * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hq : ∀ (t : ℕ), (3 : ℝ) * ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((3 : ℝ) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))).
  Uses library pow_add.
  Lean, lines 114–116
  have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by have := hr0 k rw [pow_add]; field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 114–116 at 841d05f96fd2 ↗
  hden.key.hterm.hq.this have (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves (3 : ℝ) * ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((3 : ℝ) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))), with this : (8 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 115
  have := hr0 k
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 115 at 841d05f96fd2 ↗
  hden.key.hterm.hw have ∀ (j : ℕ), (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ))
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ)) * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hw : ∀ (j : ℕ), (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ)).
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast.
  Lean, lines 117–119
  have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 117–119 at 841d05f96fd2 ↗
  hden.key.hterm.hrr have ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ ^ (n - (1 : ℕ)) * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (1 : ℝ)
  Leaves Polynomial.eval ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℝ → Polynomial ℝ) (3 : ℝ) * Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ j / ((8 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (3 : ℝ) ^ (n - (1 : ℕ)) * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))), with hrr : ((8 : ℝ) * (2 : ℝ) ^ k)⁻¹ ^ (n - (1 : ℕ)) * ((8 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)) = (1 : ℝ).
  Uses library mul_pow, inv_mul_cancel₀, one_pow.
  Lean, lines 120–121
  have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 120–121 at 841d05f96fd2 ↗
  hden.key.hc have (↑(((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ))) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i)).coeff (n - (1 : ℕ))
  Leaves (↑(((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ))) : ℝ) = (3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hc : (↑(((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i).coeff (n - (1 : ℕ))) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((Polynomial.C : ℤ → Polynomial ℤ) (3 : ℤ) * Polynomial.X - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t))) * ∏ j ∈ Finset.Icc (1 : ℕ) n, ∑ i ∈ Finset.range n, ((Polynomial.C : ℤ → Polynomial ℤ) ((8 : ℤ) * (2 : ℤ) ^ j) * Polynomial.X) ^ i)).coeff (n - (1 : ℕ)).
  Uses library Finset.Icc, Polynomial.C, Polynomial.X, Finset.range, Polynomial.map, Int.castRingHom, Polynomial.coeff_map.
  Lean, lines 130–131
  have hc : ((((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1) : ℤ) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) := by rw [Polynomial.coeff_map]; simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 130–131 at 841d05f96fd2 ↗
2. hden.hW3 have ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹
  Leaves ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with hW3 : ∀ (n : ℕ), ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow.
  Lean, lines 134–137
  have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by intro n rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)), Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 134–137 at 841d05f96fd2 ↗
3. hden.h9 have (9 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)) * (3 : ℝ) ^ n
  Leaves (((3 : ℝ) * (↑(n - (2 : ℕ)).factorial : ℝ) * ∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * ∏ x ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ x)) * ((3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) = (9 : ℝ) ^ n * ((↑(n - (2 : ℕ)).factorial : ℝ) * ((∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with h9 : (9 : ℝ) ^ n = (3 : ℝ) * (3 : ℝ) ^ (n - (1 : ℕ)) * (3 : ℝ) ^ n.
  Uses library mul_pow, mul_comm, pow_succ, Nat.sub_add_cancel.
  Lean, lines 143–144
  have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 143–144 at 841d05f96fd2 ↗
4. hden.h3 have ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = (1 : ℝ)
  Leaves (((3 : ℝ) * (↑(n - (2 : ℕ)).factorial : ℝ) * ∏ x ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ x)) * ∏ x ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ x)) * ((3 : ℝ) ^ (n - (1 : ℕ)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) = (9 : ℝ) ^ n * ((↑(n - (2 : ℕ)).factorial : ℝ) * ((∏ k ∈ Finset.Icc (1 : ℕ) n, ((3 : ℝ) - (8 : ℝ) * (2 : ℝ) ^ k)) * ((3 : ℝ) ^ n)⁻¹) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))), with h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = (1 : ℝ).
  Uses library inv_mul_cancel₀.
  Lean, line 145
  have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 145 at 841d05f96fd2 ↗
s4 obtain ∃ (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with b : ℕ → ℤ, hb : ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ))).
Lean, line 149
```
  obtain ⟨b, hb⟩ := hden
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 149 at 841d05f96fd2 ↗
hpf have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hpf : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹.
Uses library Finset.Icc, Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.
Lean, lines 151–281
```
  have hpf : ∀ n : ℕ, 1 ≤ n → ∀ m : ℕ, 1 ≤ m → (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := by
    intro n hn m hm
    -- General partial fractions, from Mathlib's Lagrange interpolation.
    have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s →
      Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) →
      Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ =
        ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
      intro s p Q hp hinj hdeg x hx
      have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by
        intro a ha b hb hab
        exact hinj ha hb (inv_injective hab)
      have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by
        intro k hk hxk
        apply hx k hk
        rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
      have hQ := Lagrange.eq_interpolate hinj' hdeg
      have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
      rw [← hQ, Lagrange.eval_nodal] at hev
      have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by
        intro k hk
        have h1 := hp k hk
        have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by
          rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
        rw [← h3, mul_inv, inv_neg]
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev]
      have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
      rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc,
        mul_inv_cancel₀ hne, one_mul, Finset.sum_mul]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk]
      have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by
        intro j hj
        have hjs := Finset.mem_of_mem_erase hj
        have hjk := Finset.ne_of_mem_erase hj
        have hpj := hp j hjs
        have hpk := hp k hk
        have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
        have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
        field_simp
        ring
      calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ *
            (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹)
          = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by
            rw [Finset.prod_mul_distrib]; ring
        _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
            rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
    have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by
      calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
        _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
    have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
    have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
    have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by
      intro k
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      nlinarith
    have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by
      intro a _ b _ hab
      have h : (2 : ℝ) ^ a = 2 ^ b := by
        have := hab; simp only at this
        exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
    have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by
      rw [Polynomial.degree_neg, Polynomial.degree_prod]
      simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc]
      rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one]
      exact_mod_cast (by omega : n - 1 < n)
    have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t)))
      (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
    simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C] at hL
    have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
    -- geometric sums
    have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by
      intro u hu hu1 N
      induction N with
      | zero => simp
      | succ N ih =>
        rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih]
        have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
        field_simp
        ring
    -- the left side, in terms of the Lagrange data
    have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by
      rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
    have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
    have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
    have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by
      have hn' : n = n - 1 + 1 := by omega
      conv_lhs => rw [hn']
      rw [Finset.prod_Icc_succ_top (by omega), ← hn']
    have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by
      rw [mul_sub, inv_mul_cancel₀ hx0]; ring
    have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hz1, hz2, hz3]
      rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib,
        Finset.prod_const, hcard, inv_pow]
      ring
    -- the right side
    have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
    have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by
      simp only [Finset.sum_mul, Finset.mul_sum]
      rw [Finset.sum_comm]
      refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_))
      rw [mul_pow, mul_inv]; ring
    have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hpk]
      rw [hswap, ← Finset.sum_add_distrib]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)]
      ring
    -- matching the coefficients
    have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by
      intro k hk
      have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by
        rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring,
          inv_mul_cancel₀ (hp0 k), mul_one]
      have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow]
      simp only [hw]
      ring
    calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
      _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
      _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [← hcoef k hk, mul_pow, mul_inv]
        ring
      _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 151–281 at 841d05f96fd2 ↗
1. hpf.lag have ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ (0 : ℝ)) → Set.InjOn p (↑s : Set ℕ) → Q.degree < (↑s.card : WithBot ℕ) → ∀ (x : ℝ), (∀ k ∈ s, (1 : ℝ) - p k * x ≠ (0 : ℝ)) → Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ (0 : ℝ)) → Set.InjOn p (↑s : Set ℕ) → Q.degree < (↑s.card : WithBot ℕ) → ∀ (x : ℝ), (∀ k ∈ s, (1 : ℝ) - p k * x ≠ (0 : ℝ)) → Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹.
  Uses library Finset, Polynomial, Set.InjOn, Lagrange.eval_nodal, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_inv_distrib, mul_assoc, mul_comm, mul_inv_cancel₀, one_mul, Finset.sum_mul, Finset.sum_congr, Lagrange.nodalWeight, Finset.mul_prod_erase.
  Lean, lines 154–196
  have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s → Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) → Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ = ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by intro s p Q hp hinj hdeg x hx have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by intro a ha b hb hab exact hinj ha hb (inv_injective hab) have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by intro k hk hxk apply hx k hk rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self] have hQ := Lagrange.eq_interpolate hinj' hdeg have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv rw [← hQ, Lagrange.eval_nodal] at hev have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by intro k hk have h1 := hp k hk have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring rw [← h3, mul_inv, inv_neg] rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev] have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi)) rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc, mul_inv_cancel₀ hne, one_mul, Finset.sum_mul] refine Finset.sum_congr rfl (fun k hk => ?_) rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk] have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by intro j hj have hjs := Finset.mem_of_mem_erase hj have hjk := Finset.ne_of_mem_erase hj have hpj := hp j hjs have hpk := hp k hk have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h) have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne) field_simp ring calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by rw [Finset.prod_mul_distrib]; ring _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 154–196 at 841d05f96fd2 ↗
  hpf.lag.hinj' have Set.InjOn (fun (k : ℕ) => (p k)⁻¹) (↑s : Set ℕ)
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hinj' : Set.InjOn (fun (k : ℕ) => (p k)⁻¹) (↑s : Set ℕ).
  Uses library Set.InjOn, inv_injective.
  Lean, lines 159–161
  have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by intro a ha b hb hab exact hinj ha hb (inv_injective hab)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 159–161 at 841d05f96fd2 ↗
  hpf.lag.hxv have ∀ k ∈ s, x ≠ (p k)⁻¹
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hxv : ∀ k ∈ s, x ≠ (p k)⁻¹.
  Uses library mul_inv_cancel₀, sub_self.
  Lean, lines 162–165
  have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by intro k hk hxk apply hx k hk rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 162–165 at 841d05f96fd2 ↗
  hpf.lag.hQ have Q = (Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hQ : Q = (Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q.
  Uses library Lagrange.eq_interpolate.
  Lean, line 166
  have hQ := Lagrange.eq_interpolate hinj' hdeg
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 166 at 841d05f96fd2 ↗
  hpf.lag.hev have Polynomial.eval x ((Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q) = Polynomial.eval x (Lagrange.nodal s fun (k : ℕ) => (p k)⁻¹) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hev : Polynomial.eval x ((Lagrange.interpolate s fun (k : ℕ) => (p k)⁻¹ : (ℕ → ℝ) → Polynomial ℝ) fun (i : ℕ) => Polynomial.eval (p i)⁻¹ Q) = Polynomial.eval x (Lagrange.nodal s fun (k : ℕ) => (p k)⁻¹) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q.
  Uses library Lagrange.eval_interpolate_not_at_node.
  Lean, line 167
  have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 167 at 841d05f96fd2 ↗
  hpf.lag.hfac have ∀ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹
  Leaves Polynomial.eval x Q * ∏ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hfac : ∀ k ∈ s, ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹.
  Uses library mul_inv, inv_neg.
  Lean, lines 169–174
  have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by intro k hk have h1 := hp k hk have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring rw [← h3, mul_inv, inv_neg]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 169–174 at 841d05f96fd2 ↗
  hpf.lag.hfac.h1 have p k ≠ (0 : ℝ)
  Leaves ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹, with h1 : p k ≠ (0 : ℝ).
  Lean, line 171
  have h1 := hp k hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 171 at 841d05f96fd2 ↗
  hpf.lag.hfac.h3 have (x - (p k)⁻¹) * -p k = (1 : ℝ) - p k * x
  Leaves ((1 : ℝ) - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * -(p k)⁻¹, with h3 : (x - (p k)⁻¹) * -p k = (1 : ℝ) - p k * x.
  Uses library sub_mul, mul_neg, inv_mul_cancel₀.
  Lean, lines 172–173
  have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 172–173 at 841d05f96fd2 ↗
  hpf.lag.hne have ∏ i ∈ s, (x - (p i)⁻¹) ≠ (0 : ℝ)
  Leaves ((∏ i ∈ s, (x - (p i)⁻¹)) * ∑ i ∈ s, Lagrange.nodalWeight s (fun (k : ℕ) => (p k)⁻¹) i * (x - (p i)⁻¹)⁻¹ * Polynomial.eval (p i)⁻¹ Q) * ((∏ x_1 ∈ s, (x - (p x_1)⁻¹)⁻¹) * ∏ x ∈ s, -(p x)⁻¹) = ∑ k ∈ s, Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ (0 : ℝ).
  Uses library Finset.prod_ne_zero_iff, sub_ne_zero.
  Lean, line 176
  have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 176 at 841d05f96fd2 ↗
  hpf.lag.hj have ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹
  Leaves (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Polynomial.eval (p k)⁻¹ Q * (-(p k)⁻¹ * ∏ x ∈ s.erase k, -(p x)⁻¹) = Polynomial.eval (p k)⁻¹ Q * (∏ j ∈ s.erase k, ((1 : ℝ) - p j / p k))⁻¹ * ((1 : ℝ) - p k * x)⁻¹, with hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹.
  Lean, lines 181–190
  have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by intro j hj have hjs := Finset.mem_of_mem_erase hj have hjk := Finset.ne_of_mem_erase hj have hpj := hp j hjs have hpk := hp k hk have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h) have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne) field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 181–190 at 841d05f96fd2 ↗
  hpf.lag.hj.hjs have j ∈ s
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hjs : j ∈ s.
  Uses library Finset.mem_of_mem_erase.
  Lean, line 183
  have hjs := Finset.mem_of_mem_erase hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 183 at 841d05f96fd2 ↗
  hpf.lag.hj.hjk have j ≠ k
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hjk : j ≠ k.
  Uses library Finset.ne_of_mem_erase.
  Lean, line 184
  have hjk := Finset.ne_of_mem_erase hj
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 184 at 841d05f96fd2 ↗
  hpf.lag.hj.hpj have p j ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hpj : p j ≠ (0 : ℝ).
  Lean, line 185
  have hpj := hp j hjs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 185 at 841d05f96fd2 ↗
  hpf.lag.hj.hpk have p k ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hpk : p k ≠ (0 : ℝ).
  Lean, line 186
  have hpk := hp k hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 186 at 841d05f96fd2 ↗
  hpf.lag.hj.hne have p j ≠ p k
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hne : p j ≠ p k.
  Lean, line 187
  have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 187 at 841d05f96fd2 ↗
  hpf.lag.hj.hsub have p k - p j ≠ (0 : ℝ)
  Leaves ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = ((1 : ℝ) - p j / p k)⁻¹, with hsub : p k - p j ≠ (0 : ℝ).
  Uses library sub_ne_zero, Ne.symm.
  Lean, line 188
  have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 188 at 841d05f96fd2 ↗
  hpf.lag.s8 calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Polynomial.eval (p k)⁻¹ Q * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = (Polynomial.eval (p k)⁻¹ Q * ∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹)
  Closes the goal.
  Uses library Finset.prod_mul_distrib.
  Lean, lines 191–194
  calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ * (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹) = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by rw [Finset.prod_mul_distrib]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 191–194 at 841d05f96fd2 ↗
  hpf.lag.s9 calc
  Closes the goal.
  Uses library Finset.prod_congr, rfl, Finset.prod_inv_distrib.
  Lean, lines 195–196
  _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 195–196 at 841d05f96fd2 ↗
2. hpf.hx2 have (2 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m.
  Lean, lines 197–199
  have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 197–199 at 841d05f96fd2 ↗
  hpf.hx2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 198
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 198 at 841d05f96fd2 ↗
  hpf.hx2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ m
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 199
  _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 199 at 841d05f96fd2 ↗
3. hpf.hx0 have (2 : ℝ) ^ m ≠ (0 : ℝ) routine: positivity
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hx0 : (2 : ℝ) ^ m ≠ (0 : ℝ).
  Lean, line 200
  have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 200 at 841d05f96fd2 ↗
4. hpf.hp0 have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hp0 : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ k ≠ (0 : ℝ).
  Lean, line 201
  have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 201 at 841d05f96fd2 ↗
5. hpf.hpx have ∀ (k : ℕ), (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hpx : ∀ (k : ℕ), (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ).
  Lean, lines 202–205
  have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by intro k have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num) nlinarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 202–205 at 841d05f96fd2 ↗
  hpf.hpx.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ k
  Leaves (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m ≠ (0 : ℝ), with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ k.
  Uses library one_le_pow₀.
  Lean, line 204
  have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 204 at 841d05f96fd2 ↗
6. hpf.hinj have Set.InjOn (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hinj : Set.InjOn (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) (↑(Finset.Icc (1 : ℕ) n) : Set ℕ).
  Uses library Set.InjOn, Finset.Icc, Set, pow_right_injective₀.
  Lean, lines 206–211
  have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by intro a _ b _ hab have h : (2 : ℝ) ^ a = 2 ^ b := by have := hab; simp only at this exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 206–211 at 841d05f96fd2 ↗
  hpf.hinj.h have (2 : ℝ) ^ a = (2 : ℝ) ^ b
  Leaves a = b, with h : (2 : ℝ) ^ a = (2 : ℝ) ^ b.
  Uses library mul_left_cancel₀.
  Lean, lines 208–210
  have h : (2 : ℝ) ^ a = 2 ^ b := by have := hab; simp only at this exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 208–210 at 841d05f96fd2 ↗
  hpf.hinj.h.this have (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) a = (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) b
  Leaves (2 : ℝ) ^ a = (2 : ℝ) ^ b, with this : (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) a = (fun (k : ℕ) => (8 / 3 : ℝ) * (2 : ℝ) ^ k) b.
  Lean, line 209
  have := hab; simp only at this
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 209 at 841d05f96fd2 ↗
7. hpf.hdeg have (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).degree < (↑(Finset.Icc (1 : ℕ) n).card : WithBot ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hdeg : (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))).degree < (↑(Finset.Icc (1 : ℕ) n).card : WithBot ℕ).
  Uses library Finset.Icc, Polynomial.X, Polynomial.C, WithBot, Polynomial.degree_neg, Polynomial.degree_prod, Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc, nsmul_one.
  Lean, lines 212–216
  have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by rw [Polynomial.degree_neg, Polynomial.degree_prod] simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc] rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one] exact_mod_cast (by omega : n - 1 < n)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 212–216 at 841d05f96fd2 ↗
8. hpf.hL have Polynomial.eval ((2 : ℝ) ^ m) (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hL : Polynomial.eval ((2 : ℝ) ^ m) (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, Polynomial.eval ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (Polynomial.X - (Polynomial.C : ℝ → Polynomial ℝ) ((2 : ℝ) ^ t))) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Polynomial.X, Polynomial.C.
  Lean, lines 217–218
  have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 217–218 at 841d05f96fd2 ↗
9. hpf.hcard have (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hcard : (Finset.Icc (1 : ℕ) (n - (1 : ℕ))).card = n - (1 : ℕ).
  Uses library Finset.Icc, Nat.card_Icc.
  Lean, line 220
  have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 220 at 841d05f96fd2 ↗
10. hpf.hgeo have ∀ (u : ℝ), u ≠ (0 : ℝ) → (1 : ℝ) - u ≠ (0 : ℝ) → ∀ (N : ℕ), ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hgeo : ∀ (u : ℝ), u ≠ (0 : ℝ) → (1 : ℝ) - u ≠ (0 : ℝ) → ∀ (N : ℕ), ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Finset.Icc.
  Lean, lines 222–230
  have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by intro u hu hu1 N induction N with | zero => simp | succ N ih => rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih] have hN : u ^ N ≠ 0 := pow_ne_zero _ hu field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 222–230 at 841d05f96fd2 ↗
  hpf.hgeo.s1 case routine: simp
  Leaves ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (0 : ℕ), (u ^ i)⁻¹ = (u ^ (0 : ℕ))⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Nat.zero.
  Lean, line 225
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 225 at 841d05f96fd2 ↗
  hpf.hgeo.s2 case
  Leaves ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (N + (1 : ℕ)), (u ^ i)⁻¹ = (u ^ (N + (1 : ℕ)))⁻¹ * ((1 : ℝ) - u)⁻¹, with N : ℕ, ih : ((1 : ℝ) - u)⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹.
  Uses library Nat.succ, Finset.sum_Icc_succ_top, add_assoc.
  Lean, lines 226–230
  | succ N ih => rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih] have hN : u ^ N ≠ 0 := pow_ne_zero _ hu field_simp ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 226–230 at 841d05f96fd2 ↗
  hpf.hgeo.s2.hN have u ^ N ≠ (0 : ℝ)
  Leaves (u ^ N)⁻¹ * ((1 : ℝ) - u)⁻¹ + (u ^ (N + (1 : ℕ)))⁻¹ = (u ^ (N + (1 : ℕ)))⁻¹ * ((1 : ℝ) - u)⁻¹, with hN : u ^ N ≠ (0 : ℝ).
  Uses library pow_ne_zero.
  Lean, line 228
  have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 228 at 841d05f96fd2 ↗
11. hpf.hz1 have ∀ (k : ℕ), (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ)) = (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz1 : ∀ (k : ℕ), (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ)) = (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹.
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast, div_eq_mul_inv.
  Lean, lines 232–233
  have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 232–233 at 841d05f96fd2 ↗
12. hpf.hz2 have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz2 : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Uses library zpow_add₀, two_ne_zero, zpow_natCast, mul_assoc.
  Lean, lines 234–235
  have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 234–235 at 841d05f96fd2 ↗
13. hpf.hz3 have (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)) = (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m.
  Uses library zpow_add₀, two_ne_zero, zpow_natCast, mul_comm, mul_assoc.
  Lean, lines 236–237
  have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 236–237 at 841d05f96fd2 ↗
14. hpf.hsplit have ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hsplit : ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Finset.prod_Icc_succ_top.
  Lean, lines 238–241
  have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by have hn' : n = n - 1 + 1 := by omega conv_lhs => rw [hn'] rw [Finset.prod_Icc_succ_top (by omega), ← hn']
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 238–241 at 841d05f96fd2 ↗
  hpf.hsplit.hn' have n = n - (1 : ℕ) + (1 : ℕ) routine: omega
  Leaves ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ n * (2 : ℝ) ^ m)⁻¹, with hn' : n = n - (1 : ℕ) + (1 : ℕ).
  Lean, line 239
  have hn' : n = n - 1 + 1 := by omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 239 at 841d05f96fd2 ↗
15. hpf.hfac have ∀ (k : ℕ), (1 : ℝ) - (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hfac : ∀ (k : ℕ), (1 : ℝ) - (2 : ℝ) ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k).
  Uses library mul_sub, inv_mul_cancel₀.
  Lean, lines 242–243
  have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by rw [mul_sub, inv_mul_cancel₀ hx0]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 242–243 at 841d05f96fd2 ↗
16. hpf.hLHS have -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹)
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hLHS : -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹).
  Uses library Finset.Icc, Finset.prod_mul_distrib, Finset.prod_congr, rfl, Finset.prod_const, inv_pow.
  Lean, lines 244–248
  have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by simp only [hz1, hz2, hz3] rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 244–248 at 841d05f96fd2 ↗
17. hpf.hpk have ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hpk : ∀ (k : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m) = (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Uses library pow_add, mul_assoc.
  Lean, line 250
  have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 250 at 841d05f96fd2 ↗
18. hpf.hswap have ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ i)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hswap : ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ i)⁻¹.
  Uses library Finset.Icc, Finset.sum_mul, Finset.mul_sum, Finset.sum_comm, Finset.sum_congr, rfl, mul_pow, mul_inv.
  Lean, lines 251–255
  have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by simp only [Finset.sum_mul, Finset.mul_sum] rw [Finset.sum_comm] refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_)) rw [mul_pow, mul_inv]; ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 251–255 at 841d05f96fd2 ↗
19. hpf.hRHS have ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hRHS : ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹.
  Uses library Finset.Icc, Finset.sum_add_distrib, Finset.sum_congr, rfl, mul_add, mul_ne_zero.
  Lean, lines 256–261
  have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by simp only [hpk] rw [hswap, ← Finset.sum_add_distrib] refine Finset.sum_congr rfl (fun k hk => ?_) rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 256–261 at 841d05f96fd2 ↗
20. hpf.hcoef have ∀ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹, with hcoef : ∀ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹.
  Uses library Finset.Icc, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, inv_pow.
  Lean, lines 263–273
  have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by intro k hk have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring, inv_mul_cancel₀ (hp0 k), mul_one] have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow] simp only [hw] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 263–273 at 841d05f96fd2 ↗
  hpf.hcoef.hq have ∀ (t : ℕ), ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))
  Leaves ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹, with hq : ∀ (t : ℕ), ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)).
  Uses library mul_sub, mul_one, pow_add, inv_mul_cancel₀.
  Lean, lines 265–267
  have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring, inv_mul_cancel₀ (hp0 k), mul_one]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 265–267 at 841d05f96fd2 ↗
  hpf.hcoef.hw have ∀ (j : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ))
  Leaves ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ (n - (1 : ℕ)))⁻¹ = (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (((8 / 3 : ℝ) * (2 : ℝ) ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k)))⁻¹, with hw : ∀ (j : ℕ), (8 / 3 : ℝ) * (2 : ℝ) ^ j / ((8 / 3 : ℝ) * (2 : ℝ) ^ k) = (2 : ℝ) ^ ((↑j : ℤ) - (↑k : ℤ)).
  Uses library zpow_sub₀, two_ne_zero, zpow_natCast.
  Lean, lines 268–270
  have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast] field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 268–270 at 841d05f96fd2 ↗
21. hpf.s21 calc -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = (((2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹)
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 274
  calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 274 at 841d05f96fd2 ↗
22. hpf.s22 calc
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 275
  _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 275 at 841d05f96fd2 ↗
23. hpf.s23 calc
  Closes the goal.
  Uses library Finset.Icc, Finset.mul_sum, Finset.sum_congr, rfl, mul_pow, mul_inv.
  Lean, lines 276–280
  _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [Finset.mul_sum] refine Finset.sum_congr rfl (fun k hk => ?_) rw [← hcoef k hk, mul_pow, mul_inv] ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 276–280 at 841d05f96fd2 ↗
24. hpf.s24 calc ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m) ^ (n - (1 : ℕ)))⁻¹ * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m)⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
  Closes the goal.
  Uses library Finset.Icc.
  Lean, line 281
  _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 281 at 841d05f96fd2 ↗
hres have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ)))
Leaves ∃ (a : ℕ → ℤ) (b : ℕ → ℤ), ∀ (n : ℕ), (1 : ℕ) ≤ n → (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) = (9 : ℝ) ^ n * (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) * ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹, with hres : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))).
Uses library Finset.Icc, zero_add, tsum_congr, Summable.tsum_finsetSum, Finset.mem_Icc, Finset.sum_congr, rfl, tsum_mul_left, mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv.
Lean, lines 283–359
```
  have hres : ∀ n : ℕ, 1 ≤ n → (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    intro n hn
    have pf := hpf n hn
    -- Summability of the two families of pieces.
    have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by
      intro k
      refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_)
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
      have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
      rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div]
      exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
    have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by
      intro i hi
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
      have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
      have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by
        intro m
        rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
      rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1]
      have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
      have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
      field_simp
    have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by
      intro i hi
      by_contra hns
      have := tsum_eq_zero_of_not_summable hns
      rw [hGeo i hi] at this
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
      contradiction
    have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by
      simpa using hT 0
    have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      induction k with
      | zero => simp
      | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
    have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      have hs := hz0.sum_add_tsum_nat_add k
      rw [hIcc k] at hs
      rw [← hs, add_sub_cancel_left]
      exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
    -- The series E(n) summed from m = 1: its first n - 1 terms vanish.
    have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) :=
      summable_sum (fun k _ => (hT k).mul_left _)
    have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) :=
      summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
    have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
    have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) :=
      hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
    have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
    have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by
      refine Finset.sum_eq_zero (fun i hi => ?_)
      have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by
        rw [Finset.mem_range] at hi
        rw [Finset.mem_Icc]; omega
      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_)
      simp
    rw [hzero, zero_add] at hshift
    have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by
      rw [← hshift]
      exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
    rw [hE, tsum_congr (fun m => pf (m + 1) (by omega)), hRs1.tsum_add hRs2,
      Summable.tsum_finsetSum (fun k _ => (hT k).mul_left _),
      Summable.tsum_finsetSum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _),
      Finset.sum_congr (s₁ := Finset.Icc 1 n) rfl (fun k _ => by rw [tsum_mul_left, hTail k]),
      Finset.sum_congr (s₁ := Finset.Icc 1 (n - 1)) rfl (fun i hi => by rw [tsum_mul_left, hGeo i (Finset.mem_Icc.mp hi).1])]
    simp only [mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv]
    ring
```
targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 283–359 at 841d05f96fd2 ↗
1. hres.pf have ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with pf : ∀ (m : ℕ), (1 : ℕ) ≤ m → -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑m : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑m : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑m : ℤ)))⁻¹ = ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹.
  Lean, line 285
  have pf := hpf n hn
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 285 at 841d05f96fd2 ↗
2. hres.hT have ∀ (k : ℕ), Summable fun (m : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hT : ∀ (k : ℕ), Summable fun (m : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹.
  Uses library Summable, Summable.of_norm_bounded, summable_geometric_of_lt_one, Real.norm_eq_abs, abs_inv, abs_of_neg, one_div_pow, one_div, inv_anti₀.
  Lean, lines 287–295
  have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by intro k refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_) have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num) have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div] exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 287–295 at 841d05f96fd2 ↗
  hres.hT.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ k
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ k.
  Uses library one_le_pow₀.
  Lean, line 290
  have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 290 at 841d05f96fd2 ↗
  hres.hT.h2 have (1 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with h2 : (1 : ℝ) ≤ (2 : ℝ) ^ m.
  Uses library one_le_pow₀.
  Lean, line 291
  have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 291 at 841d05f96fd2 ↗
  hres.hT.he have (2 : ℝ) ^ (k + (m + (1 : ℕ))) = (2 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m routine: ring
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with he : (2 : ℝ) ^ (k + (m + (1 : ℕ))) = (2 : ℝ) * (2 : ℝ) ^ k * (2 : ℝ) ^ m.
  Lean, line 292
  have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 292 at 841d05f96fd2 ↗
  hres.hT.hneg have (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))) < (0 : ℝ)
  Leaves ‖((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹‖ ≤ (1 / 2 : ℝ) ^ m, with hneg : (1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))) < (0 : ℝ).
  Lean, line 293
  have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 293 at 841d05f96fd2 ↗
3. hres.hGeo have ∀ (i : ℕ), (1 : ℕ) ≤ i → ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGeo : ∀ (i : ℕ), (1 : ℕ) ≤ i → ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹.
  Uses library tsum_congr, tsum_mul_left, tsum_geometric_of_lt_one.
  Lean, lines 296–309
  have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by intro i hi have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith) have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by intro m rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm] rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1] have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity field_simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 296–309 at 841d05f96fd2 ↗
  hres.hGeo.h2 have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h2 : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 298–300
  have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 298–300 at 841d05f96fd2 ↗
  hres.hGeo.h2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 299
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 299 at 841d05f96fd2 ↗
  hres.hGeo.h2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 300
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 300 at 841d05f96fd2 ↗
  hres.hGeo.hr0 have (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ routine: positivity
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹.
  Lean, line 301
  have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 301 at 841d05f96fd2 ↗
  hres.hGeo.hr1 have ((2 : ℝ) ^ i)⁻¹ < (1 : ℝ)
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hr1 : ((2 : ℝ) ^ i)⁻¹ < (1 : ℝ).
  Uses library inv_lt_one_of_one_lt₀.
  Lean, line 302
  have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 302 at 841d05f96fd2 ↗
  hres.hGeo.hc have ∀ (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * ((2 : ℝ) ^ i)⁻¹ ^ m
  Leaves ∑' (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with hc : ∀ (m : ℕ), (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * ((2 : ℝ) ^ i)⁻¹ ^ m.
  Uses library pow_succ', inv_pow, pow_mul, mul_comm.
  Lean, lines 303–305
  have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by intro m rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 303–305 at 841d05f96fd2 ↗
  hres.hGeo.h3 have (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ) routine: linarith
  Leaves ((2 : ℝ) ^ i)⁻¹ * ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h3 : (2 : ℝ) ^ i - (1 : ℝ) ≠ (0 : ℝ).
  Lean, line 307
  have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 307 at 841d05f96fd2 ↗
  hres.hGeo.h4 have (2 : ℝ) ^ i ≠ (0 : ℝ) routine: positivity
  Leaves ((2 : ℝ) ^ i)⁻¹ * ((1 : ℝ) - ((2 : ℝ) ^ i)⁻¹)⁻¹ = ((2 : ℝ) ^ i - (1 : ℝ))⁻¹, with h4 : (2 : ℝ) ^ i ≠ (0 : ℝ).
  Lean, line 308
  have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 308 at 841d05f96fd2 ↗
4. hres.hGs have ∀ (i : ℕ), (1 : ℕ) ≤ i → Summable fun (m : ℕ) => (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGs : ∀ (i : ℕ), (1 : ℕ) ≤ i → Summable fun (m : ℕ) => (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable.
  Lean, lines 310–319
  have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by intro i hi by_contra hns have := tsum_eq_zero_of_not_summable hns rw [hGeo i hi] at this have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith) contradiction
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 310–319 at 841d05f96fd2 ↗
  hres.hGs.s1 have ∑' (b : ℕ), (((2 : ℝ) ^ (b + (1 : ℕ))) ^ i)⁻¹ = (0 : ℝ)
  Leaves False, with this : ∑' (b : ℕ), (((2 : ℝ) ^ (b + (1 : ℕ))) ^ i)⁻¹ = (0 : ℝ).
  Uses library tsum_eq_zero_of_not_summable.
  Lean, line 313
  have := tsum_eq_zero_of_not_summable hns
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 313 at 841d05f96fd2 ↗
  hres.hGs.h2 have (2 : ℝ) ≤ (2 : ℝ) ^ i
  Leaves False, with h2 : (2 : ℝ) ≤ (2 : ℝ) ^ i.
  Lean, lines 315–317
  have h2 : (2 : ℝ) ≤ 2 ^ i := by calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 315–317 at 841d05f96fd2 ↗
  hres.hGs.h2.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 316
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 316 at 841d05f96fd2 ↗
  hres.hGs.h2.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ i
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 317
  _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 317 at 841d05f96fd2 ↗
  hres.hGs.s3 have ((2 : ℝ) ^ i - (1 : ℝ))⁻¹ ≠ (0 : ℝ)
  Leaves False, with this : ((2 : ℝ) ^ i - (1 : ℝ))⁻¹ ≠ (0 : ℝ).
  Uses library inv_ne_zero.
  Lean, line 318
  have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 318 at 841d05f96fd2 ↗
5. hres.hz0 have Summable fun (j : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hz0 : Summable fun (j : ℕ) => ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹.
  Uses library Summable.
  Lean, lines 320–321
  have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by simpa using hT 0
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 320–321 at 841d05f96fd2 ↗
6. hres.hIcc have ∀ (k : ℕ), ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hIcc : ∀ (k : ℕ), ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.range, Finset.Icc.
  Lean, lines 322–326
  have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by intro k induction k with | zero => simp | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 322–326 at 841d05f96fd2 ↗
  hres.hIcc.s1 case routine: simp
  Leaves ∑ i ∈ Finset.range (0 : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) (0 : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.zero.
  Lean, line 325
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 325 at 841d05f96fd2 ↗
  hres.hIcc.s2 case
  Leaves ∑ i ∈ Finset.range (k + (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) (k + (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with k : ℕ, ih : ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ = ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Nat.succ, Finset.sum_range_succ, Finset.sum_Icc_succ_top.
  Lean, line 326
  | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 326 at 841d05f96fd2 ↗
7. hres.hTail have ∀ (k : ℕ), ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hTail : ∀ (k : ℕ), ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹.
  Uses library Finset.Icc, add_sub_cancel_left, tsum_congr.
  Lean, lines 327–332
  have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by intro k have hs := hz0.sum_add_tsum_nat_add k rw [hIcc k] at hs rw [← hs, add_sub_cancel_left] exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 327–332 at 841d05f96fd2 ↗
  hres.hTail.hs have ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ + ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + k + (1 : ℕ)))⁻¹ = ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹
  Leaves ∑' (m : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ = ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹, with hs : ∑ i ∈ Finset.range k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹ + ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + k + (1 : ℕ)))⁻¹ = ∑' (i : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + (1 : ℕ)))⁻¹.
  Lean, line 329
  have hs := hz0.sum_add_tsum_nat_add k
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 329 at 841d05f96fd2 ↗
8. hres.hRs1 have Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs1 : Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹.
  Uses library Summable, Finset.Icc, summable_sum.
  Lean, lines 334–335
  have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := summable_sum (fun k _ => (hT k).mul_left _)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 334–335 at 841d05f96fd2 ↗
9. hres.hRs2 have Summable fun (m : ℕ) => ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs2 : Summable fun (m : ℕ) => ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable, Finset.Icc, summable_sum, Finset.mem_Icc.
  Lean, lines 336–337
  have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 336–337 at 841d05f96fd2 ↗
10. hres.hRs have Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hRs : Summable fun (m : ℕ) => ∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + (1 : ℕ))))⁻¹ + ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + (1 : ℕ))) ^ i)⁻¹.
  Uses library Summable, Finset.Icc.
  Lean, line 338
  have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 338 at 841d05f96fd2 ↗
11. hres.hGsum have Summable fun (m : ℕ) => -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hGsum : Summable fun (m : ℕ) => -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹.
  Uses library Summable, Finset.Icc.
  Lean, lines 339–340
  have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) := hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 339–340 at 841d05f96fd2 ↗
12. hres.hshift have ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ + ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ)))⁻¹ = ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hshift : ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ + ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (n - (1 : ℕ)) + (1 : ℕ)) : ℤ)))⁻¹ = ∑' (i : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹.
  Lean, line 341
  have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, line 341 at 841d05f96fd2 ↗
13. hres.hzero have ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ)
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hzero : ∑ i ∈ Finset.range (n - (1 : ℕ)), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ).
  Uses library Finset.range, Finset.Icc, Finset.sum_eq_zero, mul_eq_zero_of_right, Finset.prod_eq_zero.
  Lean, lines 342–348
  have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by refine Finset.sum_eq_zero (fun i hi => ?_) have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by rw [Finset.mem_range] at hi rw [Finset.mem_Icc]; omega refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_) simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 342–348 at 841d05f96fd2 ↗
  hres.hzero.hi' have i + (1 : ℕ) ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ))
  Leaves -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(i + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(i + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(i + (1 : ℕ)) : ℤ)))⁻¹ = (0 : ℝ), with hi' : i + (1 : ℕ) ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)).
  Uses library Finset.Icc, Finset.mem_range, Finset.mem_Icc.
  Lean, lines 344–346
  have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by rw [Finset.mem_range] at hi rw [Finset.mem_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 344–346 at 841d05f96fd2 ↗
14. hres.hE have ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = ∑' (m : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹
  Leaves ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = (∑ k ∈ Finset.Icc (1 : ℕ) n, (-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * ∑ i ∈ Finset.Icc (1 : ℕ) k, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ - ∑ i ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), (∑ k ∈ Finset.Icc (1 : ℕ) n, ((-∏ t ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc (1 : ℕ) n).erase k, ((1 : ℝ) - (2 : ℝ) ^ ((↑l : ℤ) - (↑k : ℤ)))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - (1 : ℝ))), with hE : ∑' (j : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ = ∑' (m : ℕ), -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(m + (1 : ℕ)) : ℤ) + (↑n : ℤ)))⁻¹ * ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)), ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(m + (1 : ℕ)) : ℤ))) * ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(m + (1 : ℕ)) : ℤ)))⁻¹.
  Uses library Finset.Icc, tsum_congr.
  Lean, lines 350–352
  have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by rw [← hshift] exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/attempts/20260929T194647Z-t0929-1-partial.lean, lines 350–352 at 841d05f96fd2 ↗

import Mathlib
import Nodes.«erdos-1050--h1-v2--h2».Context

/-! Hole `integrality` of a merged partial proof, as a node (D-12 #5, D-29).
Statement generated by the gate from the assembly; immutable (D-3). -/

theorem erdos_1050__h1_v2__h2 : ∃ (a : ℕ → ℤ) (b : ℕ → ℤ),
  ∀ (n : ℕ),
    (1 : ℕ) ≤ n →
      (↑(b n) : ℝ) * ∑' (j : ℕ), ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + (1 : ℕ)))⁻¹ - (↑(a n) : ℝ) =
        (9 : ℝ) ^ n *
            (((↑(n - (2 : ℕ)).factorial : ℝ) * ∏ k ∈ Finset.Icc (1 : ℕ) n, ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) *
              ∏ k ∈ Finset.Icc ((n + (1 : ℕ)) / (2 : ℕ)) n, ((1 : ℝ) - (2 : ℝ) ^ k)) *
          ∑' (j : ℕ),
            -((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑(n + j) : ℤ) + (↑n : ℤ)))⁻¹ *
              ∏ k ∈ Finset.Icc (1 : ℕ) (n - (1 : ℕ)),
                ((1 : ℝ) - (2 : ℝ) ^ ((↑k : ℤ) - (↑(n + j) : ℤ))) *
                  ((1 : ℝ) - (8 / 3 : ℝ) * (2 : ℝ) ^ ((↑k : ℤ) + (↑(n + j) : ℤ)))⁻¹ := by
  -- E(n) = beta0(n) * z - alpha0(n) by partial fractions in x = 2^m (proved below, via Lagrange
  -- interpolation); b(n) = 9^n W(n) beta0(n) is an integer (proved below); a(n) = 9^n W(n) alpha0(n)
  -- is the one hole. Informal argument: annex 959e0d5b10994049b5f27ad77d9d2d8535be9f817bf4de69d747a1e05b9db44b on this node.
  -- Borwein Lemma 3 (numerator integrality): the one hole.
  have numerator_integral : ∃ a : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (a n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := sorry
  obtain ⟨a, ha⟩ := numerator_integral
  -- Denominator integrality (proved): 3^(n-1) beta0(n) is a coefficient of an integer polynomial.
  have hden : ∃ b : ℕ → ℤ, ∀ n : ℕ, 1 ≤ n → (b n : ℝ) = (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
    have key : ∀ n : ℕ, 1 ≤ n → ∃ z : ℤ, (z : ℝ) = (3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) := by
      intro n hn
      have hr0 : ∀ k : ℕ, ((8 : ℝ) * 2 ^ k) ≠ 0 := fun k => by positivity
      have hvinj : Set.InjOn (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹) (Finset.Icc 1 n : Set ℕ) := by
        intro a _ b _ hab
        have h : (2 : ℝ) ^ a = 2 ^ b := by
          have := inv_injective hab
          exact mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this
        exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
      -- (1) the polynomial partial-fraction identity, by agreement at the n nodes 1/r_k
      have hid : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) = ∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) := by
        refine Polynomial.eq_of_natDegree_lt_card_of_eval_eq' _ _ ((Finset.Icc 1 n).image (fun k : ℕ => ((8 : ℝ) * 2 ^ k)⁻¹)) ?_ ?_
        · intro y hy
          obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
          rw [Polynomial.eval_finsetSum, Finset.sum_eq_single_of_mem k hk]
          · rw [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod]
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            have hne : ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) ≠ 0 := by
              refine Finset.prod_ne_zero_iff.mpr (fun j hj => ?_)
              have hjk : j ≠ k := Finset.ne_of_mem_erase hj
              intro h0
              have : ((8 : ℝ) * 2 ^ j) = ((8 : ℝ) * 2 ^ k) := by
                have := hr0 k
                field_simp at h0
                linarith
              exact hjk (pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) (mul_left_cancel₀ (by norm_num : (8 : ℝ) ≠ 0) this))
            rw [mul_assoc, ← Finset.prod_congr rfl (fun j _ => by rw [div_eq_mul_inv, mul_comm (((8 : ℝ) * 2 ^ j)) _] : ∀ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)) = 1 - ((8 : ℝ) * 2 ^ j) * ((8 : ℝ) * 2 ^ k)⁻¹), inv_mul_cancel₀ hne, mul_one]
          · intro i hi hik
            rw [Polynomial.eval_mul, Polynomial.eval_prod]
            refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hik, hk⟩) ?_)
            simp only [Polynomial.eval_sub, Polynomial.eval_one, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_X]
            rw [mul_inv_cancel₀ (hr0 k), sub_self]
        · rw [Finset.card_image_of_injOn hvinj, Nat.card_Icc]
          have hQdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).natDegree ≤ n - 1 := by
            rw [Polynomial.natDegree_neg]
            refine (Polynomial.natDegree_prod_le _ _).trans ?_
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun t _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
          have hRdeg : (∑ k ∈ Finset.Icc 1 n, Polynomial.C ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X))).natDegree ≤ n - 1 := by
            refine Polynomial.natDegree_sum_le_of_forall_le _ _ (fun k hk => ?_)
            refine (Polynomial.natDegree_C_mul_le _ _).trans ((Polynomial.natDegree_prod_le _ _).trans ?_)
            refine (Finset.sum_le_sum (g := fun _ => 1) (fun j _ => ?_)).trans (le_of_eq ?_)
            · compute_degree
            · simp only [Finset.sum_const, smul_eq_mul, mul_one]
              rw [Finset.card_erase_of_mem hk, Nat.card_Icc]; omega
          omega
      -- (2) coefficient n - 1 of T_k * P is r_k ^ (n - 1)
      have hU : ∀ (S : Finset ℕ), ∀ d < n, (∏ j ∈ S, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)).coeff d = if d = 0 then 1 else 0 := by
        intro S
        induction S using Finset.induction_on with
        | empty => intro d hd; simp [Polynomial.coeff_one]
        | insert a S ha ih =>
          intro d hd
          rw [Finset.prod_insert ha, sub_mul, one_mul, Polynomial.coeff_sub, mul_pow, ← Polynomial.C_pow, mul_assoc,
            Polynomial.coeff_C_mul, Polynomial.coeff_X_pow_mul', if_neg (by omega), mul_zero, sub_zero, ih d hd]
      have hG : ∀ k : ℕ, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k
        simp only [mul_pow, ← Polynomial.C_pow, Polynomial.finsetSum_coeff, Polynomial.coeff_C_mul_X_pow]
        rw [Finset.sum_ite_eq (Finset.range n) (n - 1), if_pos (Finset.mem_range.mpr (by omega))]
      have hTP : ∀ k ∈ Finset.Icc 1 n, ((∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        intro k hk
        have hsplitP : (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i) :=
          (Finset.mul_prod_erase _ (fun j => (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) hk).symm
        have hprod : (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X)) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ k) * Polynomial.X) ^ i) * ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) := by
          rw [hsplitP, mul_left_comm, ← Finset.prod_mul_distrib]
          congr 1
          refine Finset.prod_congr rfl (fun j _ => ?_)
          rw [mul_neg_geom_sum]
        obtain ⟨E, hE⟩ : Polynomial.X ^ n ∣ (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n)) - 1 :=
          Polynomial.X_pow_dvd_iff.mpr (fun d hd => by
          rw [Polynomial.coeff_sub, hU _ d hd, Polynomial.coeff_one]; split_ifs <;> simp_all)
        rw [hprod, show ∏ j ∈ (Finset.Icc 1 n).erase k, (1 - (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ n) = 1 + Polynomial.X ^ n * E by rw [← hE]; ring,
          mul_add, mul_one, Polynomial.coeff_add, mul_left_comm, Polynomial.coeff_X_pow_mul', if_neg (by omega), add_zero, hG]
      -- (3) so coefficient n - 1 of Qt * P is the sum of C_k r_k^(n-1)
      have hQP : ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) = ∑ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) := by
        nth_rw 1 [hid]
        rw [Finset.sum_mul, Polynomial.finsetSum_coeff]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [mul_assoc, Polynomial.coeff_C_mul, hTP k hk]
      -- (4) and it is an integer, since Qt * P has integer coefficients
      have hmap : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, (∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℝ) * 2 ^ j) * Polynomial.X) ^ i)) = Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)) := by
        simp [Polynomial.map_prod, Polynomial.map_sum, Polynomial.C_mul, Polynomial.C_pow, map_ofNat]
      -- (5) C_k r_k^(n-1) = 3^(n-1) A(n,k)
      have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
      have hterm : ∀ k ∈ Finset.Icc 1 n, ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℝ) * Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).eval ((8 : ℝ) * 2 ^ k)⁻¹ * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k)))⁻¹) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = (3 : ℝ) ^ (n - 1) * (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) := by
        intro k hk
        have hq : ∀ t : ℕ, (3 : ℝ) * ((8 : ℝ) * 2 ^ k)⁻¹ - 2 ^ t = ((8 : ℝ) * 2 ^ k)⁻¹ * (3 * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) := fun t => by
          have := hr0 k
          rw [pow_add]; field_simp
        have hw : ∀ j : ℕ, ((8 : ℝ) * 2 ^ j) / ((8 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
          rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
          field_simp
        have hrr : (((8 : ℝ) * 2 ^ k)⁻¹) ^ (n - 1) * ((8 : ℝ) * 2 ^ k) ^ (n - 1) = 1 := by
          rw [← mul_pow, inv_mul_cancel₀ (hr0 k), one_pow]
        simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_mul, Polynomial.eval_C,
          Polynomial.eval_X]
        rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
          Finset.prod_const, hcard]
        simp only [hw]
        linear_combination (-((3 : ℝ) ^ (n - 1) * ∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) *
          (∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - (k : ℤ))))⁻¹) * hrr
      refine ⟨((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1), ?_⟩
      have hc : ((((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i)).coeff (n - 1) : ℤ) : ℝ) = (Polynomial.map (Int.castRingHom ℝ) ((-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.C (3 : ℤ) * Polynomial.X - Polynomial.C ((2 : ℤ) ^ t))) * (∏ j ∈ Finset.Icc 1 n, ∑ i ∈ Finset.range n, (Polynomial.C ((8 : ℤ) * 2 ^ j) * Polynomial.X) ^ i))).coeff (n - 1) := by
        rw [Polynomial.coeff_map]; simp
      rw [hc, ← hmap, hQP, Finset.mul_sum]
      exact Finset.sum_congr rfl hterm
    have hW3 : ∀ n : ℕ, ∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) * ((3 : ℝ) ^ n)⁻¹ := by
      intro n
      rw [Finset.prod_congr rfl (fun k _ => (by ring : (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k) = ((3 : ℝ) - 8 * 2 ^ k) * (3 : ℝ)⁻¹)),
        Finset.prod_mul_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, inv_pow]
    refine ⟨fun n => if h : 1 ≤ n then 3 * (Nat.factorial (n - 2) : ℤ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℤ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℤ) - 2 ^ k)) * Classical.choose (key n h) else 0, fun n hn => ?_⟩
    simp only [dif_pos hn]
    push_cast
    rw [Classical.choose_spec (key n hn), hW3 n]
    have h9 : (9 : ℝ) ^ n = 3 * (3 : ℝ) ^ (n - 1) * (3 : ℝ) ^ n := by
      rw [show (9 : ℝ) = 3 * 3 by norm_num, mul_pow, mul_comm (3 : ℝ) _, ← pow_succ, Nat.sub_add_cancel hn]
    have h3 : ((3 : ℝ) ^ n)⁻¹ * (3 : ℝ) ^ n = 1 := inv_mul_cancel₀ (by positivity)
    rw [h9]
    linear_combination (-(3 : ℝ) * (Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, ((3 : ℝ) - 8 * 2 ^ k)) *
        (∏ k ∈ Finset.Icc ((n + 1) / 2) n, ((1 : ℝ) - 2 ^ k)) * ((3 : ℝ) ^ (n - 1) * (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))))) * h3
  obtain ⟨b, hb⟩ := hden
  -- The partial-fraction identity at x = 2^m (proved).
  have hpf : ∀ n : ℕ, 1 ≤ n → ∀ m : ℕ, 1 ≤ m → (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := by
    intro n hn m hm
    -- General partial fractions, from Mathlib's Lagrange interpolation.
    have lag : ∀ (s : Finset ℕ) (p : ℕ → ℝ) (Q : Polynomial ℝ), (∀ k ∈ s, p k ≠ 0) → Set.InjOn p s →
      Q.degree < s.card → ∀ x : ℝ, (∀ k ∈ s, 1 - p k * x ≠ 0) →
      Q.eval x * ∏ k ∈ s, (1 - p k * x)⁻¹ =
        ∑ k ∈ s, Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
      intro s p Q hp hinj hdeg x hx
      have hinj' : Set.InjOn (fun k => (p k)⁻¹) s := by
        intro a ha b hb hab
        exact hinj ha hb (inv_injective hab)
      have hxv : ∀ k ∈ s, x ≠ (p k)⁻¹ := by
        intro k hk hxk
        apply hx k hk
        rw [hxk, mul_inv_cancel₀ (hp k hk), sub_self]
      have hQ := Lagrange.eq_interpolate hinj' hdeg
      have hev := Lagrange.eval_interpolate_not_at_node (s := s) (v := fun k => (p k)⁻¹) (fun i => Q.eval (p i)⁻¹) hxv
      rw [← hQ, Lagrange.eval_nodal] at hev
      have hfac : ∀ k ∈ s, (1 - p k * x)⁻¹ = (x - (p k)⁻¹)⁻¹ * (-(p k)⁻¹) := by
        intro k hk
        have h1 := hp k hk
        have h3 : (x - (p k)⁻¹) * (-(p k)) = 1 - p k * x := by
          rw [sub_mul, mul_neg, mul_neg, inv_mul_cancel₀ h1]; ring
        rw [← h3, mul_inv, inv_neg]
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, hev]
      have hne : ∏ i ∈ s, (x - (p i)⁻¹) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => sub_ne_zero.mpr (hxv i hi))
      rw [Finset.prod_inv_distrib, mul_assoc, ← mul_assoc (∑ i ∈ s, _), mul_comm (∑ i ∈ s, _), ← mul_assoc, ← mul_assoc,
        mul_inv_cancel₀ hne, one_mul, Finset.sum_mul]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [Lagrange.nodalWeight, ← Finset.mul_prod_erase s (fun i => -(p i)⁻¹) hk]
      have hj : ∀ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹ = (1 - p j / p k)⁻¹ := by
        intro j hj
        have hjs := Finset.mem_of_mem_erase hj
        have hjk := Finset.ne_of_mem_erase hj
        have hpj := hp j hjs
        have hpk := hp k hk
        have hne : p j ≠ p k := fun h => hjk (hinj hjs hk h)
        have hsub : p k - p j ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
        field_simp
        ring
      calc (∏ j ∈ s.erase k, ((p k)⁻¹ - (p j)⁻¹)⁻¹) * (x - (p k)⁻¹)⁻¹ * Q.eval (p k)⁻¹ *
            (-(p k)⁻¹ * ∏ j ∈ s.erase k, -(p j)⁻¹)
          = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (((p k)⁻¹ - (p j)⁻¹)⁻¹ * -(p j)⁻¹)) * ((x - (p k)⁻¹)⁻¹ * -(p k)⁻¹) := by
            rw [Finset.prod_mul_distrib]; ring
        _ = Q.eval (p k)⁻¹ * (∏ j ∈ s.erase k, (1 - p j / p k))⁻¹ * (1 - p k * x)⁻¹ := by
            rw [Finset.prod_congr rfl hj, ← Finset.prod_inv_distrib, hfac k hk]
    have hx2 : (2 : ℝ) ≤ (2 : ℝ) ^ m := by
      calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
        _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
    have hx0 : (2 : ℝ) ^ m ≠ 0 := by positivity
    have hp0 : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ k ≠ 0 := fun k => by positivity
    have hpx : ∀ k : ℕ, 1 - (8 / 3 : ℝ) * 2 ^ k * 2 ^ m ≠ 0 := by
      intro k
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      nlinarith
    have hinj : Set.InjOn (fun k : ℕ => (8 / 3 : ℝ) * 2 ^ k) (Finset.Icc 1 n : Set ℕ) := by
      intro a _ b _ hab
      have h : (2 : ℝ) ^ a = 2 ^ b := by
        have := hab; simp only at this
        exact mul_left_cancel₀ (by norm_num : (8 / 3 : ℝ) ≠ 0) this
      exact pow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h
    have hdeg : (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t))).degree < ((Finset.Icc 1 n).card : WithBot ℕ) := by
      rw [Polynomial.degree_neg, Polynomial.degree_prod]
      simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Icc]
      rw [show n - 1 + 1 - 1 = n - 1 by omega, show n + 1 - 1 = n by omega, nsmul_one]
      exact_mod_cast (by omega : n - 1 < n)
    have hL := lag (Finset.Icc 1 n) (fun k => (8 / 3 : ℝ) * 2 ^ k) (-∏ t ∈ Finset.Icc 1 (n - 1), (Polynomial.X - Polynomial.C ((2 : ℝ) ^ t)))
      (fun k _ => hp0 k) hinj hdeg ((2 : ℝ) ^ m) (fun k _ => hpx k)
    simp only [Polynomial.eval_neg, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C] at hL
    have hcard : (Finset.Icc 1 (n - 1)).card = n - 1 := by rw [Nat.card_Icc]; omega
    -- geometric sums
    have hgeo : ∀ u : ℝ, u ≠ 0 → 1 - u ≠ 0 → ∀ N : ℕ, (1 - u)⁻¹ + ∑ i ∈ Finset.Icc 1 N, (u ^ i)⁻¹ = (u ^ N)⁻¹ * (1 - u)⁻¹ := by
      intro u hu hu1 N
      induction N with
      | zero => simp
      | succ N ih =>
        rw [Finset.sum_Icc_succ_top (by omega), ← add_assoc, ih]
        have hN : u ^ N ≠ 0 := pow_ne_zero _ hu
        field_simp
        ring
    -- the left side, in terms of the Lagrange data
    have hz1 : ∀ k : ℕ, (2 : ℝ) ^ ((k : ℤ) - (m : ℤ)) = 2 ^ k * ((2 : ℝ) ^ m)⁻¹ := fun k => by
      rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast, div_eq_mul_inv]
    have hz2 : ∀ k : ℕ, (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℤ)) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_assoc]
    have hz3 : (8 / 3 : ℝ) * (2 : ℝ) ^ ((m : ℤ) + (n : ℤ)) = ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m := by
      rw [zpow_add₀ two_ne_zero, zpow_natCast, zpow_natCast, mul_comm ((2 : ℝ) ^ m), mul_assoc]
    have hsplit : ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹ = (∏ k ∈ Finset.Icc 1 (n - 1), (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) * (1 - ((8 / 3 : ℝ) * 2 ^ n) * (2 : ℝ) ^ m)⁻¹ := by
      have hn' : n = n - 1 + 1 := by omega
      conv_lhs => rw [hn']
      rw [Finset.prod_Icc_succ_top (by omega), ← hn']
    have hfac : ∀ k : ℕ, 1 - 2 ^ k * ((2 : ℝ) ^ m)⁻¹ = ((2 : ℝ) ^ m)⁻¹ * ((2 : ℝ) ^ m - (2 : ℝ) ^ k) := fun k => by
      rw [mul_sub, inv_mul_cancel₀ hx0]; ring
    have hLHS : (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hz1, hz2, hz3]
      rw [Finset.prod_mul_distrib, hsplit, Finset.prod_congr rfl (fun k _ => hfac k), Finset.prod_mul_distrib,
        Finset.prod_const, hcard, inv_pow]
      ring
    -- the right side
    have hpk : ∀ k : ℕ, (8 / 3 : ℝ) * 2 ^ (k + m) = ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m := fun k => by rw [pow_add, mul_assoc]
    have hswap : ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹ = ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ∑ i ∈ Finset.Icc 1 (n - 1), ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ i)⁻¹ := by
      simp only [Finset.sum_mul, Finset.mul_sum]
      rw [Finset.sum_comm]
      refine Finset.sum_congr rfl (fun k _ => Finset.sum_congr rfl (fun i _ => ?_))
      rw [mul_pow, mul_inv]; ring
    have hRHS : ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
      simp only [hpk]
      rw [hswap, ← Finset.sum_add_distrib]
      refine Finset.sum_congr rfl (fun k hk => ?_)
      rw [← mul_add, hgeo _ (mul_ne_zero (hp0 k) hx0) (hpx k) (n - 1)]
      ring
    -- matching the coefficients
    have hcoef : ∀ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * 2 ^ k) ^ (n - 1))⁻¹ = (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ := by
      intro k hk
      have hq : ∀ t : ℕ, ((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t = ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k)) := fun t => by
        rw [mul_sub, mul_one, pow_add, show ((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * (2 ^ t * 2 ^ k)) = 2 ^ t * (((8 / 3 : ℝ) * 2 ^ k)⁻¹ * ((8 / 3 : ℝ) * 2 ^ k)) by ring,
          inv_mul_cancel₀ (hp0 k), mul_one]
      have hw : ∀ j : ℕ, ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k) = (2 : ℝ) ^ ((j : ℤ) - (k : ℤ)) := fun j => by
        rw [zpow_sub₀ two_ne_zero, zpow_natCast, zpow_natCast]
        field_simp
      rw [Finset.prod_congr rfl (fun t _ => hq t), Finset.prod_mul_distrib, Finset.prod_const, hcard, inv_pow]
      simp only [hw]
      ring
    calc (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((m : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (m : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (m : ℕ)))⁻¹) = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * ((-∏ t ∈ Finset.Icc 1 (n - 1), ((2 : ℝ) ^ m - (2 : ℝ) ^ t)) * ∏ k ∈ Finset.Icc 1 n, (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := hLHS
      _ = (((2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (∑ k ∈ Finset.Icc 1 n, (-∏ t ∈ Finset.Icc 1 (n - 1), (((8 / 3 : ℝ) * 2 ^ k)⁻¹ - (2 : ℝ) ^ t)) * (∏ j ∈ (Finset.Icc 1 n).erase k, (1 - ((8 / 3 : ℝ) * 2 ^ j) / ((8 / 3 : ℝ) * 2 ^ k)))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by rw [hL]
      _ = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * ((((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m) ^ (n - 1))⁻¹ * (1 - ((8 / 3 : ℝ) * 2 ^ k) * (2 : ℝ) ^ m)⁻¹) := by
        rw [Finset.mul_sum]
        refine Finset.sum_congr rfl (fun k hk => ?_)
        rw [← hcoef k hk, mul_pow, mul_inv]
        ring
      _ = ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + m))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ m) ^ i)⁻¹) := hRHS.symm
  -- Borwein Lemma 1 (proved): sum over m >= 1; the first n - 1 terms vanish.
  have hres : ∀ n : ℕ, 1 ≤ n → (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k)))) * (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹)) - ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) / ((2 : ℝ) ^ i - 1)) := by
    intro n hn
    have pf := hpf n hn
    -- Summability of the two families of pieces.
    have hT : ∀ k : ℕ, Summable (fun m : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) := by
      intro k
      refine Summable.of_norm_bounded (summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num)) (fun m => ?_)
      have h1 : (1 : ℝ) ≤ 2 ^ k := one_le_pow₀ (by norm_num)
      have h2 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have he : (2 : ℝ) ^ (k + (m + 1)) = 2 * 2 ^ k * 2 ^ m := by ring
      have hneg : (1 : ℝ) - 8 / 3 * 2 ^ (k + (m + 1)) < 0 := by rw [he]; nlinarith
      rw [Real.norm_eq_abs, abs_inv, abs_of_neg hneg, one_div_pow, one_div]
      exact inv_anti₀ (by positivity) (by rw [he]; nlinarith)
    have hGeo : ∀ i : ℕ, 1 ≤ i → ∑' m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i - 1)⁻¹ := by
      intro i hi
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have hr0 : (0 : ℝ) ≤ ((2 : ℝ) ^ i)⁻¹ := by positivity
      have hr1 : ((2 : ℝ) ^ i)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
      have hc : ∀ m : ℕ, (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹ = ((2 : ℝ) ^ i)⁻¹ * (((2 : ℝ) ^ i)⁻¹) ^ m := by
        intro m
        rw [← pow_succ', inv_pow, ← pow_mul, ← pow_mul, mul_comm]
      rw [tsum_congr hc, tsum_mul_left, tsum_geometric_of_lt_one hr0 hr1]
      have h3 : (2 : ℝ) ^ i - 1 ≠ 0 := by linarith
      have h4 : (2 : ℝ) ^ i ≠ 0 := by positivity
      field_simp
    have hGs : ∀ i : ℕ, 1 ≤ i → Summable (fun m : ℕ => (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) := by
      intro i hi
      by_contra hns
      have := tsum_eq_zero_of_not_summable hns
      rw [hGeo i hi] at this
      have h2 : (2 : ℝ) ≤ 2 ^ i := by
        calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
          _ ≤ 2 ^ i := pow_le_pow_right₀ (by norm_num) hi
      have : ((2 : ℝ) ^ i - 1)⁻¹ ≠ 0 := inv_ne_zero (by linarith)
      contradiction
    have hz0 : Summable (fun j : ℕ => (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) := by
      simpa using hT 0
    have hIcc : ∀ k : ℕ, ∑ i ∈ Finset.range k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (i + 1))⁻¹ = ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      induction k with
      | zero => simp
      | succ k ih => rw [Finset.sum_range_succ, ih, Finset.sum_Icc_succ_top (by omega)]
    have hTail : ∀ k : ℕ, ∑' m : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹ = (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) - ∑ i ∈ Finset.Icc 1 k, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ i)⁻¹ := by
      intro k
      have hs := hz0.sum_add_tsum_nat_add k
      rw [hIcc k] at hs
      rw [← hs, add_sub_cancel_left]
      exact tsum_congr (fun m => by rw [show k + (m + 1) = m + k + 1 by omega])
    -- The series E(n) summed from m = 1: its first n - 1 terms vanish.
    have hRs1 : Summable (fun m : ℕ => ∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) :=
      summable_sum (fun k _ => (hT k).mul_left _)
    have hRs2 : Summable (fun m : ℕ => ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹) :=
      summable_sum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _)
    have hRs : Summable (fun m : ℕ => ((∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (k + (m + 1)))⁻¹) + ∑ i ∈ Finset.Icc 1 (n - 1), (∑ k ∈ Finset.Icc 1 n, (-(∏ t ∈ Finset.Icc 1 (n - 1), (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (t + k))) / ∏ l ∈ (Finset.Icc 1 n).erase k, (1 - (2 : ℝ) ^ ((l : ℤ) - k))) * (((8 / 3 : ℝ) * (2 : ℝ) ^ k) ^ i)⁻¹) * (((2 : ℝ) ^ (m + 1)) ^ i)⁻¹)) := hRs1.add hRs2
    have hGsum : Summable (fun m : ℕ => (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹)) :=
      hRs.congr (fun m => (pf (m + 1) (by omega)).symm)
    have hshift := hGsum.sum_add_tsum_nat_add (n - 1)
    have hzero : ∑ i ∈ Finset.range (n - 1), (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((i + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((i + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((i + 1) : ℕ)))⁻¹) = 0 := by
      refine Finset.sum_eq_zero (fun i hi => ?_)
      have hi' : i + 1 ∈ Finset.Icc 1 (n - 1) := by
        rw [Finset.mem_range] at hi
        rw [Finset.mem_Icc]; omega
      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero hi' ?_)
      simp
    rw [hzero, zero_add] at hshift
    have hE : (∑' j : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (((n + j : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - (n + j : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + (n + j : ℕ)))⁻¹)) = ∑' m : ℕ, (-(1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((((m + 1) : ℕ) : ℤ) + n))⁻¹ * ∏ k ∈ Finset.Icc 1 (n - 1), (1 - (2 : ℝ) ^ ((k : ℤ) - ((m + 1) : ℕ))) * (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ ((k : ℤ) + ((m + 1) : ℕ)))⁻¹) := by
      rw [← hshift]
      exact tsum_congr (fun j => by rw [show j + (n - 1) + 1 = n + j by omega])
    rw [hE, tsum_congr (fun m => pf (m + 1) (by omega)), hRs1.tsum_add hRs2,
      Summable.tsum_finsetSum (fun k _ => (hT k).mul_left _),
      Summable.tsum_finsetSum (fun i hi => (hGs i (Finset.mem_Icc.mp hi).1).mul_left _),
      Finset.sum_congr (s₁ := Finset.Icc 1 n) rfl (fun k _ => by rw [tsum_mul_left, hTail k]),
      Finset.sum_congr (s₁ := Finset.Icc 1 (n - 1)) rfl (fun i hi => by rw [tsum_mul_left, hGeo i (Finset.mem_Icc.mp hi).1])]
    simp only [mul_sub, Finset.sum_sub_distrib, Finset.sum_mul, div_eq_mul_inv]
    ring
  refine ⟨a, b, fun n hn => ?_⟩
  linear_combination (∑' j : ℕ, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ (j + 1))⁻¹) * hb n hn - ha n hn - (9 : ℝ) ^ n * ((Nat.factorial (n - 2) : ℝ) * (∏ k ∈ Finset.Icc 1 n, (1 - (8 / 3 : ℝ) * (2 : ℝ) ^ k)) * (∏ k ∈ Finset.Icc ((n + 1) / 2) n, (1 - (2 : ℝ) ^ k))) * hres n hn

Annexes (3), in full

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by t1001-1050b, 2026-10-01T12:18:39Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/annex/448e8c7a99a2e7ff3ff2821ed238a0ba9ad0cdd5bbca9da532566e8be33151fb.md.

Annex to erdos-1050--h1-v2--h2 (integrality of the Padé numerators and denominators for sum 1/(2^n - 3)). How the sorry-free Lean proof submitted on this node is organised, for readers and for anyone reusing its lemmas. The mathematics of the numerator half is in annex a03db62f0576f792454c3f95d14fb02d193662242ec562819da63116009d326b on the child node erdos-1050--h1-v2--h2--h1; the partial-fraction and residue steps (hpf, hres) are unchanged from the earlier partial on this node (annexes 959e0d5b... and ab20b788...).

Notation: for a finite set s of naturals and k in s, Pi(s,k) = prod over l in s minus k of (1 - 2^(l-k)); with s = {1..n} this is the denominator D_k of the Padé coefficients A_k.

Shared lemmas, stated once at the top of the proof (all over the reals, nodes 2^k):

h0: for nonempty s, sum_{k in s} 1/Pi(s,k) = 1. This is Mathlib's Lagrange.sum_basis for the nodes 2^(-k), evaluated at 0.
hrec: for a in s and any weight f, sum_{k in s} 2^k f(k)/Pi(s,k) = sum_{k in s minus a} 2^k f(k)/Pi(s minus a,k) + 2^a sum_{k in s} f(k)/Pi(s,k). (Split off the factor l = a of each product; one line of algebra per term.)
hW: sum_{k in s} (2^k)^e / Pi(s,k) is an integer for every e >= 0 (it is the complete homogeneous symmetric polynomial h_e of the nodes). Induction on |s| + e with hrec, base cases h0 and the empty set.
hV: sum_{k in s} (2^k)^(-m) / Pi(s,k) = 0 for 1 <= m < |s|. Induction on m with hrec and h0.
hpoly: prod_{t=1..n-1} (1 - 2^t y) = sum_{j<n} alpha_j y^j with integer alpha_j (coefficients of an integer polynomial of degree < n).
hgeo: sum_{i=1..k} 2^(-im) = (1 - 2^(-km))/(2^m - 1); hdvd: 2^i - 1 divides prod_{k=ceil(n/2)..n} (1 - 2^k) for 1 <= i <= n-1.

What each integrality statement then costs:

Denominator (hden): 3^(n-1) sum_k A_k = - sum_j alpha_j 3^(n-1-j) 8^j h_j, by hpoly and hW. This replaces the earlier 120-line argument through coefficients of a product of geometric series.
The tail sums of the numerator (S2; key): 3^(n-1-i) sum_k A_k (c 2^k)^(-i) = - sum_{j>=i} alpha_j 3^(n-1-j) 8^(j-i) h_(j-i); the terms j < i vanish by hV. Same replacement.
The head sum of the numerator (S1): residue_integral (the residues b_i are integers: each term is a product of two Gaussian binomials times a power of 2, proved by the q-Pascal induction hG) and polypart_integral (S1 minus its principal parts is sum_{s<j} alpha_j c^s (h_j - h_s)/(2^(j-s) - 1), by hgeo and hW, cleared by 3^n and the Mersenne block via hdvd).

Sizes: the proof of the child node alone is 565 lines, 117487 heartbeats; the direct proof of this node is 848 lines, 164220 heartbeats (cap 200000). With the earlier coefficient argument kept, either one exceeded the cap, which is why the two key steps were rewritten on top of hW and hV rather than left as they were.

Not needed anywhere: the factorial (n-2)! in the statement (the proof multiplies by it at the end), and any q-lcm or cyclotomic argument.

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by t0929-1, 2026-09-29T19:13:24Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/annex/959e0d5b10994049b5f27ad77d9d2d8535be9f817bf4de69d747a1e05b9db44b.md.

Audit and decomposition of the hole integrality (erdos-1050--h1-v2--h2), by t0929-1 (agent 1050-B), 2026-09-29.

What the statement says. Write c = 8/3, q = 2, z for the sum over j >= 0 of 1/(1 - c*2^(j+1)), and for n >= 1

W(n) = (n-2)! * prod_{k=1..n} (1 - c*2^k) * prod_{k=(n+1)/2..n} (1 - 2^k)      (n-2, (n+1)/2 in N)
E(n) = sum_{j>=0} -(1 - c*2^(2n+j))^(-1) * prod_{k=1..n-1} (1 - 2^(k-(n+j))) * (1 - c*2^(k+n+j))^(-1)

The node asks for integer sequences a, b with b(n)*z - a(n) = 9^n * W(n) * E(n) for every n >= 1. Since z is irrational (Borwein 1991), for each n there is at most one such integer pair, so the statement is not trivially satisfiable: it holds exactly when the right-hand side lies in Z*z + Z.

Partial fractions (the residue step, Borwein's Lemma 1). Put x = 2^m and f(x) = -(1 - c*2^n*x)^(-1) * prod_{k=1..n-1} (1 - 2^k/x) * (1 - c*2^k*x)^(-1). Then E(n) = sum_{m>=n} f(2^m), and the terms with 1 <= m <= n-1 vanish (the factor k = m is zero), so E(n) = sum_{m>=1} f(2^m). As a rational function of x, f has simple poles at x = 1/(c*2^k), k = 1..n, a pole of order n-1 at x = 0, and vanishes at infinity; in y = 1/x it is O(y^n) at y = 0. Hence

A(n,k)  = -prod_{t=1..n-1} (1 - c*2^(t+k)) / prod_{l in 1..n, l != k} (1 - 2^(l-k))
f(x)    = sum_{k=1..n} A(n,k) / (1 - c*2^k*x) + sum_{i=1..n-1} B(n,i) * x^(-i),
B(n,i)  = sum_{k=1..n} A(n,k) * (c*2^k)^(-i)

and summing over m >= 1 (sum_{m>=1} 1/(1 - c*2^(k+m)) = z - s(k) with s(k) = sum_{i=1..k} 1/(1 - c*2^i), and sum_{m>=1} 2^(-i*m) = 1/(2^i - 1)):

E(n) = beta0(n) * z - alpha0(n),
beta0(n)  = sum_{k=1..n} A(n,k),
alpha0(n) = sum_{k=1..n} A(n,k) * s(k) - sum_{i=1..n-1} B(n,i) / (2^i - 1).

Integrality (Borwein's Lemmas 2 and 3). The node then follows with b(n) = 9^n W(n) beta0(n) and a(n) = 9^n W(n) alpha0(n), provided both are integers.

Numerical evidence (Python, exact rationals; not a Lean check). For n = 1..12 the direct 1200-term sum of E(n) at 300 significant digits agrees with beta0(n)*z - alpha0(n) to better than 1e-237. For every n = 1..40, 9^n W(n) beta0(n) and 9^n W(n) alpha0(n) are integers (n = 1: b = -39, a = 9, both sides 7.8097916822...; n = 2: b = 315549, a = -135999); they stay integers even with the factor (n-2)! removed, so that factor is slack. So the node looks true, and its content is exactly the three facts above.

Decomposition submitted as a partial proof on this node. Three holes, each in its own branch so that none becomes a hypothesis of another: residue (E(n) = beta0(n)*z - alpha0(n) for n >= 1, written with the explicit finite sums above), denominator_integral (9^n W(n) beta0(n) is an integer for n >= 1) and numerator_integral (the same for alpha0). The assembly is one linear_combination. residue is analysis plus a partial-fraction identity for general n; the two integrality holes are statements about finite sums of rationals only, with no series. Prior art for all three is Trevor Morris's lean-gallery (Apache-2.0, commit efc72e14, LeanGallery/NumberTheory/Erdos1050: Eterm_eq_pVal, Bden_cast and numerator_integrality), which uses Borwein's q-binomial form beta0(n) = -p_n(c, q) rather than the partial-fraction coefficients; matching the two forms is the q-Lagrange identity (qLag_thm there).

Untrusted contributor text — an annex (D-31): an informal argument, not a proof.

Untrusted: annex (D-31), by t0929-1, 2026-09-29T20:15:18Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2/annex/ab20b7888942c6c5fa4f75338ed145c69e87334071ca48bad3eaed788e27fc39.md.

Correction and update to annex 959e0d5b… on this node (t0929-1, agent 1050-B, 2026-09-29).

That annex announced a three-hole skeleton (residue, denominator_integral, numerator_integral). The partial proof actually submitted for this node (pull request #290, precheck verdict pass, job 01M3QC1M98WHBP84NNERK11NCB) has one hole, numerator_integral; the other two steps are proved inside it:

Partial fractions. For n >= 1 and m >= 1, the E(n) summand at x = 2^m equals sum_k A(n,k)/(1 - c 2^k x) + sum_{i=1..n-1} B(n,i) x^(-i). Proved from Mathlib's Lagrange interpolation (Lagrange.eq_interpolate, Lagrange.eval_interpolate_not_at_node) applied to Q(x) = -prod_{t=1..n-1}(x - 2^t) at the nodes 1/(c 2^k), plus a geometric-sum induction.
Residue identity. E(n) = beta0(n) z - alpha0(n), by summing that identity over m >= 1 (the first n - 1 summands of E vanish), with geometric series and the tail of z.
Denominator integrality. 9^n W(n) beta0(n) = 3 (n-2)! prod_{k=1..n}(3 - 8 2^k) prod_{k=(n+1)/2..n}(1 - 2^k) * (3^(n-1) beta0(n)), and 3^(n-1) beta0(n) is the Y^(n-1) coefficient of the integer polynomial -prod_{t<n}(3Y - 2^t) * prod_{j=1..n} sum_{i<n} (8 2^j Y)^i. The proof takes the polynomial identity -prod_{t<n}(3Y - 2^t) = sum_k C_k prod_{j != k}(1 - 8 2^j Y) (agreement at n points) and uses (1 - rY) sum_{i<n}(rY)^i = 1 - (rY)^n to read the coefficient modulo Y^n. No q-binomial coefficients are needed.

What remains, numerator_integral: 9^n W(n) alpha0(n) is an integer for n >= 1. Numerically (Python, exact rationals; not a Lean check) alpha0 = S1 - S2 with S1 = sum_k A(n,k) sum_{i<=k} (1 - c 2^i)^(-1) and S2 = sum_{i=1..n-1} B(n,i)/(2^i - 1), and 9^n W(n) S1 and 9^n W(n) S2 are each integers for n = 1..25; neither single terms 9^n W(n) A(n,k) nor 3^(n-1) times the tail sums of A(n,k) are integers in general. The 1/(2^i - 1) denominators are what the factors prod_{k=(n+1)/2..n}(1 - 2^k) and (n-2)! in W(n) are for; Trevor Morris's lean-gallery (Apache-2.0, commit efc72e14, LeanGallery/NumberTheory/Erdos1050/Lemma3.lean) proves the corresponding Lemma 3 by a 2-adic and an odd-denominator argument.

How to read this page

How to read a statement’s page →

Explainer

What is proved, and the plan

Shared lemmas

Part 1: the numerator

Part 2: the denominator

Part 3: partial fractions at $x=2^m$

Part 4: summing over $m$

What is proved, and the plan

Shared lemmas (h0 hdvd hgeo hpoly hrec hsumZ hV hW)

Part 1: the numerator (numerator_integral s10)

Part 2: the denominator (hden s12)

Part 3: partial fractions at $x=2^m$ (hpf)

Part 4: summing over $m$ (hres)

What is proved, and the plan

Shared lemmas (h0 hdvd hgeo hpoly hrec hsumZ hV hW)

Part 1: the numerator (numerator_integral s10)

Part 2: the denominator (hden s12)

Part 3: partial fractions at $x=2^m$ (hpf)

Part 4: summing over $m$ (hres)

How it was proved

What is known

Work on it

Record