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

Numerator integrality (Borwein Lemma 3)

erdos-1050--h1-v2--h2--h1

proved

Hole numerator_integral of erdos-1050--h1-v2--h2 (split by 20260929T194647Z-t0929-1-partial.lean).

$\displaystyle \exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$

There is a sequence of integers $a(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, that is, $9^n W(n)(S_1(n)-S_2(n))$ is an integer, where, with $c=8/3$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$ for $1\le k\le n$, $S_1(n)=\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$ and $S_2(n)=\sum_{i=1}^{n-1}\dfrac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$, all sums finite. Here $n-2$ is natural-number subtraction (so $(n-2)!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponent $l-k$ is an integer (so $2^{l-k}$ may be $1/2$ and so on), at $n=1$ the product over $t$ is empty and $S_2(1)=0$, and nothing is required of $a(0)$.

unverified · history

Lean

theorem erdos_1050__h1_v2__h2__h1 : ∃ (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))) := 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/dbda1b664f81….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/dec3b03a5a2b….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 $c = 8/3$. For $n \ge 1$ and $1 \le k \le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k(x)=-\frac{\prod_{t=1}^{n-1}\bigl(1-x\,2^{\,t+k}\bigr)}{D_k},\qquad h_i=\bigl(1-c\,2^{i}\bigr)^{-1},$$ $$P(n)=\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr),\qquad M_n=\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),\qquad W(n)=(n-2)!\,P(n)\,M_n .$$ The statement asks for integers $a(n)$ with $a(n)=9^nW(n)\bigl(S_1(n)-S_2(n)\bigr)$ for $n\ge1$, where $S_1=\sum_{k=1}^n A_k(c)\sum_{i=1}^k h_i$ and $S_2=\sum_{i=1}^{n-1} B_i/(2^i-1)$ with $B_i=\sum_{k=1}^n A_k(c)\,(c\,2^k)^{-i}$.

The proof shows separately that $9^nW(n)S_1(n)$ is an integer (head_integral) and that $9^nW(n)S_2(n)$ is an integer (hmer), and takes $a(n)=a_1(n)-a_2(n)$; the last four lines of the proof check the identity by ring. The factor $(n-2)!$ is carried along as an integer and plays no other part. The proof uses no other node of the graph (the outline lists none), and its comments point to annex a03db62f… on this node for the informal argument.

Throughout, $H_e=\sum_{k=1}^{n}(2^k)^e/D_k$ for $e \ge 0$ (a sum the proof shows to be an integer).

Read it step by step (7 more sections)

Finite sums of integers

Draft · drafted, by the model named above

hsumZ is bookkeeping: if every term $g(i)$, $i\in s$, of a finite sum of reals is (the image of) an integer, so is the sum. It is proved by induction on the finite set $s$, and is used whenever a sum is shown to be an integer term by term.

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)
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)))
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)))
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)

Lagrange sums at the nodes $2^k$

Draft · drafted, by the model named above

For a finite set $s$ of naturals and $k\in s$ write $D_s(k)=\prod_{l\in s,\ l\ne k}(1-2^{\,l-k})$; for $s=\{1,\dots,n\}$ this is $D_k$.

h0: for nonempty $s$, $\sum_{k\in s}1/D_s(k)=1$. Mathlib's Lagrange basis polynomials for the nodes $2^{-k}$ sum to $1$; evaluating that identity at $0$ gives exactly these terms.

hrec: for $a\in s$ and any weight $f$,

$$\sum_{k\in s}\frac{2^k f(k)}{D_s(k)}=\sum_{k\in s\setminus\{a\}}\frac{2^k f(k)}{D_{s\setminus\{a\}}(k)}+2^a\sum_{k\in s}\frac{f(k)}{D_s(k)},$$ by splitting off the factor $l=a$ from each $D_s(k)$.

hW: for every $s$ and every $e\ge0$, $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer. Induction on $|s|+e$: the empty set gives $0$, $e=0$ gives $1$ by h0, and otherwise hrec with $f(k)=(2^k)^{e-1}$ writes the sum as the sum for $(s\setminus\{a\},e)$ plus $2^a$ times the sum for $(s,e-1)$. In particular every $H_e$ is an integer.

hV: for $m+1<|s|$, $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$. Induction on $m$, again from hrec, with h0 in the base case.

Geometric sums, a divisibility, and an integer polynomial

Draft · drafted, by the model named above

hgeo: for $m\ge1$ and every $k$, $\sum_{i=1}^{k}2^{-im}=\dfrac{1-2^{-km}}{2^m-1}$, by induction on $k$.

hdvd: for $1\le i\le n-1$, the integer $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$. The multiple $k_0=i\lfloor n/i\rfloor$ of $i$ lies in $[\lceil n/2\rceil,n]$, and $2^i-1$ divides $2^{k_0}-1$. This is what the factor $M_n$ in the statement is for.

hpoly: for $n\ge1$ there are integers $\alpha_0,\dots,\alpha_{n-1}$ with $\prod_{t=1}^{n-1}(1-2^t y)=\sum_{j<n}\alpha_j y^j$ for every real $y$ (the coefficients of that integer polynomial, of degree less than $n$). Consequently $A_k(x)=-\frac1{D_k}\sum_{j<n}\alpha_j\,(x\,2^k)^j$.

The residues are integers

Draft · drafted, by the model named above

residue_integral: for $1\le i\le n$, the number $b_i=\sum_{k=i}^{n}A_k(2^{-i})$ is an integer. In fact each term is. Write $k=i+d$ and $Q(m)=\prod_{s=1}^{m}(1-2^s)$, which is never $0$.

At $x=2^{-i}$ the numerator of $A_k$ is the integer $\prod_{t=1}^{n-1}(1-2^{\,t+d})$.

$D_k$ splits into the factors with $l<k$ and $l>k$; the first part times $\prod_{s=1}^{k-1}(-2^s)$ is $Q(k-1)$, and the second part is $Q(n-k)$.

Two identities about $Q$: $Q(a+b)$ is an integer multiple of $Q(a)Q(b)$ (the Gaussian binomial coefficient at $q=2$ is an integer), and $Q(d)\prod_{t=1}^{m}(1-2^{\,t+d})=Q(d+m)$.

Together these give numerator $=z_1z_2\,Q(k-1)Q(n-k)$ for integers $z_1,z_2$, so the term $A_k(2^{-i})$ equals the integer $-z_1z_2\prod_{s=1}^{k-1}(-2^s)$.

Steps

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)

The polynomial part

Draft · drafted, by the model named above

polypart_integral: for $n\ge1$, with $R(n)=\sum_{i=1}^{n}b_i\,h_i$, the number $3^n M_n\bigl(S_1(n)-R(n)\bigr)$ is an integer. So $S_1$ differs from its "residue part" $R$ by something whose denominators are only powers of $3$ and divisors of $M_n$.

Exchanging the order of summation, $R=\sum_{k=1}^{n}\sum_{i=1}^{k}A_k(2^{-i})\,h_i$, so $S_1-R=\sum_{k}\sum_{i\le k}\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i$.

For each $k$ and $i\ge1$, using hpoly and the factorisation of $y^j-c^j$ with $y=2^{-i}$:

$$\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i=\frac1{D_k}\sum_{j<n}\sum_{s<j}\alpha_j\,2^{kj}\,c^{s}\,2^{-i(j-s)} .$$

Summing over $i=1,\dots,k$ with hgeo turns $2^{kj}\sum_i 2^{-i(j-s)}$ into $\bigl((2^k)^j-(2^k)^s\bigr)/(2^{j-s}-1)$.

Summing over $k$ then gives $\sum_{j<n}\sum_{s<j}\alpha_j\,c^s\,(H_j-H_s)/(2^{j-s}-1)$. Here $H_j,H_s$ are integers by hW, $2^{j-s}-1$ divides $M_n$ by hdvd (as $1\le j-s\le n-1$), and $3^n c^s=3^{n-s}8^s$. Each term times $3^nM_n$ is therefore an integer, and so is the sum.

Steps

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
s12 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

The first sum $S_1$

Draft · drafted, by the model named above

head_integral produces integers $a_1(n)$ with $a_1(n)=9^nW(n)S_1(n)$ for $n\ge1$. Writing $S_1=(S_1-R)+R$: $$9^nW(n)S_1=(n-2)!\Bigl[\bigl(3^nP(n)\bigr)\cdot 3^nM_n(S_1-R)+3^nM_n\cdot\bigl(3^nP(n)R\bigr)\Bigr].$$ Here $3^nP(n)=\prod_{k=1}^n(3-8\cdot2^k)$ is an integer, $3^nM_n(S_1-R)$ is an integer by polypart_integral, and $3^nP(n)R$ is an integer because $3^nP(n)\,h_i=3\prod_{j\ne i}(3-8\cdot2^j)$ and each $b_i$ is an integer by residue_integral. The sequence $a_1$ is then chosen from these integers (with value $0$ at $n=0$); s12 unpacks it.

Steps

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))
s14 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))

The second sum $S_2$

Draft · drafted, by the model named above

hmer produces integers $a_2(n)$ with $a_2(n)=9^nW(n)S_2(n)$ for $n\ge1$. Its step key shows that for $1\le i\le n-1$ the number $3^{\,n-1-i}B_i$ is an integer: expanding $A_k(c)$ with hpoly, $$B_i=-\sum_{j<n}\alpha_j\sum_{k=1}^{n}\frac{(c\,2^k)^{\,j-i}}{D_k}.$$ For $j<i$ the inner sum is $c^{-(i-j)}$ times a sum that vanishes by hV (since $1\le i-j<n$); for $j\ge i$ it is $c^{\,j-i}H_{j-i}$, and $3^{\,n-1-i}c^{\,j-i}=3^{\,n-1-j}8^{\,j-i}$.

Then each term of $9^nW(n)S_2(n)$ is an integer: write $9^n=3^{i+1}\cdot3^{\,n-1-i}\cdot3^n$, use $3^nP(n)=\prod_k(3-8\cdot 2^k)$, and use hdvd to write $M_n=(2^i-1)q$, which cancels the denominator $2^i-1$. The term becomes $3^{i+1}(n-2)!\prod_k(3-8\cdot2^k)\,q$ times the integer from key. (Inside hmer the proof restates hdvd and hsumZ as local copies.) s14 unpacks $a_2$, and the proof ends with $a=a_1-a_2$.

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/dbda1b664f81….md · the first version of its chain
Read this version
What is proved, and the plan
Fix $c = 8/3$. For $n \ge 1$ and $1 \le k \le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k(x)=-\frac{\prod_{t=1}^{n-1}\bigl(1-x\,2^{\,t+k}\bigr)}{D_k},\qquad h_i=\bigl(1-c\,2^{i}\bigr)^{-1},$$ $$P(n)=\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr),\qquad M_n=\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),\qquad W(n)=(n-2)!\,P(n)\,M_n .$$ The statement asks for integers $a(n)$ with $a(n)=9^nW(n)\bigl(S_1(n)-S_2(n)\bigr)$ for $n\ge1$, where $S_1=\sum_{k=1}^n A_k(c)\sum_{i=1}^k h_i$ and $S_2=\sum_{i=1}^{n-1} B_i/(2^i-1)$ with $B_i=\sum_{k=1}^n A_k(c)\,(c\,2^k)^{-i}$.

The proof shows separately that $9^nW(n)S_1(n)$ is an integer (head_integral) and that $9^nW(n)S_2(n)$ is an integer (hmer), and takes $a(n)=a_1(n)-a_2(n)$; the last four lines of the proof check the identity by ring. The factor $(n-2)!$ is carried along as an integer and plays no other part. The proof uses no other node of the graph (the outline lists none), and its comments point to annex a03db62f… on this node for the informal argument.

Throughout, $H_e=\sum_{k=1}^{n}(2^k)^e/D_k$ for $e \ge 0$ (a sum the proof shows to be an integer).
Finite sums of integers (hsumZ)
hsumZ is bookkeeping: if every term $g(i)$, $i\in s$, of a finite sum of reals is (the image of) an integer, so is the sum. It is proved by induction on the finite set $s$, and is used whenever a sum is shown to be an integer term by term.
Lagrange sums at the nodes $2^k$ (h0 hrec hW hV)
For a finite set $s$ of naturals and $k\in s$ write $D_s(k)=\prod_{l\in s,\ l\ne k}(1-2^{\,l-k})$; for $s=\{1,\dots,n\}$ this is $D_k$.
- h0: for nonempty $s$, $\sum_{k\in s}1/D_s(k)=1$. Mathlib's Lagrange basis polynomials for the nodes $2^{-k}$ sum to $1$; evaluating that identity at $0$ gives exactly these terms.
- hrec: for $a\in s$ and any weight $f$,
$$\sum_{k\in s}\frac{2^k f(k)}{D_s(k)}=\sum_{k\in s\setminus\{a\}}\frac{2^k f(k)}{D_{s\setminus\{a\}}(k)}+2^a\sum_{k\in s}\frac{f(k)}{D_s(k)},$$ by splitting off the factor $l=a$ from each $D_s(k)$.
- hW: for every $s$ and every $e\ge0$, $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer. Induction on $|s|+e$: the empty set gives $0$, $e=0$ gives $1$ by h0, and otherwise hrec with $f(k)=(2^k)^{e-1}$ writes the sum as the sum for $(s\setminus\{a\},e)$ plus $2^a$ times the sum for $(s,e-1)$. In particular every $H_e$ is an integer.
- hV: for $m+1<|s|$, $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$. Induction on $m$, again from hrec, with h0 in the base case.
Geometric sums, a divisibility, and an integer polynomial (hdvd hgeo hpoly)
- hgeo: for $m\ge1$ and every $k$, $\sum_{i=1}^{k}2^{-im}=\dfrac{1-2^{-km}}{2^m-1}$, by induction on $k$.
- hdvd: for $1\le i\le n-1$, the integer $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$. The multiple $k_0=i\lfloor n/i\rfloor$ of $i$ lies in $[\lceil n/2\rceil,n]$, and $2^i-1$ divides $2^{k_0}-1$. This is what the factor $M_n$ in the statement is for.
- hpoly: for $n\ge1$ there are integers $\alpha_0,\dots,\alpha_{n-1}$ with $\prod_{t=1}^{n-1}(1-2^t y)=\sum_{j<n}\alpha_j y^j$ for every real $y$ (the coefficients of that integer polynomial, of degree less than $n$). Consequently $A_k(x)=-\frac1{D_k}\sum_{j<n}\alpha_j\,(x\,2^k)^j$.
The residues are integers (residue_integral)
residue_integral: for $1\le i\le n$, the number $b_i=\sum_{k=i}^{n}A_k(2^{-i})$ is an integer. In fact each term is. Write $k=i+d$ and $Q(m)=\prod_{s=1}^{m}(1-2^s)$, which is never $0$.
- At $x=2^{-i}$ the numerator of $A_k$ is the integer $\prod_{t=1}^{n-1}(1-2^{\,t+d})$.
- $D_k$ splits into the factors with $l<k$ and $l>k$; the first part times $\prod_{s=1}^{k-1}(-2^s)$ is $Q(k-1)$, and the second part is $Q(n-k)$.
- Two identities about $Q$: $Q(a+b)$ is an integer multiple of $Q(a)Q(b)$ (the Gaussian binomial coefficient at $q=2$ is an integer), and $Q(d)\prod_{t=1}^{m}(1-2^{\,t+d})=Q(d+m)$.
- Together these give numerator $=z_1z_2\,Q(k-1)Q(n-k)$ for integers $z_1,z_2$, so the term $A_k(2^{-i})$ equals the integer $-z_1z_2\prod_{s=1}^{k-1}(-2^s)$.
The polynomial part (polypart_integral)
polypart_integral: for $n\ge1$, with $R(n)=\sum_{i=1}^{n}b_i\,h_i$, the number $3^n M_n\bigl(S_1(n)-R(n)\bigr)$ is an integer. So $S_1$ differs from its "residue part" $R$ by something whose denominators are only powers of $3$ and divisors of $M_n$.
- Exchanging the order of summation, $R=\sum_{k=1}^{n}\sum_{i=1}^{k}A_k(2^{-i})\,h_i$, so $S_1-R=\sum_{k}\sum_{i\le k}\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i$.
- For each $k$ and $i\ge1$, using hpoly and the factorisation of $y^j-c^j$ with $y=2^{-i}$:
$$\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i=\frac1{D_k}\sum_{j<n}\sum_{s<j}\alpha_j\,2^{kj}\,c^{s}\,2^{-i(j-s)} .$$
- Summing over $i=1,\dots,k$ with hgeo turns $2^{kj}\sum_i 2^{-i(j-s)}$ into $\bigl((2^k)^j-(2^k)^s\bigr)/(2^{j-s}-1)$.
- Summing over $k$ then gives $\sum_{j<n}\sum_{s<j}\alpha_j\,c^s\,(H_j-H_s)/(2^{j-s}-1)$. Here $H_j,H_s$ are integers by hW, $2^{j-s}-1$ divides $M_n$ by hdvd (as $1\le j-s\le n-1$), and $3^n c^s=3^{n-s}8^s$. Each term times $3^nM_n$ is therefore an integer, and so is the sum.
The first sum $S_1$ (head_integral s12)
head_integral produces integers $a_1(n)$ with $a_1(n)=9^nW(n)S_1(n)$ for $n\ge1$. Writing $S_1=(S_1-R)+R$: $$9^nW(n)S_1=(n-2)!\Bigl[\bigl(3^nP(n)\bigr)\cdot 3^nM_n(S_1-R)+3^nM_n\cdot\bigl(3^nP(n)R\bigr)\Bigr].$$ Here $3^nP(n)=\prod_{k=1}^n(3-8\cdot2^k)$ is an integer, $3^nM_n(S_1-R)$ is an integer by polypart_integral, and $3^nP(n)R$ is an integer because $3^nP(n)\,h_i=3\prod_{j\ne i}(3-8\cdot2^j)$ and each $b_i$ is an integer by residue_integral. The sequence $a_1$ is then chosen from these integers (with value $0$ at $n=0$); s12 unpacks it.
The second sum $S_2$ (hmer s14)
hmer produces integers $a_2(n)$ with $a_2(n)=9^nW(n)S_2(n)$ for $n\ge1$. Its step key shows that for $1\le i\le n-1$ the number $3^{\,n-1-i}B_i$ is an integer: expanding $A_k(c)$ with hpoly, $$B_i=-\sum_{j<n}\alpha_j\sum_{k=1}^{n}\frac{(c\,2^k)^{\,j-i}}{D_k}.$$ For $j<i$ the inner sum is $c^{-(i-j)}$ times a sum that vanishes by hV (since $1\le i-j<n$); for $j\ge i$ it is $c^{\,j-i}H_{j-i}$, and $3^{\,n-1-i}c^{\,j-i}=3^{\,n-1-j}8^{\,j-i}$.

Then each term of $9^nW(n)S_2(n)$ is an integer: write $9^n=3^{i+1}\cdot3^{\,n-1-i}\cdot3^n$, use $3^nP(n)=\prod_k(3-8\cdot 2^k)$, and use hdvd to write $M_n=(2^i-1)q$, which cancels the denominator $2^i-1$. The term becomes $3^{i+1}(n-2)!\prod_k(3-8\cdot2^k)\,q$ times the integer from key. (Inside hmer the proof restates hdvd and hsumZ as local copies.) s14 unpacks $a_2$, and the proof ends with $a=a_1-a_2$.
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/dec3b03a5a2b….md · diff to its predecessor
Read this version
What is proved, and the plan
Fix $c = 8/3$. For $n \ge 1$ and $1 \le k \le n$ write $$D_k=\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr),\qquad A_k(x)=-\frac{\prod_{t=1}^{n-1}\bigl(1-x\,2^{\,t+k}\bigr)}{D_k},\qquad h_i=\bigl(1-c\,2^{i}\bigr)^{-1},$$ $$P(n)=\prod_{k=1}^{n}\bigl(1-c\,2^k\bigr),\qquad M_n=\prod_{k=\lceil n/2\rceil}^{n}\bigl(1-2^k\bigr),\qquad W(n)=(n-2)!\,P(n)\,M_n .$$ The statement asks for integers $a(n)$ with $a(n)=9^nW(n)\bigl(S_1(n)-S_2(n)\bigr)$ for $n\ge1$, where $S_1=\sum_{k=1}^n A_k(c)\sum_{i=1}^k h_i$ and $S_2=\sum_{i=1}^{n-1} B_i/(2^i-1)$ with $B_i=\sum_{k=1}^n A_k(c)\,(c\,2^k)^{-i}$.

The proof shows separately that $9^nW(n)S_1(n)$ is an integer (head_integral) and that $9^nW(n)S_2(n)$ is an integer (hmer), and takes $a(n)=a_1(n)-a_2(n)$; the last four lines of the proof check the identity by ring. The factor $(n-2)!$ is carried along as an integer and plays no other part. The proof uses no other node of the graph (the outline lists none), and its comments point to annex a03db62f… on this node for the informal argument.

Throughout, $H_e=\sum_{k=1}^{n}(2^k)^e/D_k$ for $e \ge 0$ (a sum the proof shows to be an integer).
Finite sums of integers (hsumZ)
hsumZ is bookkeeping: if every term $g(i)$, $i\in s$, of a finite sum of reals is (the image of) an integer, so is the sum. It is proved by induction on the finite set $s$, and is used whenever a sum is shown to be an integer term by term.
Lagrange sums at the nodes $2^k$ (h0 hrec hW hV)
For a finite set $s$ of naturals and $k\in s$ write $D_s(k)=\prod_{l\in s,\ l\ne k}(1-2^{\,l-k})$; for $s=\{1,\dots,n\}$ this is $D_k$.

h0: for nonempty $s$, $\sum_{k\in s}1/D_s(k)=1$. Mathlib's Lagrange basis polynomials for the nodes $2^{-k}$ sum to $1$; evaluating that identity at $0$ gives exactly these terms.

hrec: for $a\in s$ and any weight $f$,

$$\sum_{k\in s}\frac{2^k f(k)}{D_s(k)}=\sum_{k\in s\setminus\{a\}}\frac{2^k f(k)}{D_{s\setminus\{a\}}(k)}+2^a\sum_{k\in s}\frac{f(k)}{D_s(k)},$$ by splitting off the factor $l=a$ from each $D_s(k)$.

hW: for every $s$ and every $e\ge0$, $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer. Induction on $|s|+e$: the empty set gives $0$, $e=0$ gives $1$ by h0, and otherwise hrec with $f(k)=(2^k)^{e-1}$ writes the sum as the sum for $(s\setminus\{a\},e)$ plus $2^a$ times the sum for $(s,e-1)$. In particular every $H_e$ is an integer.

hV: for $m+1<|s|$, $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$. Induction on $m$, again from hrec, with h0 in the base case.
Geometric sums, a divisibility, and an integer polynomial (hdvd hgeo hpoly)
hgeo: for $m\ge1$ and every $k$, $\sum_{i=1}^{k}2^{-im}=\dfrac{1-2^{-km}}{2^m-1}$, by induction on $k$.

hdvd: for $1\le i\le n-1$, the integer $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$. The multiple $k_0=i\lfloor n/i\rfloor$ of $i$ lies in $[\lceil n/2\rceil,n]$, and $2^i-1$ divides $2^{k_0}-1$. This is what the factor $M_n$ in the statement is for.

hpoly: for $n\ge1$ there are integers $\alpha_0,\dots,\alpha_{n-1}$ with $\prod_{t=1}^{n-1}(1-2^t y)=\sum_{j<n}\alpha_j y^j$ for every real $y$ (the coefficients of that integer polynomial, of degree less than $n$). Consequently $A_k(x)=-\frac1{D_k}\sum_{j<n}\alpha_j\,(x\,2^k)^j$.
The residues are integers (residue_integral)
residue_integral: for $1\le i\le n$, the number $b_i=\sum_{k=i}^{n}A_k(2^{-i})$ is an integer. In fact each term is. Write $k=i+d$ and $Q(m)=\prod_{s=1}^{m}(1-2^s)$, which is never $0$.

At $x=2^{-i}$ the numerator of $A_k$ is the integer $\prod_{t=1}^{n-1}(1-2^{\,t+d})$.

$D_k$ splits into the factors with $l<k$ and $l>k$; the first part times $\prod_{s=1}^{k-1}(-2^s)$ is $Q(k-1)$, and the second part is $Q(n-k)$.

Two identities about $Q$: $Q(a+b)$ is an integer multiple of $Q(a)Q(b)$ (the Gaussian binomial coefficient at $q=2$ is an integer), and $Q(d)\prod_{t=1}^{m}(1-2^{\,t+d})=Q(d+m)$.

Together these give numerator $=z_1z_2\,Q(k-1)Q(n-k)$ for integers $z_1,z_2$, so the term $A_k(2^{-i})$ equals the integer $-z_1z_2\prod_{s=1}^{k-1}(-2^s)$.
The polynomial part (polypart_integral)
polypart_integral: for $n\ge1$, with $R(n)=\sum_{i=1}^{n}b_i\,h_i$, the number $3^n M_n\bigl(S_1(n)-R(n)\bigr)$ is an integer. So $S_1$ differs from its "residue part" $R$ by something whose denominators are only powers of $3$ and divisors of $M_n$.

Exchanging the order of summation, $R=\sum_{k=1}^{n}\sum_{i=1}^{k}A_k(2^{-i})\,h_i$, so $S_1-R=\sum_{k}\sum_{i\le k}\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i$.

For each $k$ and $i\ge1$, using hpoly and the factorisation of $y^j-c^j$ with $y=2^{-i}$:

$$\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i=\frac1{D_k}\sum_{j<n}\sum_{s<j}\alpha_j\,2^{kj}\,c^{s}\,2^{-i(j-s)} .$$

Summing over $i=1,\dots,k$ with hgeo turns $2^{kj}\sum_i 2^{-i(j-s)}$ into $\bigl((2^k)^j-(2^k)^s\bigr)/(2^{j-s}-1)$.

Summing over $k$ then gives $\sum_{j<n}\sum_{s<j}\alpha_j\,c^s\,(H_j-H_s)/(2^{j-s}-1)$. Here $H_j,H_s$ are integers by hW, $2^{j-s}-1$ divides $M_n$ by hdvd (as $1\le j-s\le n-1$), and $3^n c^s=3^{n-s}8^s$. Each term times $3^nM_n$ is therefore an integer, and so is the sum.
The first sum $S_1$ (head_integral s12)
head_integral produces integers $a_1(n)$ with $a_1(n)=9^nW(n)S_1(n)$ for $n\ge1$. Writing $S_1=(S_1-R)+R$: $$9^nW(n)S_1=(n-2)!\Bigl[\bigl(3^nP(n)\bigr)\cdot 3^nM_n(S_1-R)+3^nM_n\cdot\bigl(3^nP(n)R\bigr)\Bigr].$$ Here $3^nP(n)=\prod_{k=1}^n(3-8\cdot2^k)$ is an integer, $3^nM_n(S_1-R)$ is an integer by polypart_integral, and $3^nP(n)R$ is an integer because $3^nP(n)\,h_i=3\prod_{j\ne i}(3-8\cdot2^j)$ and each $b_i$ is an integer by residue_integral. The sequence $a_1$ is then chosen from these integers (with value $0$ at $n=0$); s12 unpacks it.
The second sum $S_2$ (hmer s14)
hmer produces integers $a_2(n)$ with $a_2(n)=9^nW(n)S_2(n)$ for $n\ge1$. Its step key shows that for $1\le i\le n-1$ the number $3^{\,n-1-i}B_i$ is an integer: expanding $A_k(c)$ with hpoly, $$B_i=-\sum_{j<n}\alpha_j\sum_{k=1}^{n}\frac{(c\,2^k)^{\,j-i}}{D_k}.$$ For $j<i$ the inner sum is $c^{-(i-j)}$ times a sum that vanishes by hV (since $1\le i-j<n$); for $j\ge i$ it is $c^{\,j-i}H_{j-i}$, and $3^{\,n-1-i}c^{\,j-i}=3^{\,n-1-j}8^{\,j-i}$.

Then each term of $9^nW(n)S_2(n)$ is an integer: write $9^n=3^{i+1}\cdot3^{\,n-1-i}\cdot3^n$, use $3^nP(n)=\prod_k(3-8\cdot 2^k)$, and use hdvd to write $M_n=(2^i-1)q$, which cancels the denominator $2^i-1$. The term becomes $3^{i+1}(n-2)!\prod_k(3-8\cdot2^k)\,q$ times the integer from key. (Inside hmer the proof restates hdvd and hsumZ as local copies.) s14 unpacks $a_2$, and the proof ends with $a=a_1-a_2$.

Diff of dec3b03a5a2b against dbda1b664f81

@@ -19,16 +19,22 @@
 For a finite set $s$ of naturals and $k\in s$ write $D_s(k)=\prod_{l\in s,\ l\ne k}(1-2^{\,l-k})$; for $s=\{1,\dots,n\}$ this is $D_k$.
 
-- `h0`: for nonempty $s$, $\sum_{k\in s}1/D_s(k)=1$. Mathlib's Lagrange basis polynomials for the nodes $2^{-k}$ sum to $1$; evaluating that identity at $0$ gives exactly these terms.
-- `hrec`: for $a\in s$ and any weight $f$,
+`h0`: for nonempty $s$, $\sum_{k\in s}1/D_s(k)=1$. Mathlib's Lagrange basis polynomials for the nodes $2^{-k}$ sum to $1$; evaluating that identity at $0$ gives exactly these terms.
+
+`hrec`: for $a\in s$ and any weight $f$,
+
 $$\sum_{k\in s}\frac{2^k f(k)}{D_s(k)}=\sum_{k\in s\setminus\{a\}}\frac{2^k f(k)}{D_{s\setminus\{a\}}(k)}+2^a\sum_{k\in s}\frac{f(k)}{D_s(k)},$$
 by splitting off the factor $l=a$ from each $D_s(k)$.
-- `hW`: for every $s$ and every $e\ge0$, $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer. Induction on $|s|+e$: the empty set gives $0$, $e=0$ gives $1$ by `h0`, and otherwise `hrec` with $f(k)=(2^k)^{e-1}$ writes the sum as the sum for $(s\setminus\{a\},e)$ plus $2^a$ times the sum for $(s,e-1)$. In particular every $H_e$ is an integer.
-- `hV`: for $m+1<|s|$, $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$. Induction on $m$, again from `hrec`, with `h0` in the base case.
+
+`hW`: for every $s$ and every $e\ge0$, $\sum_{k\in s}(2^k)^e/D_s(k)$ is an integer. Induction on $|s|+e$: the empty set gives $0$, $e=0$ gives $1$ by `h0`, and otherwise `hrec` with $f(k)=(2^k)^{e-1}$ writes the sum as the sum for $(s\setminus\{a\},e)$ plus $2^a$ times the sum for $(s,e-1)$. In particular every $H_e$ is an integer.
+
+`hV`: for $m+1<|s|$, $\sum_{k\in s}(2^k)^{-(m+1)}/D_s(k)=0$. Induction on $m$, again from `hrec`, with `h0` in the base case.
 
 ## Geometric sums, a divisibility, and an integer polynomial {steps: hdvd hgeo hpoly}
 
-- `hgeo`: for $m\ge1$ and every $k$, $\sum_{i=1}^{k}2^{-im}=\dfrac{1-2^{-km}}{2^m-1}$, by induction on $k$.
-- `hdvd`: for $1\le i\le n-1$, the integer $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$. The multiple $k_0=i\lfloor n/i\rfloor$ of $i$ lies in $[\lceil n/2\rceil,n]$, and $2^i-1$ divides $2^{k_0}-1$. This is what the factor $M_n$ in the statement is for.
-- `hpoly`: for $n\ge1$ there are integers $\alpha_0,\dots,\alpha_{n-1}$ with $\prod_{t=1}^{n-1}(1-2^t y)=\sum_{j<n}\alpha_j y^j$ for every real $y$ (the coefficients of that integer polynomial, of degree less than $n$). Consequently $A_k(x)=-\frac1{D_k}\sum_{j<n}\alpha_j\,(x\,2^k)^j$.
+`hgeo`: for $m\ge1$ and every $k$, $\sum_{i=1}^{k}2^{-im}=\dfrac{1-2^{-km}}{2^m-1}$, by induction on $k$.
+
+`hdvd`: for $1\le i\le n-1$, the integer $2^i-1$ divides $\prod_{k=\lceil n/2\rceil}^{n}(1-2^k)$. The multiple $k_0=i\lfloor n/i\rfloor$ of $i$ lies in $[\lceil n/2\rceil,n]$, and $2^i-1$ divides $2^{k_0}-1$. This is what the factor $M_n$ in the statement is for.
+
+`hpoly`: for $n\ge1$ there are integers $\alpha_0,\dots,\alpha_{n-1}$ with $\prod_{t=1}^{n-1}(1-2^t y)=\sum_{j<n}\alpha_j y^j$ for every real $y$ (the coefficients of that integer polynomial, of degree less than $n$). Consequently $A_k(x)=-\frac1{D_k}\sum_{j<n}\alpha_j\,(x\,2^k)^j$.
 
 ## The residues are integers {steps: residue_integral}
@@ -36,8 +42,11 @@
 `residue_integral`: for $1\le i\le n$, the number $b_i=\sum_{k=i}^{n}A_k(2^{-i})$ is an integer. In fact each term is. Write $k=i+d$ and $Q(m)=\prod_{s=1}^{m}(1-2^s)$, which is never $0$.
 
-- At $x=2^{-i}$ the numerator of $A_k$ is the integer $\prod_{t=1}^{n-1}(1-2^{\,t+d})$.
-- $D_k$ splits into the factors with $l<k$ and $l>k$; the first part times $\prod_{s=1}^{k-1}(-2^s)$ is $Q(k-1)$, and the second part is $Q(n-k)$.
-- Two identities about $Q$: $Q(a+b)$ is an integer multiple of $Q(a)Q(b)$ (the Gaussian binomial coefficient at $q=2$ is an integer), and $Q(d)\prod_{t=1}^{m}(1-2^{\,t+d})=Q(d+m)$.
-- Together these give numerator $=z_1z_2\,Q(k-1)Q(n-k)$ for integers $z_1,z_2$, so the term $A_k(2^{-i})$ equals the integer $-z_1z_2\prod_{s=1}^{k-1}(-2^s)$.
+At $x=2^{-i}$ the numerator of $A_k$ is the integer $\prod_{t=1}^{n-1}(1-2^{\,t+d})$.
+
+$D_k$ splits into the factors with $l<k$ and $l>k$; the first part times $\prod_{s=1}^{k-1}(-2^s)$ is $Q(k-1)$, and the second part is $Q(n-k)$.
+
+Two identities about $Q$: $Q(a+b)$ is an integer multiple of $Q(a)Q(b)$ (the Gaussian binomial coefficient at $q=2$ is an integer), and $Q(d)\prod_{t=1}^{m}(1-2^{\,t+d})=Q(d+m)$.
+
+Together these give numerator $=z_1z_2\,Q(k-1)Q(n-k)$ for integers $z_1,z_2$, so the term $A_k(2^{-i})$ equals the integer $-z_1z_2\prod_{s=1}^{k-1}(-2^s)$.
 
 ## The polynomial part {steps: polypart_integral}
@@ -45,9 +54,13 @@
 `polypart_integral`: for $n\ge1$, with $R(n)=\sum_{i=1}^{n}b_i\,h_i$, the number $3^n M_n\bigl(S_1(n)-R(n)\bigr)$ is an integer. So $S_1$ differs from its "residue part" $R$ by something whose denominators are only powers of $3$ and divisors of $M_n$.
 
-- Exchanging the order of summation, $R=\sum_{k=1}^{n}\sum_{i=1}^{k}A_k(2^{-i})\,h_i$, so $S_1-R=\sum_{k}\sum_{i\le k}\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i$.
-- For each $k$ and $i\ge1$, using `hpoly` and the factorisation of $y^j-c^j$ with $y=2^{-i}$:
+Exchanging the order of summation, $R=\sum_{k=1}^{n}\sum_{i=1}^{k}A_k(2^{-i})\,h_i$, so $S_1-R=\sum_{k}\sum_{i\le k}\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i$.
+
+For each $k$ and $i\ge1$, using `hpoly` and the factorisation of $y^j-c^j$ with $y=2^{-i}$:
+
 $$\bigl(A_k(c)-A_k(2^{-i})\bigr)h_i=\frac1{D_k}\sum_{j<n}\sum_{s<j}\alpha_j\,2^{kj}\,c^{s}\,2^{-i(j-s)} .$$
-- Summing over $i=1,\dots,k$ with `hgeo` turns $2^{kj}\sum_i 2^{-i(j-s)}$ into $\bigl((2^k)^j-(2^k)^s\bigr)/(2^{j-s}-1)$.
-- Summing over $k$ then gives $\sum_{j<n}\sum_{s<j}\alpha_j\,c^s\,(H_j-H_s)/(2^{j-s}-1)$. Here $H_j,H_s$ are integers by `hW`, $2^{j-s}-1$ divides $M_n$ by `hdvd` (as $1\le j-s\le n-1$), and $3^n c^s=3^{n-s}8^s$. Each term times $3^nM_n$ is therefore an integer, and so is the sum.
+
+Summing over $i=1,\dots,k$ with `hgeo` turns $2^{kj}\sum_i 2^{-i(j-s)}$ into $\bigl((2^k)^j-(2^k)^s\bigr)/(2^{j-s}-1)$.
+
+Summing over $k$ then gives $\sum_{j<n}\sum_{s<j}\alpha_j\,c^s\,(H_j-H_s)/(2^{j-s}-1)$. Here $H_j,H_s$ are integers by `hW`, $2^{j-s}-1$ divides $M_n$ by `hdvd` (as $1\le j-s\le n-1$), and $3^n c^s=3^{n-s}8^s$. Each term times $3^nM_n$ is therefore an integer, and so is the sum.
 
 ## The first sum $S_1$ {steps: head_integral s12}

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, 000332, records the gate’s run.

The proof step by step (outline)

Checked by the kernel — an outline of the attested proof below: attestations/000332.json, merge commit b88a90ea8e90.

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), ∀ (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 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 28–36
```
  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--h1/Proof.lean, lines 28–36 at b88a90ea8e90 ↗
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--h1/Proof.lean, line 31 at b88a90ea8e90 ↗
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 32–36
  | 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--h1/Proof.lean, lines 32–36 at b88a90ea8e90 ↗
  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 34
  obtain ⟨za, hza⟩ := h a (Finset.mem_insert_self a s)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 34 at b88a90ea8e90 ↗
  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 35
  obtain ⟨zs, hzs⟩ := ih (fun i hi => h i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 35 at b88a90ea8e90 ↗
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), ∀ (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 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 38–54
```
  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--h1/Proof.lean, lines 38–54 at b88a90ea8e90 ↗
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 40–42
  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--h1/Proof.lean, lines 40–42 at b88a90ea8e90 ↗
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 43
  have hsum := congrArg (Polynomial.eval (0 : ℝ)) (Lagrange.sum_basis hv hs)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 43 at b88a90ea8e90 ↗
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 49
  have ha : (2 : ℝ) ^ k ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 49 at b88a90ea8e90 ↗
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 50
  have hb : (2 : ℝ) ^ l ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 50 at b88a90ea8e90 ↗
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), ∀ (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 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 56–78
```
  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--h1/Proof.lean, lines 56–78 at b88a90ea8e90 ↗
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 60
  · ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 60 at b88a90ea8e90 ↗
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 61–78
  · 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--h1/Proof.lean, lines 61–78 at b88a90ea8e90 ↗
  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 62
  have hka : k ≠ a := Finset.ne_of_mem_erase hk
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 62 at b88a90ea8e90 ↗
  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 63
  have hmem : a ∈ s.erase k := Finset.mem_erase.mpr ⟨Ne.symm hka, ha⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 63 at b88a90ea8e90 ↗
  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 64
  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--h1/Proof.lean, line 64 at b88a90ea8e90 ↗
  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 67–71
  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--h1/Proof.lean, lines 67–71 at b88a90ea8e90 ↗
  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 69
  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--h1/Proof.lean, line 69 at b88a90ea8e90 ↗
  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 70
  have h3 := zpow_right_injective₀ (by norm_num : (0 : ℝ) < 2) (by norm_num) h2
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 70 at b88a90ea8e90 ↗
  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 72
  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--h1/Proof.lean, line 72 at b88a90ea8e90 ↗
  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 73–75
  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--h1/Proof.lean, lines 73–75 at b88a90ea8e90 ↗
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), ∀ (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 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 80–103
```
  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--h1/Proof.lean, lines 80–103 at b88a90ea8e90 ↗
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 83–87
  | 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--h1/Proof.lean, lines 83–87 at b88a90ea8e90 ↗
  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 85
  have hs : s = ∅ := Finset.card_eq_zero.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 85 at b88a90ea8e90 ↗
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 88–103
  | 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--h1/Proof.lean, lines 88–103 at b88a90ea8e90 ↗
  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 91
  · exact ⟨0, by simp⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 91 at b88a90ea8e90 ↗
  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 93
  · 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--h1/Proof.lean, line 93 at b88a90ea8e90 ↗
  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 94
  obtain ⟨e', rfl⟩ : ∃ e', e = e' + 1 := ⟨e - 1, by omega⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 94 at b88a90ea8e90 ↗
  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 95
  obtain ⟨a, ha⟩ := hs
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 95 at b88a90ea8e90 ↗
  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 96
  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--h1/Proof.lean, line 96 at b88a90ea8e90 ↗
  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 96
  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--h1/Proof.lean, line 96 at b88a90ea8e90 ↗
  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 97
  obtain ⟨z2, hz2⟩ := ih s e' (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 97 at b88a90ea8e90 ↗
  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 99
  have hr := hrec s a (fun k => ((2 : ℝ) ^ k) ^ e') ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 99 at b88a90ea8e90 ↗
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), ∀ (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 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 105–132
```
  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--h1/Proof.lean, lines 105–132 at b88a90ea8e90 ↗
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 108–120
  | 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--h1/Proof.lean, lines 108–120 at b88a90ea8e90 ↗
  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 110
  obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 110 at b88a90ea8e90 ↗
  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 111
  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--h1/Proof.lean, line 111 at b88a90ea8e90 ↗
  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 112
  have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (0 + 1))⁻¹) ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 112 at b88a90ea8e90 ↗
  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 113–114
  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--h1/Proof.lean, lines 113–114 at b88a90ea8e90 ↗
  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 117
  have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 117 at b88a90ea8e90 ↗
  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 119
  · exact absurd h5 h2a
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 119 at b88a90ea8e90 ↗
  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 120
  · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 120 at b88a90ea8e90 ↗
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 121–132
  | 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--h1/Proof.lean, lines 121–132 at b88a90ea8e90 ↗
  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 123
  obtain ⟨a, ha⟩ : s.Nonempty := Finset.card_pos.mp (by omega)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 123 at b88a90ea8e90 ↗
  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 124
  have h := hrec s a (fun k => (((2 : ℝ) ^ k) ^ (m + 1 + 1))⁻¹) ha
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 124 at b88a90ea8e90 ↗
  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 125–126
  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--h1/Proof.lean, lines 125–126 at b88a90ea8e90 ↗
  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 129
  have h2a : (2 : ℝ) ^ a ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 129 at b88a90ea8e90 ↗
  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 131
  · exact absurd h5 h2a
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 131 at b88a90ea8e90 ↗
  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 132
  · exact h5
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 132 at b88a90ea8e90 ↗
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), ∀ (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 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 134–150
```
  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--h1/Proof.lean, lines 134–150 at b88a90ea8e90 ↗
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 136–140
  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--h1/Proof.lean, lines 136–140 at b88a90ea8e90 ↗
  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 137–139
  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--h1/Proof.lean, lines 137–139 at b88a90ea8e90 ↗
  hgeo.hB.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ)
  Closes the goal.
  Uses library pow_one.
  Lean, line 138
  calc (2 : ℝ) = 2 ^ 1 := (pow_one 2).symm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 138 at b88a90ea8e90 ↗
  hgeo.hB.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ m
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 139
  _ ≤ 2 ^ m := pow_le_pow_right₀ (by norm_num) hm
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 139 at b88a90ea8e90 ↗
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 142
  | zero => simp
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 142 at b88a90ea8e90 ↗
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 143–150
  | 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--h1/Proof.lean, lines 143–150 at b88a90ea8e90 ↗
  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 145
  have hX : ((2 : ℝ) ^ k) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 145 at b88a90ea8e90 ↗
  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 146
  have hY : (2 : ℝ) ^ m ≠ 0 := by positivity
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 146 at b88a90ea8e90 ↗
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), ∀ (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 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 151–165
```
  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--h1/Proof.lean, lines 151–165 at b88a90ea8e90 ↗
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 153
  obtain ⟨k0, hk0⟩ : ∃ k0, k0 = i * (n / i) := ⟨_, rfl⟩
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 153 at b88a90ea8e90 ↗
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 154
  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--h1/Proof.lean, line 154 at b88a90ea8e90 ↗
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 155
  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--h1/Proof.lean, line 155 at b88a90ea8e90 ↗
  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 155
  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--h1/Proof.lean, line 155 at b88a90ea8e90 ↗
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 156
  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--h1/Proof.lean, line 156 at b88a90ea8e90 ↗
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 157
  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--h1/Proof.lean, line 157 at b88a90ea8e90 ↗
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 158–161
  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--h1/Proof.lean, lines 158–161 at b88a90ea8e90 ↗
  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 159
  have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ i) 1 (n / i)
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 159 at b88a90ea8e90 ↗
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 162–164
  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--h1/Proof.lean, lines 162–164 at b88a90ea8e90 ↗
  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 163
  have : (1 : ℤ) - 2 ^ k0 = -((2 : ℤ) ^ k0 - 1) := by ring
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 163 at b88a90ea8e90 ↗
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), ∀ (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 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 166–183
```
  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--h1/Proof.lean, lines 166–183 at b88a90ea8e90 ↗
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 169–174
  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--h1/Proof.lean, lines 169–174 at b88a90ea8e90 ↗
  hpoly.hdeg.s1 case
  Leaves ((1 : Polynomial ℤ) - (Polynomial.C : ℤ → Polynomial ℤ) ((2 : ℤ) ^ t) * Polynomial.X).natDegree ≤ (1 : ℕ).
  Lean, line 173
  · compute_degree
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 173 at b88a90ea8e90 ↗
  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 174
  · simp only [Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]; omega
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 174 at b88a90ea8e90 ↗
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 175
  have hev := Polynomial.eval_eq_sum_range' hdeg y
  targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Proof.lean, line 175 at b88a90ea8e90 ↗
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 176–179
  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--h1/Proof.lean, lines 176–179 at b88a90ea8e90 ↗
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--h1/Proof.lean, lines 184–311 at b88a90ea8e90 ↗
1. 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--h1/Proof.lean, lines 185–192 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 188 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 189–191 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 190 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 191 at b88a90ea8e90 ↗
2. 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--h1/Proof.lean, lines 193–220 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 196–199 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 198 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 200–220 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 203 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 205 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 206 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 207 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 208 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 209 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 211 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 212 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 213 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 214 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 215 at b88a90ea8e90 ↗
3. 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--h1/Proof.lean, lines 221–227 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 224 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 225–227 at b88a90ea8e90 ↗
4. 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--h1/Proof.lean, lines 228–236 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 231 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 232–236 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 234 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 235 at b88a90ea8e90 ↗
5. 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--h1/Proof.lean, line 239 at b88a90ea8e90 ↗
6. 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--h1/Proof.lean, line 240 at b88a90ea8e90 ↗
7. 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--h1/Proof.lean, lines 242–246 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 245 at b88a90ea8e90 ↗
8. 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--h1/Proof.lean, lines 248–251 at b88a90ea8e90 ↗
9. 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--h1/Proof.lean, lines 252–256 at b88a90ea8e90 ↗
10. 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--h1/Proof.lean, lines 257–276 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 258–269 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 260 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 261 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 262 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 263 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 264–269 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 274 at b88a90ea8e90 ↗
11. 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--h1/Proof.lean, lines 277–289 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 280 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 281 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 282 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 283 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 284–289 at b88a90ea8e90 ↗
12. 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--h1/Proof.lean, line 290 at b88a90ea8e90 ↗
13. 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--h1/Proof.lean, line 291 at b88a90ea8e90 ↗
14. 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--h1/Proof.lean, lines 293–296 at b88a90ea8e90 ↗
15. 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--h1/Proof.lean, lines 297–298 at b88a90ea8e90 ↗
16. 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--h1/Proof.lean, line 299 at b88a90ea8e90 ↗
17. 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--h1/Proof.lean, line 300 at b88a90ea8e90 ↗
18. 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--h1/Proof.lean, lines 301–303 at b88a90ea8e90 ↗
19. 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--h1/Proof.lean, lines 304–307 at b88a90ea8e90 ↗
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--h1/Proof.lean, lines 312–397 at b88a90ea8e90 ↗
1. 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--h1/Proof.lean, line 314 at b88a90ea8e90 ↗
2. 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--h1/Proof.lean, line 315 at b88a90ea8e90 ↗
3. 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--h1/Proof.lean, lines 316–320 at b88a90ea8e90 ↗
4. 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--h1/Proof.lean, lines 323–327 at b88a90ea8e90 ↗
5. 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--h1/Proof.lean, lines 333–359 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 335 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 336–341 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 337–339 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 338 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 339 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 342–343 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 344–352 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 346 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 347–348 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 351 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 353–354 at b88a90ea8e90 ↗
6. 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--h1/Proof.lean, lines 361–373 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 367 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 369–370 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 371 at b88a90ea8e90 ↗
7. 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--h1/Proof.lean, line 378 at b88a90ea8e90 ↗
8. 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--h1/Proof.lean, line 379 at b88a90ea8e90 ↗
9. 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--h1/Proof.lean, line 380 at b88a90ea8e90 ↗
10. 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--h1/Proof.lean, line 381 at b88a90ea8e90 ↗
11. 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--h1/Proof.lean, line 382 at b88a90ea8e90 ↗
12. 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--h1/Proof.lean, line 383 at b88a90ea8e90 ↗
13. 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--h1/Proof.lean, lines 384–388 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 385–387 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 386 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 387 at b88a90ea8e90 ↗
14. 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--h1/Proof.lean, line 389 at b88a90ea8e90 ↗
15. 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--h1/Proof.lean, line 390 at b88a90ea8e90 ↗
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--h1/Proof.lean, lines 398–454 at b88a90ea8e90 ↗
1. 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--h1/Proof.lean, lines 399–407 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 402 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 403–407 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 405 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 406 at b88a90ea8e90 ↗
2. 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--h1/Proof.lean, lines 408–451 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 410 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 411–439 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 414 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 415 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 417–422 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 418–420 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 419 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 420 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 423–424 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 425–427 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 428–429 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 430 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 431 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 435–438 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 440 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 441–445 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 444 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 446–447 at b88a90ea8e90 ↗
s12 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--h1/Proof.lean, line 455 at b88a90ea8e90 ↗
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--h1/Proof.lean, lines 456–556 at b88a90ea8e90 ↗
1. 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--h1/Proof.lean, lines 457–501 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 459 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 460–464 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 466–471 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 474 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 476 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 478–487 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 479–485 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 482–484 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 488–501 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 488 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 490–496 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 493–495 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 497 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 498 at b88a90ea8e90 ↗
2. 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--h1/Proof.lean, lines 502–505 at b88a90ea8e90 ↗
3. 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--h1/Proof.lean, lines 506–520 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 508 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 509 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 510 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 510 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 511 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 512 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 513–516 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 514 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 517–519 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 518 at b88a90ea8e90 ↗
4. 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--h1/Proof.lean, lines 521–529 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 524 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 525–529 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 527 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 528 at b88a90ea8e90 ↗
5. 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--h1/Proof.lean, lines 530–553 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 534 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 535 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 536 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 537 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 538–542 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 539–541 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 540 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 541 at b88a90ea8e90 ↗
  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--h1/Proof.lean, lines 543–548 at b88a90ea8e90 ↗
  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--h1/Proof.lean, line 552 at b88a90ea8e90 ↗
s14 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--h1/Proof.lean, line 557 at b88a90ea8e90 ↗

The proof in Lean (Proof.lean)

Checked by the kernel — these bytes are the ones its attestation names: attestations/000332.json, merge commit b88a90ea8e90.

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--h1/Proof.lean, sha256 4829240ca0f3.

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

/-! Hole `numerator_integral` 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__h1 : ∃ (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))) := by
  -- 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 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
  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

What is known

Notes

Annex to erdos-1050--h1-v2--h2--h1 (the numerator integrality of the Padé construction for sum 1/(2^n - 3)). full text →

Work on it

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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--h1».Context

/-! Hole `numerator_integral` 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__h1 : ∃ (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))) := 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

$\displaystyle \exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$

Rendered from gloss/1f2c89f09924….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/d62bc02fda73….md · the first version of its chain
Read this version
There is a sequence of integers $a(n)$, indexed by the natural numbers, such that for every $n \ge 1$ $$a(n) \;=\; 9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr),$$ where, with $c = 8/3$ and all arithmetic in the real numbers:
- $W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
- for $1\le k\le n$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$;
- $S_1(n)=\displaystyle\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$;
- $S_2(n)=\displaystyle\sum_{i=1}^{n-1}\frac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$.
In other words: for each $n\ge1$ the real number $9^n W(n)(S_1(n)-S_2(n))$ is an integer. All sums here are finite.

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 the denominator of $A_k$ the exponent $l-k$ is an integer and may be negative, so $2^{\,l-k}$ is a real power such as $1/2$.
- For $n=1$ the product over $t$ in $A_k$ is empty (equal to $1$) and $S_2(1)=0$ (an empty sum).
- Nothing is required of $a(0)$.
The file's comment records that this statement is the hole numerator_integral 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/6013312970f3….md · diff to its predecessor
Read this version
There is a sequence of integers $a(n)$, indexed by the natural numbers, such that for every $n \ge 1$ $$a(n) \;=\; 9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr),$$ where, with $c = 8/3$ and all arithmetic in the real numbers:

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

for $1\le k\le n$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$;

$S_1(n)=\displaystyle\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$;

$S_2(n)=\displaystyle\sum_{i=1}^{n-1}\frac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$.

In other words: for each $n\ge1$ the real number $9^n W(n)(S_1(n)-S_2(n))$ is an integer. All sums here are finite.

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 the denominator of $A_k$ the exponent $l-k$ is an integer and may be negative, so $2^{\,l-k}$ is a real power such as $1/2$.

For $n=1$ the product over $t$ in $A_k$ is empty (equal to $1$) and $S_2(1)=0$ (an empty sum).

Nothing is required of $a(0)$.

The file's comment records that this statement is the hole numerator_integral 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/1f2c89f09924….md · diff to its predecessor
Read this version
$\displaystyle \exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$

There is a sequence of integers $a(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, that is, $9^n W(n)(S_1(n)-S_2(n))$ is an integer, where, with $c=8/3$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$ for $1\le k\le n$, $S_1(n)=\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$ and $S_2(n)=\sum_{i=1}^{n-1}\dfrac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$, all sums finite. Here $n-2$ is natural-number subtraction (so $(n-2)!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponent $l-k$ is an integer (so $2^{l-k}$ may be $1/2$ and so on), at $n=1$ the product over $t$ is empty and $S_2(1)=0$, and nothing is required of $a(0)$.

Diff of 6013312970f3 against d62bc02fda73

@@ -3,8 +3,11 @@
 where, with $c = 8/3$ and all arithmetic in the real numbers:
 
-- $W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
-- for $1\le k\le n$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$;
-- $S_1(n)=\displaystyle\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$;
-- $S_2(n)=\displaystyle\sum_{i=1}^{n-1}\frac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$.
+$W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
+
+for $1\le k\le n$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$;
+
+$S_1(n)=\displaystyle\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$;
+
+$S_2(n)=\displaystyle\sum_{i=1}^{n-1}\frac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$.
 
 In other words: for each $n\ge1$ the real number $9^n W(n)(S_1(n)-S_2(n))$ is an integer. All sums here are finite.
@@ -12,8 +15,11 @@
 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 the denominator of $A_k$ the exponent $l-k$ is an integer and may be negative, so $2^{\,l-k}$ is a real power such as $1/2$.
-- For $n=1$ the product over $t$ in $A_k$ is empty (equal to $1$) and $S_2(1)=0$ (an empty sum).
-- Nothing is required of $a(0)$.
+$n-2$ 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 the denominator of $A_k$ the exponent $l-k$ is an integer and may be negative, so $2^{\,l-k}$ is a real power such as $1/2$.
+
+For $n=1$ the product over $t$ in $A_k$ is empty (equal to $1$) and $S_2(1)=0$ (an empty sum).
+
+Nothing is required of $a(0)$.
 
 The file's comment records that this statement is the hole `numerator_integral` of a merged partial proof, written into its own node by the gate.

Diff of 1f2c89f09924 against 6013312970f3

@@ -1,25 +1,3 @@
-There is a sequence of integers $a(n)$, indexed by the natural numbers, such that for every $n \ge 1$
-$$a(n) \;=\; 9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr),$$
-where, with $c = 8/3$ and all arithmetic in the real numbers:
+$$\exists\, a:\mathbb{N}\to\mathbb{Z}\ \ \forall n\ge 1:\qquad a(n)\;=\;9^n\,W(n)\,\bigl(S_1(n)-S_2(n)\bigr)$$
 
-$W(n)=\displaystyle (n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\;\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$;
-
-for $1\le k\le n$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$;
-
-$S_1(n)=\displaystyle\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$;
-
-$S_2(n)=\displaystyle\sum_{i=1}^{n-1}\frac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$.
-
-In other words: for each $n\ge1$ the real number $9^n W(n)(S_1(n)-S_2(n))$ is an integer. All sums here are finite.
-
-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 the denominator of $A_k$ the exponent $l-k$ is an integer and may be negative, so $2^{\,l-k}$ is a real power such as $1/2$.
-
-For $n=1$ the product over $t$ in $A_k$ is empty (equal to $1$) and $S_2(1)=0$ (an empty sum).
-
-Nothing is required of $a(0)$.
-
-The file's comment records that this statement is the hole `numerator_integral` of a merged partial proof, written into its own node by the gate.
+There is a sequence of integers $a(n)$ such that for every $n\ge 1$ the identity holds in $\mathbb{R}$, that is, $9^n W(n)(S_1(n)-S_2(n))$ is an integer, where, with $c=8/3$, $W(n)=(n-2)!\;\prod_{k=1}^{n}\bigl(1-c\,2^{k}\bigr)\prod_{k=\lfloor (n+1)/2\rfloor}^{n}\bigl(1-2^{k}\bigr)$, $A_k=\dfrac{-\prod_{t=1}^{n-1}\bigl(1-c\,2^{\,t+k}\bigr)}{\prod_{1\le l\le n,\ l\ne k}\bigl(1-2^{\,l-k}\bigr)}$ for $1\le k\le n$, $S_1(n)=\sum_{k=1}^{n} A_k\sum_{i=1}^{k}\bigl(1-c\,2^{i}\bigr)^{-1}$ and $S_2(n)=\sum_{i=1}^{n-1}\dfrac{\sum_{k=1}^{n} A_k\,\bigl(c\,2^{k}\bigr)^{-i}}{2^{i}-1}$, all sums finite. Here $n-2$ is natural-number subtraction (so $(n-2)!=1$ at $n=1$), $\lfloor (n+1)/2\rfloor$ is natural-number division, the exponent $l-k$ is an integer (so $2^{l-k}$ may be $1/2$ and so on), at $n=1$ the product over $t$ is empty and $S_2(1)=0$, and nothing is required of $a(0)$.

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--h1/Statement.lean. Origin: compiler-derived.

Attestation: pass, 7 of 7 checks

Checked by the kernel — the gate’s record of the run: attestations/000332.json, merge commit b88a90ea8e90.

Verdict pass, from attestations/000332.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/000332.json, merge commit b88a90ea8e90.

Witness.lean — checked at step 7 of the run recorded below: pass. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/Witness.lean, sha256 227f693807cf.

import Mathlib
import Nodes.«erdos-1050--h1-v2--h2--h1».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/0700da0ffa50….md · Improve these words →

Annexes (1), in full

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

Untrusted: annex (D-31), by t1001-1050b, 2026-10-01T11:48:26Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h2--h1/annex/a03db62f0576f792454c3f95d14fb02d193662242ec562819da63116009d326b.md.

Annex to erdos-1050--h1-v2--h2--h1 (the numerator integrality of the Padé construction for sum 1/(2^n - 3)). A complete informal proof of the one part earlier annexes and logs left open (the "head" sum S1, called head_integral in the partial proofs), by partial fractions in the parameter c. Nothing here is checked in Lean; every identity below was checked in exact rational arithmetic for n = 1..13.

Notation. Fix n >= 1 and c = 8/3. D_k = prod over l in [1,n], l != k, of (1 - 2^(l-k)). A_k(x) = - prod_{t=1..n-1} (1 - x 2^(t+k)) / D_k, a polynomial in x of degree n-1. h_i(x) = 1/(1 - x 2^i). The node's right-hand side is 9^n (n-2)! P(n) M_n (S1 - S2), with P(n) = prod_{k=1..n} (1 - c 2^k), M_n = prod_{k=ceil(n/2)..n} (1 - 2^k), S1 = sum_{k=1..n} A_k(c) sum_{i=1..k} h_i(c), and S2 the sum with the factors 1/(2^i - 1). The S2 half is already proved in Lean (the partial submitted with this annex). This annex proves: 9^n P(n) M_n S1 is an integer (the factorial is not needed).

Step 0, a Lagrange fact (L). For distinct nodes y_1..y_n and every integer e >= -(n-1): sum_k y_k^e / prod_{l != k} (1 - y_l / y_k) = H_e(y_1..y_n), the complete homogeneous symmetric polynomial of degree e, read as 0 for e < 0. (Equivalently sum_k y_k^(e+n-1) / prod_{l != k} (y_k - y_l) = H_e: a divided difference of a monomial.) With y_k = 2^k this reads sum_k 2^(ke) / D_k = H_e, an integer. The recursion W(s, e) = W(s minus a, e-1) + y_a W(s, e-1), obtained from y_k^e = y_k^(e-1) (y_k - y_a) + y_a y_k^(e-1), proves integrality for integer nodes by induction.

Step 1, the identity. Write prod_{t=1..n-1} (1 - 2^t y) = sum_j alpha_j y^j (integer alpha_j), so A_k(x) = -(1/D_k) sum_j alpha_j 2^(kj) x^j. For each i and j: x^j h_i(x) = 2^(-ij) h_i(x) - sum_{s<j} 2^(-i(j-s)) x^s (a finite geometric sum in x 2^i). Summing over i <= k and then over k:

S1(x) = E(x) + sum_{i=1..n} b_i h_i(x), where b_i = sum_{k=i..n} A_k(2^(-i)) = sum_{k=1..n} A_k(2^(-i)) (the terms k < i vanish, since the factor t = i-k of A_k(2^(-i)) is 1 - 2^0 = 0), and E(x) = sum_{j=1..n-1} (2^j - 1)^(-1) sum_{j'=j..n-1} alpha_{j'} (H_{j'} - H_{j'-j}) x^(j'-j). The formula for E uses sum_{i=1..k} 2^(-i m) = (1 - 2^(-k m)) / (2^m - 1) and then (L) twice (exponents j' and j'-m... all exponents that occur are >= -(n-1)). In words: E is the polynomial part of S1 at infinity and b_i the residues; the truncation i <= k of the harmonic-type sums costs nothing at the poles x = 2^(-i), because A_k vanishes there for k < i.

Step 2, integrality. (a) E(c): each inner sum is an integer polynomial in x of degree <= n-2, so 3^(n-2) times it at x = 8/3 is an integer; each 2^j - 1 with 1 <= j <= n-1 divides M_n (take the multiple of j in [ceil(n/2), n]; this is hdvd in the Lean partial). So 3^(n-2) M_n E(c) is an integer. (b) b_i is an integer. With y_k = 2^(k-i): b_i = - sum_k g(y_k) y_k^(n-1) / prod_{l != k} (y_k - y_l), g(y) = prod_{t=1..n-1} (1 - 2^t y), a divided difference of g(y) y^(n-1) over the n nodes. The nodes with k < i are the roots 2^(-t), t = 1..i-1, of g; cancelling them leaves the divided difference of the integer polynomial y^(n-1) prod_{t=i..n-1} (1 - 2^t y), times prod_{t=1..i-1} (-2^t), over the integer nodes 1, 2, ..., 2^(n-i). By (L) that is an integer. (Without the cancellation one only gets b_i in Z[1/2].) (c) 3^n P(n) h_i(c) = 3 prod_{i' != i} (3 - 8 2^(i')) is an integer for i <= n. Hence 9^n P(n) M_n S1(c) is an integer, and so is the node's expression (multiply by (n-2)!).

Relation to earlier notes on this target: the earlier logs split S1 by the weight k (a q-derivative, needing the q-lcm prod_{d<n} Phi_d(2)) and left a remainder R(n) open. Expanding at infinity instead of at 0 avoids the q-derivative term altogether: only the denominators 2^j - 1, j <= n-1, appear, which is exactly why the Mersenne block M_n is in the statement.

Suggested Lean route: (1) (L) as integrality of divided differences of monomials at integer nodes, by the recursion in Step 0; (2) the per-term identity for x^j h_i; (3) assemble. The existing partial's key step is the special case of (L) used for S2.

How to read this page

How to read a statement’s page →

Explainer

What is proved, and the plan

Finite sums of integers

Lagrange sums at the nodes $2^k$

Geometric sums, a divisibility, and an integer polynomial

The residues are integers

The polynomial part

The first sum $S_1$

The second sum $S_2$

What is proved, and the plan

Finite sums of integers (hsumZ)

Lagrange sums at the nodes $2^k$ (h0 hrec hW hV)

Geometric sums, a divisibility, and an integer polynomial (hdvd hgeo hpoly)

The residues are integers (residue_integral)

The polynomial part (polypart_integral)

The first sum $S_1$ (head_integral s12)

The second sum $S_2$ (hmer s14)

What is proved, and the plan

Finite sums of integers (hsumZ)

Lagrange sums at the nodes $2^k$ (h0 hrec hW hV)

Geometric sums, a divisibility, and an integer polynomial (hdvd hgeo hpoly)

The residues are integers (residue_integral)

The polynomial part (polypart_integral)

The first sum $S_1$ (head_integral s12)

The second sum $S_2$ (hmer s14)

How it was proved

What is known

Work on it

Record