Padé remainder bound without the denominator hypothesis

spec-180d8b72 erdos_1050_pade_remainder_bound

proved

Proposed by a contributor.

$\displaystyle \forall n\in\mathbb{N}:\qquad R_n>0\quad\wedge\quad R_n^{2}\,2^{4n^2}\le 2^{2n+4},\qquad R_n=3T\,Q_x(n)-A_q(n),\quad T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$

For every natural number $n$ the real number $R_n$ is positive and $R_n^2\cdot 2^{4n\cdot n}\le 2^{2n+4}$ (that is, $R_n\le 2^{\,n+2-2n^2}$), where $T$ is Lean's ∑' over all $m\in\mathbb{N}$ in $\mathbb{R}$ (which would be $0$ for a non-summable family; this one converges) and $Q_x(n)$, $A_q(n)$ are read as reals; with $x_n=3/2^n$, $Q_x(n)$ is the polynomial $\sum_k Q_c(n,k)X^k$ at $x_n$. The functions $Q_c:\mathbb{N}\times\mathbb{N}\to\mathbb{Q}$, $Q_x:\mathbb{N}\to\mathbb{Q}$ and $A_q:\mathbb{N}\to\mathbb{Q}$ are fixed by hypotheses as $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\prod_{i=0}^{k-1}\frac{2^{n-i}-1}{2^{i+1}-1}\prod_{i=0}^{n-1}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$, $Q_x(n)=\sum_{k=0}^{n}Q_c(n,k)\,(3/2^n)^k$ and $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. Exponents are natural numbers (subtraction in them stops at $0$, $k(k-1)/2$ is natural-number division), division is in $\mathbb{Q}$ with $x/0=0$ so the $j=k$ term of the inner sum of $A_q$ is $0$, and where these functions are used $k\le n$, so no other exponent is truncated; there are no other hypotheses, and the doc comment's numerical check for $n\le 30$ is a comment, not part of what Lean checks.

unverified · history

Lean

theorem erdos_1050_pade_remainder_bound : ∀ (Qc : ℕ → ℕ → ℚ),
  (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) →
    ∀ (Qx : ℕ → ℚ),
      (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) →
        ∀ (Aq : ℕ → ℚ),
          (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) →
              ∀ (n : ℕ),
                (0 : ℝ) <
                    (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                      (↑(Aq n) : ℝ) ∧
                  ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                          (↑(Aq n) : ℝ)) ^
                        (2 : ℕ) *
                      (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤
                    (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) := 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/fe09e6640e14….md, written by t1006-w3, drafted with Claude Opus 5.5 (claude-opus-5-5) in a Claude Code agent session, 2026-10-06.

How the proof goes

Draft · drafted, by the model named above

Write $T=\sum_{m\ge0}1/(2^{m+3}-3)$, $x_n=3/2^n$ and $R_n=3T\,Q_x(n)-A_q(n)$, as in the statement. The statement has no hypothesis about denominators; apart from that one missing hypothesis (and so one argument fewer in intro Qc hQc Qx hQx Aq hAq n) this proof's tactic script is the same, line for line, as that of erdos-1050--h1-v2--h4.

The idea is that $R_n$ is the sum of a convergent series in $x_n$ whose first $2n+1$ coefficients vanish: $$R_n=\sum_{K>2n}\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\,x_n^K\qquad(n\ge1).$$ The vanishing comes from Lagrange interpolation: the numbers $2^jQ_c(n,j)$ are the interpolation weights of $N(X)=\prod_{n<L\le 2n}(X-2^L)$ at the nodes $2^0,\dots,2^n$, and $N(2^K)=0$ for $n<K\le 2n$. Each remaining coefficient lies in $(0,\,2\cdot2^{-K}]$, so $R_n$ is positive and at most a geometric tail, $R_n\le 8\,(x_n/2)^{2n+1}$ for $n\ge2$; the stated bound is then an inequality between powers of $2$ and $3$. The cases $n=0$ and $n=1$ are computed directly, using $0<T\le 2/5$.

Read it step by step (7 more sections)

Steps

id1_test have ∀ (n j : ℕ), j ≤ n → (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
lower_test have ∀ (j : ℕ), ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
upper_test have ∀ (j n : ℕ), j ≤ n → ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (-1 : ℚ) ^ (n - j) * (2 : ℚ) ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
rhs_test have ∀ (j n : ℕ), j ≤ n → (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
id1_full have ∀ (n j : ℕ), j ≤ n → (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)

The four identities for products of $2^a-2^b$

Draft · drafted, by the model named above

These five steps have the same statements and proofs as the first five steps of the proof of erdos-1050--h1-v2--h3, where they are explained in more detail. Write $P_m=\prod_{s<m}(2^{s+1}-1)$.

id1_test: for $j\le n$, $\big(\prod_{i<j}(2^{n-i}-1)\big)P_{n-j}=P_n$ (induction on $j$).
lower_test: $\prod_{i<j}(2^j-2^i)=2^{j(j-1)/2}P_j$.
upper_test: for $j\le n$, $\prod_{j<i\le n}(2^j-2^i)=(-1)^{n-j}2^{j(n-j)}P_{n-j}$.
rhs_test: for $j\le n$, $\big(\prod_{n<K\le 2n}(2^j-2^K)\big)P_{n-j}=(-1)^n2^{jn}P_{2n-j}$.
id1_full combines them: for $j\le n$,

$$Q_c(n,j)\,2^j\prod_{i\le n,\ i\ne j}(2^j-2^i)=\prod_{n<K\le 2n}(2^j-2^K),$$ with $Q_c(n,j)$ written out as in the statement.

In other words, the numbers $2^jQ_c(n,j)$ are the Lagrange interpolation weights of the polynomial $N(X)=\prod_{n<L\le 2n}(X-2^L)$ at the nodes $2^0,2^1,\dots,2^n$.

Steps

lagr_eval have ∀ (n : ℕ) (x : ℕ → ℚ), (∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j) → ∀ (N : Polynomial ℚ), N.degree ≤ (↑n : WithBot ℕ) → ∀ (r : ℕ → ℚ), (∀ j ≤ n, r j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (x j - x i) = Polynomial.eval (x j) N) → N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))
tail_coeff have ∀ (n K : ℕ), n < K → ∀ (Qc : ℕ → ℕ → ℚ), (∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ L)) → ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i)

Interpolation, and the coefficients beyond $n$

Draft · drafted, by the model named above

lagr_eval is Lagrange interpolation over $\mathbb Q$: if $x_0,\dots,x_n$ are distinct, $\deg N\le n$, and $r_j\prod_{i\ne j}(x_j-x_i)=N(x_j)$ for every $j\le n$, then $N=\sum_{j\le n}r_j\prod_{i\le n,\,i\ne j}(X-x_i)$. It is proved by showing the difference has degree at most $n$ and vanishes at the $n+1$ nodes.

tail_coeff is the identity the rest of the proof is built on. For $K>n$ and any $Q_c$ satisfying the conclusion of id1_full for all $j\le n$: $$\sum_{j=0}^{n}\frac{Q_c(n,j)}{2^{K-j}-1}=\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}.$$ The proof takes the nodes $x_i=2^i$ and the polynomial $N$ above, checks that $\deg N\le n$ (tail_coeff.hNdeg) and the weight condition (tail_coeff.hr), applies lagr_eval, and evaluates both sides at $X=2^K$ (tail_coeff.hev). Dividing by $\prod_{i\le n}(2^K-2^i)$, which is nonzero because $K>n$, the $j$-th term becomes $Q_c(n,j)\,2^j/(2^K-2^j)=Q_c(n,j)/(2^{K-j}-1)$.

Splitting the constant $3T$

Draft · drafted, by the model named above

Let $T=\sum_{m\ge 0}1/(2^{m+3}-3)$ as in the statement.

split3T states that for every $n$ $$3T=\sum_{j<n}\frac{3}{2^{j+1}-3}+\sum_{m\ge 0}\frac{3}{2^{m+n+1}-3}.$$ With $g(m)=3/(2^{m+1}-3)$, $3T=\sum_{m\ge0}g(m+2)$, and $g(0)+g(1)=-3+3=0$ (split3T.hfirst), so $3T=\sum_{m\ge 0}g(m)$; this sum is then split after its first $n$ terms.

tail3 states that for $n\ge 1$ $$\sum_{m\ge0}\frac{3}{2^{m+n+1}-3}=\sum_{s\ge0}\frac{(3/2^n)^{s+1}}{2^{s+1}-1}.$$ Both sides are the double sum of the nonnegative numbers $f(m,s)=\big(3/2^{m+n+1}\big)^{s+1}$: summing over $s$ first gives the geometric series $3/(2^{m+n+1}-3)$ (tail3.inner1), and summing over $m$ first gives $(3/2^n)^{s+1}/(2^{s+1}-1)$ (tail3.inner2). The order of summation is exchanged because all terms are nonnegative and the double series converges (tail3.hU, tail3.hcomm).

Steps

cauchy_fin have ∀ (n : ℕ) (q a : ℕ → ℝ) (x A : ℝ), HasSum (fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))) A → HasSum (fun (K : ℕ) => ∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then q j * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), q j * x ^ j) * A)
h4_series have ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (1 : ℕ) ≤ n → HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ))

The remainder as a power series

Draft · drafted, by the model named above

cauchy_fin is the product of a polynomial with a power series: if $\sum_{s\ge0}a_s x^{s+1}=A$, then the series over $K$ of $\sum_{j\le n,\ j<K}q_j\,a_{K-j-1}\,x^K$ has sum $\big(\sum_{j\le n}q_jx^j\big)A$.

h4_series states that for $n\ge 1$, with $x=3/2^n$, the remainder $R_n=3T\,Q_x(n)-A_q(n)$ is the sum of the series $$R_n=\sum_{K>2n}\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\,x^K .$$ The steps are these.

Put $a_s=1/(2^{s+1}-1)$ and $A=\sum_{s\ge0}a_sx^{s+1}$, which converges because $x/2<1$ (h4_series.hx2, h4_series.hAsum).
By split3T and tail3, $3T=c_n+A$ with $c_n=\sum_{j<n}3/(2^{j+1}-3)$ (h4_series.h3T).
The term $Q_x(n)\,c_n$ in $A_q(n)$ then cancels, leaving $R_n=Q_x(n)\,A-P$, where $P=\sum_{k\le n}\big(\sum_{j\le k}\frac{Q_c(n,j)}{2^{k-j}-1}\big)x^k$ (h4_series.hval).
cauchy_fin with $q_j=Q_c(n,j)$ expands $Q_x(n)\,A$ as a series in $K$, and $P$ is a finite series (h4_series.hP); their difference is computed coefficient by coefficient (h4_series.hfun):
- for $K\le n$ the two coefficients are equal, because the term $j=K$ of $P$'s inner sum is $Q_c(n,K)/0=0$;
- for $K>n$ the coefficient is $\sum_{j\le n}Q_c(n,j)/(2^{K-j}-1)$, which tail_coeff turns into the quotient of products, using id1_full for the concrete $Q_c$ (h4_series.hid1);
- for $n<K\le 2n$ that quotient is $0$, because the numerator contains the factor $2^K-2^K$.

Bounding each term

Draft · drafted, by the model named above

prod_one_sub_ge is the inequality $1-\sum_{i\in s}a_i\le\prod_{i\in s}(1-a_i)$ for $0\le a_i\le 1$, by induction on the finite set $s$.

nd_bound states that for $n\ge 1$ and $K\ge 2n+1$ the coefficient in h4_series satisfies $$0<\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\le 2\cdot 2^{-K}.$$ Every factor is positive since $L,i<K$. The numerator is at most $(2^K)^n$ (nd_bound.hN). The denominator is at least $(2^K)^{n+1}/2$ (nd_bound.hD): each factor is $2^K(1-2^{-(K-i)})$, the numbers $2^{-(K-i)}$ for $i\le n$ sum to at most $1/2$ because $K-n\ge 2$, and prod_one_sub_ge applies.

Steps

ineq_pow have ∀ (n : ℕ), (2 : ℕ) ≤ n → (9 : ℕ) * (81 : ℕ) ^ n ≤ (256 : ℕ) ^ n
h4_big have ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (2 : ℕ) ≤ n → (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ))

The bound for $n\ge 2$

Draft · drafted, by the model named above

ineq_pow states $9\cdot 81^n\le 256^n$ for $n\ge 2$ (induction on $n$).

h4_big is the conclusion of the statement for $n\ge 2$. Write $x=3/2^n$, so $x/2\le 3/8$ (h4_big.hx2), and let $G_K$ be the $K$-th term of the series in h4_series.

Positivity. Every $G_K$ is $\ge 0$ and $G_{2n+1}>0$ (h4_big.hGnn, h4_big.hGpos, both from nd_bound), so the sum $R_n$ is positive.
Upper bound. By nd_bound, $G_K\le 2(x/2)^K$ for $K>2n$. Summing the geometric series, $R_n\le 2(x/2)^{2n+1}/(1-x/2)\le 8(x/2)^{2n+1}$, using $1/(1-x/2)\le 4$. Squaring, $R_n^2\le 64\,(x/2)^{4n+2}$, and $x/2=3/2^{n+1}$.
Arithmetic. It remains that $64\cdot 3^{4n+2}\cdot 2^{4n^2}\le 2^{2n+4}\,(2^{n+1})^{4n+2}$. The right side is $2^{4n^2}\cdot 64\cdot 256^n$ and the left side is $2^{4n^2}\cdot 64\cdot 9\cdot 81^n$, so this is ineq_pow. It is proved in $\mathbb N$ and cast to $\mathbb R$.

Steps

7 steps

T_facts have (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ)
s17 obtain (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ)
hQx0 have Qx (0 : ℕ) = (1 : ℚ)
hAq0 have Aq (0 : ℕ) = (0 : ℚ)
e have ((1 : ℝ) * ((3 : ℝ) * T) - (0 : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (0 : ℕ) * (0 : ℕ)) = (9 : ℝ) * (T * T)
hQx1 have Qx (1 : ℕ) = (3 / 2 : ℚ)
hAq1 have Aq (1 : ℕ) = (0 : ℚ)

The constant $T$, and the cases $n=0$ and $n=1$

Draft · drafted, by the model named above

T_facts states that the series defining $T$ converges, $T>0$ and $T\le 2/5$, from $2^{m+3}-3\ge 5\cdot 2^m$, which bounds the $m$-th term by $\frac15 2^{-m}$. The step s17 takes $T>0$ and $T\le 2/5$ out of it.

The proof then introduces the hypotheses of the statement and splits on $n$.

$n=0$: $Q_x(0)=1$ (hQx0) and $A_q(0)=0$ (hAq0), so $R_0=3T>0$, and $R_0^2\cdot 2^0=9T^2$ (e), which is at most $2^4$ since $T\le 2/5$.
$n=1$: $Q_x(1)=3/2$ (hQx1) and $A_q(1)=0$ (hAq1), so $R_1=\frac92T>0$, and $R_1^2\cdot 2^4\le 2^6$ again follows from $T\le 2/5$.
$n\ge 2$: h4_big.

Improve these words →

History: 1 version on record

2026-10-06 · written by t1006-w3, drafted with Claude Opus 5.5 (claude-opus-5-5) in a Claude Code agent session · not signed · current · explainer/fe09e6640e14….md · the first version of its chain
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How the proof goes
Write $T=\sum_{m\ge0}1/(2^{m+3}-3)$, $x_n=3/2^n$ and $R_n=3T\,Q_x(n)-A_q(n)$, as in the statement. The statement has no hypothesis about denominators; apart from that one missing hypothesis (and so one argument fewer in intro Qc hQc Qx hQx Aq hAq n) this proof's tactic script is the same, line for line, as that of erdos-1050--h1-v2--h4.

The idea is that $R_n$ is the sum of a convergent series in $x_n$ whose first $2n+1$ coefficients vanish: $$R_n=\sum_{K>2n}\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\,x_n^K\qquad(n\ge1).$$ The vanishing comes from Lagrange interpolation: the numbers $2^jQ_c(n,j)$ are the interpolation weights of $N(X)=\prod_{n<L\le 2n}(X-2^L)$ at the nodes $2^0,\dots,2^n$, and $N(2^K)=0$ for $n<K\le 2n$. Each remaining coefficient lies in $(0,\,2\cdot2^{-K}]$, so $R_n$ is positive and at most a geometric tail, $R_n\le 8\,(x_n/2)^{2n+1}$ for $n\ge2$; the stated bound is then an inequality between powers of $2$ and $3$. The cases $n=0$ and $n=1$ are computed directly, using $0<T\le 2/5$.
The four identities for products of $2^a-2^b$ (id1_test lower_test upper_test rhs_test id1_full)
These five steps have the same statements and proofs as the first five steps of the proof of erdos-1050--h1-v2--h3, where they are explained in more detail. Write $P_m=\prod_{s<m}(2^{s+1}-1)$.
- id1_test: for $j\le n$, $\big(\prod_{i<j}(2^{n-i}-1)\big)P_{n-j}=P_n$ (induction on $j$).
- lower_test: $\prod_{i<j}(2^j-2^i)=2^{j(j-1)/2}P_j$.
- upper_test: for $j\le n$, $\prod_{j<i\le n}(2^j-2^i)=(-1)^{n-j}2^{j(n-j)}P_{n-j}$.
- rhs_test: for $j\le n$, $\big(\prod_{n<K\le 2n}(2^j-2^K)\big)P_{n-j}=(-1)^n2^{jn}P_{2n-j}$.
- id1_full combines them: for $j\le n$,
$$Q_c(n,j)\,2^j\prod_{i\le n,\ i\ne j}(2^j-2^i)=\prod_{n<K\le 2n}(2^j-2^K),$$ with $Q_c(n,j)$ written out as in the statement.

In other words, the numbers $2^jQ_c(n,j)$ are the Lagrange interpolation weights of the polynomial $N(X)=\prod_{n<L\le 2n}(X-2^L)$ at the nodes $2^0,2^1,\dots,2^n$.
Interpolation, and the coefficients beyond $n$ (lagr_eval tail_coeff)
lagr_eval is Lagrange interpolation over $\mathbb Q$: if $x_0,\dots,x_n$ are distinct, $\deg N\le n$, and $r_j\prod_{i\ne j}(x_j-x_i)=N(x_j)$ for every $j\le n$, then $N=\sum_{j\le n}r_j\prod_{i\le n,\,i\ne j}(X-x_i)$. It is proved by showing the difference has degree at most $n$ and vanishes at the $n+1$ nodes.

tail_coeff is the identity the rest of the proof is built on. For $K>n$ and any $Q_c$ satisfying the conclusion of id1_full for all $j\le n$: $$\sum_{j=0}^{n}\frac{Q_c(n,j)}{2^{K-j}-1}=\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}.$$ The proof takes the nodes $x_i=2^i$ and the polynomial $N$ above, checks that $\deg N\le n$ (tail_coeff.hNdeg) and the weight condition (tail_coeff.hr), applies lagr_eval, and evaluates both sides at $X=2^K$ (tail_coeff.hev). Dividing by $\prod_{i\le n}(2^K-2^i)$, which is nonzero because $K>n$, the $j$-th term becomes $Q_c(n,j)\,2^j/(2^K-2^j)=Q_c(n,j)/(2^{K-j}-1)$.
Splitting the constant $3T$ (tail3 split3T)
Let $T=\sum_{m\ge 0}1/(2^{m+3}-3)$ as in the statement.

split3T states that for every $n$ $$3T=\sum_{j<n}\frac{3}{2^{j+1}-3}+\sum_{m\ge 0}\frac{3}{2^{m+n+1}-3}.$$ With $g(m)=3/(2^{m+1}-3)$, $3T=\sum_{m\ge0}g(m+2)$, and $g(0)+g(1)=-3+3=0$ (split3T.hfirst), so $3T=\sum_{m\ge 0}g(m)$; this sum is then split after its first $n$ terms.

tail3 states that for $n\ge 1$ $$\sum_{m\ge0}\frac{3}{2^{m+n+1}-3}=\sum_{s\ge0}\frac{(3/2^n)^{s+1}}{2^{s+1}-1}.$$ Both sides are the double sum of the nonnegative numbers $f(m,s)=\big(3/2^{m+n+1}\big)^{s+1}$: summing over $s$ first gives the geometric series $3/(2^{m+n+1}-3)$ (tail3.inner1), and summing over $m$ first gives $(3/2^n)^{s+1}/(2^{s+1}-1)$ (tail3.inner2). The order of summation is exchanged because all terms are nonnegative and the double series converges (tail3.hU, tail3.hcomm).
The remainder as a power series (cauchy_fin h4_series)
cauchy_fin is the product of a polynomial with a power series: if $\sum_{s\ge0}a_s x^{s+1}=A$, then the series over $K$ of $\sum_{j\le n,\ j<K}q_j\,a_{K-j-1}\,x^K$ has sum $\big(\sum_{j\le n}q_jx^j\big)A$.

h4_series states that for $n\ge 1$, with $x=3/2^n$, the remainder $R_n=3T\,Q_x(n)-A_q(n)$ is the sum of the series $$R_n=\sum_{K>2n}\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\,x^K .$$ The steps are these.
1. Put $a_s=1/(2^{s+1}-1)$ and $A=\sum_{s\ge0}a_sx^{s+1}$, which converges because $x/2<1$ (h4_series.hx2, h4_series.hAsum).
2. By split3T and tail3, $3T=c_n+A$ with $c_n=\sum_{j<n}3/(2^{j+1}-3)$ (h4_series.h3T).
3. The term $Q_x(n)\,c_n$ in $A_q(n)$ then cancels, leaving $R_n=Q_x(n)\,A-P$, where $P=\sum_{k\le n}\big(\sum_{j\le k}\frac{Q_c(n,j)}{2^{k-j}-1}\big)x^k$ (h4_series.hval).
4. cauchy_fin with $q_j=Q_c(n,j)$ expands $Q_x(n)\,A$ as a series in $K$, and $P$ is a finite series (h4_series.hP); their difference is computed coefficient by coefficient (h4_series.hfun):
  - for $K\le n$ the two coefficients are equal, because the term $j=K$ of $P$'s inner sum is $Q_c(n,K)/0=0$;
  - for $K>n$ the coefficient is $\sum_{j\le n}Q_c(n,j)/(2^{K-j}-1)$, which tail_coeff turns into the quotient of products, using id1_full for the concrete $Q_c$ (h4_series.hid1);
  - for $n<K\le 2n$ that quotient is $0$, because the numerator contains the factor $2^K-2^K$.
Bounding each term (prod_one_sub_ge nd_bound)
prod_one_sub_ge is the inequality $1-\sum_{i\in s}a_i\le\prod_{i\in s}(1-a_i)$ for $0\le a_i\le 1$, by induction on the finite set $s$.

nd_bound states that for $n\ge 1$ and $K\ge 2n+1$ the coefficient in h4_series satisfies $$0<\frac{\prod_{n<L\le 2n}(2^K-2^L)}{\prod_{i=0}^{n}(2^K-2^i)}\le 2\cdot 2^{-K}.$$ Every factor is positive since $L,i<K$. The numerator is at most $(2^K)^n$ (nd_bound.hN). The denominator is at least $(2^K)^{n+1}/2$ (nd_bound.hD): each factor is $2^K(1-2^{-(K-i)})$, the numbers $2^{-(K-i)}$ for $i\le n$ sum to at most $1/2$ because $K-n\ge 2$, and prod_one_sub_ge applies.
The bound for $n\ge 2$ (ineq_pow h4_big)
ineq_pow states $9\cdot 81^n\le 256^n$ for $n\ge 2$ (induction on $n$).

h4_big is the conclusion of the statement for $n\ge 2$. Write $x=3/2^n$, so $x/2\le 3/8$ (h4_big.hx2), and let $G_K$ be the $K$-th term of the series in h4_series.
- Positivity. Every $G_K$ is $\ge 0$ and $G_{2n+1}>0$ (h4_big.hGnn, h4_big.hGpos, both from nd_bound), so the sum $R_n$ is positive.
- Upper bound. By nd_bound, $G_K\le 2(x/2)^K$ for $K>2n$. Summing the geometric series, $R_n\le 2(x/2)^{2n+1}/(1-x/2)\le 8(x/2)^{2n+1}$, using $1/(1-x/2)\le 4$. Squaring, $R_n^2\le 64\,(x/2)^{4n+2}$, and $x/2=3/2^{n+1}$.
- Arithmetic. It remains that $64\cdot 3^{4n+2}\cdot 2^{4n^2}\le 2^{2n+4}\,(2^{n+1})^{4n+2}$. The right side is $2^{4n^2}\cdot 64\cdot 256^n$ and the left side is $2^{4n^2}\cdot 64\cdot 9\cdot 81^n$, so this is ineq_pow. It is proved in $\mathbb N$ and cast to $\mathbb R$.
The constant $T$, and the cases $n=0$ and $n=1$ (T_facts s17 hQx0 hAq0 e hQx1 hAq1)
T_facts states that the series defining $T$ converges, $T>0$ and $T\le 2/5$, from $2^{m+3}-3\ge 5\cdot 2^m$, which bounds the $m$-th term by $\frac15 2^{-m}$. The step s17 takes $T>0$ and $T\le 2/5$ out of it.

The proof then introduces the hypotheses of the statement and splits on $n$.
- $n=0$: $Q_x(0)=1$ (hQx0) and $A_q(0)=0$ (hAq0), so $R_0=3T>0$, and $R_0^2\cdot 2^0=9T^2$ (e), which is at most $2^4$ since $T\le 2/5$.
- $n=1$: $Q_x(1)=3/2$ (hQx1) and $A_q(1)=0$ (hAq1), so $R_1=\frac92T>0$, and $R_1^2\cdot 2^4\le 2^6$ again follows from $T\le 2/5$.
- $n\ge 2$: h4_big.

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, 000327, records the gate’s run.

The proof step by step (outline)

Checked by the kernel — an outline of the attested proof below: attestations/000327.json, merge commit dfc7b5cc45a9.

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.

id1_test have ∀ (n j : ℕ), j ≤ n → (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with id1_test : ∀ (n j : ℕ), j ≤ n → (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
Uses library Finset.range.
Lean, lines 28–39
```
  have id1_test : ∀ (n j : ℕ) (hj : j ≤ n),
      (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) =
        ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1) := by
    intro n j hj
    induction j with
    | zero => simp
    | succ j ih =>
      have ih' := ih (by omega)
      rw [Finset.prod_range_succ, ← ih']
      obtain ⟨m, hm⟩ : ∃ m, n - j = m + 1 := ⟨n - j - 1, by omega⟩
      rw [hm, Finset.prod_range_succ, show n - (j + 1) = m by omega]
      ring
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 28–39 at dfc7b5cc45a9 ↗
1. id1_test.s1 case routine: simp
  Leaves (∏ i ∈ Finset.range (0 : ℕ), ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - (0 : ℕ)), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with hj : (0 : ℕ) ≤ n.
  Uses library Nat.zero.
  Lean, line 33
  | zero => simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 33 at dfc7b5cc45a9 ↗
2. id1_test.s2 case
  Leaves (∏ i ∈ Finset.range (j + (1 : ℕ)), ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - (j + (1 : ℕ))), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with j : ℕ, ih : j ≤ n → (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), hj : j + (1 : ℕ) ≤ n.
  Uses library Nat.succ, Finset.prod_range_succ.
  Lean, lines 34–39
  | succ j ih => have ih' := ih (by omega) rw [Finset.prod_range_succ, ← ih'] obtain ⟨m, hm⟩ : ∃ m, n - j = m + 1 := ⟨n - j - 1, by omega⟩ rw [hm, Finset.prod_range_succ, show n - (j + 1) = m by omega] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 34–39 at dfc7b5cc45a9 ↗
  id1_test.s2.ih' have (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
  Leaves (∏ i ∈ Finset.range (j + (1 : ℕ)), ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - (j + (1 : ℕ))), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with ih' : (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
  Lean, line 35
  have ih' := ih (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 35 at dfc7b5cc45a9 ↗
  id1_test.s2.s2 obtain ∃ (m : ℕ), n - j = m + (1 : ℕ)
  Leaves (∏ x ∈ Finset.range j, ((2 : ℚ) ^ (n - x) - (1 : ℚ))) * ((2 : ℚ) ^ (n - j) - (1 : ℚ)) * ∏ s ∈ Finset.range (n - (j + (1 : ℕ))), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with m : ℕ, hm : n - j = m + (1 : ℕ).
  Uses library Exists.intro.
  Lean, line 37
  obtain ⟨m, hm⟩ : ∃ m, n - j = m + 1 := ⟨n - j - 1, by omega⟩
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 37 at dfc7b5cc45a9 ↗
lower_test have ∀ (j : ℕ), ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with lower_test : ∀ (j : ℕ), ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
Uses library Finset.range, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, Finset.prod_range_reflect.
Lean, lines 41–55
```
  have lower_test : ∀ (j : ℕ),
      ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - 2 ^ i) =
        2 ^ (j * (j - 1) / 2) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j
    have h1 : ∀ i ∈ Finset.range j, ((2 : ℚ) ^ j - 2 ^ i) = 2 ^ i * (2 ^ (j - 1 - i + 1) - 1) := by
      intro i hi
      have hi' := Finset.mem_range.mp hi
      rw [mul_sub, ← pow_add, show i + (j - 1 - i + 1) = j by omega]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum,
      Finset.prod_range_reflect (fun s => ((2 : ℚ) ^ (s + 1) - 1)) j]
    congr 1
    congr 1
    have := Finset.sum_range_id_mul_two j
    omega
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 41–55 at dfc7b5cc45a9 ↗
1. lower_test.h1 have ∀ i ∈ Finset.range j, (2 : ℚ) ^ j - (2 : ℚ) ^ i = (2 : ℚ) ^ i * ((2 : ℚ) ^ (j - (1 : ℕ) - i + (1 : ℕ)) - (1 : ℚ))
  Leaves ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with h1 : ∀ i ∈ Finset.range j, (2 : ℚ) ^ j - (2 : ℚ) ^ i = (2 : ℚ) ^ i * ((2 : ℚ) ^ (j - (1 : ℕ) - i + (1 : ℕ)) - (1 : ℚ)).
  Uses library Finset.range, mul_sub, pow_add.
  Lean, lines 45–49
  have h1 : ∀ i ∈ Finset.range j, ((2 : ℚ) ^ j - 2 ^ i) = 2 ^ i * (2 ^ (j - 1 - i + 1) - 1) := by intro i hi have hi' := Finset.mem_range.mp hi rw [mul_sub, ← pow_add, show i + (j - 1 - i + 1) = j by omega] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 45–49 at dfc7b5cc45a9 ↗
  lower_test.h1.hi' have i < j
  Leaves (2 : ℚ) ^ j - (2 : ℚ) ^ i = (2 : ℚ) ^ i * ((2 : ℚ) ^ (j - (1 : ℕ) - i + (1 : ℕ)) - (1 : ℚ)), with hi' : i < j.
  Uses library Finset.mem_range.
  Lean, line 47
  have hi' := Finset.mem_range.mp hi
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 47 at dfc7b5cc45a9 ↗
2. lower_test.this have (∑ i ∈ Finset.range j, i) * (2 : ℕ) = j * (j - (1 : ℕ))
  Leaves ∑ i ∈ Finset.range j, i = j * (j - (1 : ℕ)) / (2 : ℕ), with this : (∑ i ∈ Finset.range j, i) * (2 : ℕ) = j * (j - (1 : ℕ)).
  Uses library Finset.sum_range_id_mul_two.
  Lean, line 54
  have := Finset.sum_range_id_mul_two j
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 54 at dfc7b5cc45a9 ↗
upper_test have ∀ (j n : ℕ), j ≤ n → ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (-1 : ℚ) ^ (n - j) * (2 : ℚ) ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with upper_test : ∀ (j n : ℕ), j ≤ n → ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (-1 : ℚ) ^ (n - j) * (2 : ℚ) ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
Uses library Finset.Ioc, Finset.range, Finset.prod_Ico_eq_prod_range, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, pow_mul.
Lean, lines 57–69
```
  have upper_test : ∀ (j n : ℕ) (hj : j ≤ n),
      ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - 2 ^ i) =
        (-1) ^ (n - j) * 2 ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j n hj
    have e : Finset.Ioc j n = Finset.Ico (j + 1) (n + 1) := by
      ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
    rw [e, Finset.prod_Ico_eq_prod_range, show n + 1 - (j + 1) = n - j by omega]
    have h1 : ∀ t ∈ Finset.range (n - j), ((2 : ℚ) ^ j - 2 ^ (j + 1 + t)) = (-1) * 2 ^ j * (2 ^ (t + 1) - 1) := by
      intro t _
      rw [show j + 1 + t = j + (t + 1) by ring, pow_add]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
      Finset.prod_const, Finset.card_range, ← pow_mul]
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 57–69 at dfc7b5cc45a9 ↗
1. upper_test.e have Finset.Ioc j n = Finset.Ico (j + (1 : ℕ)) (n + (1 : ℕ))
  Leaves ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = (-1 : ℚ) ^ (n - j) * (2 : ℚ) ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with e : Finset.Ioc j n = Finset.Ico (j + (1 : ℕ)) (n + (1 : ℕ)).
  Uses library Finset.Ioc, Finset.Ico, Finset.mem_Ioc, Finset.mem_Ico.
  Lean, lines 61–62
  have e : Finset.Ioc j n = Finset.Ico (j + 1) (n + 1) := by ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 61–62 at dfc7b5cc45a9 ↗
2. upper_test.h1 have ∀ t ∈ Finset.range (n - j), (2 : ℚ) ^ j - (2 : ℚ) ^ (j + (1 : ℕ) + t) = (-1 : ℚ) * (2 : ℚ) ^ j * ((2 : ℚ) ^ (t + (1 : ℕ)) - (1 : ℚ))
  Leaves ∏ k ∈ Finset.range (n - j), ((2 : ℚ) ^ j - (2 : ℚ) ^ (j + (1 : ℕ) + k)) = (-1 : ℚ) ^ (n - j) * (2 : ℚ) ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with h1 : ∀ t ∈ Finset.range (n - j), (2 : ℚ) ^ j - (2 : ℚ) ^ (j + (1 : ℕ) + t) = (-1 : ℚ) * (2 : ℚ) ^ j * ((2 : ℚ) ^ (t + (1 : ℕ)) - (1 : ℚ)).
  Uses library Finset.range, pow_add.
  Lean, lines 64–67
  have h1 : ∀ t ∈ Finset.range (n - j), ((2 : ℚ) ^ j - 2 ^ (j + 1 + t)) = (-1) * 2 ^ j * (2 ^ (t + 1) - 1) := by intro t _ rw [show j + 1 + t = j + (t + 1) by ring, pow_add] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 64–67 at dfc7b5cc45a9 ↗
rhs_test have ∀ (j n : ℕ), j ≤ n → (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with rhs_test : ∀ (j n : ℕ), j ≤ n → (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
Uses library Finset.Ioc, Finset.range, Finset.prod_Ico_eq_prod_range, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, pow_mul, Finset.prod_range_add.
Lean, lines 71–87
```
  have rhs_test : ∀ (j n : ℕ) (hj : j ≤ n),
      (∏ K ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ K)) *
        ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) =
        (-1) ^ n * 2 ^ (j * n) * ∏ s ∈ Finset.range (2 * n - j), ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j n hj
    have e : Finset.Ioc n (2 * n) = Finset.Ico (n + 1) (2 * n + 1) := by
      ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
    rw [e, Finset.prod_Ico_eq_prod_range, show 2 * n + 1 - (n + 1) = n by omega]
    have h1 : ∀ t ∈ Finset.range n, ((2 : ℚ) ^ j - 2 ^ (n + 1 + t)) =
        (-1) * 2 ^ j * (2 ^ ((n - j) + t + 1) - 1) := by
      intro t _
      rw [show n + 1 + t = j + ((n - j) + t + 1) by omega, pow_add]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
      Finset.prod_const, Finset.card_range, ← pow_mul,
      show 2 * n - j = (n - j) + n by omega, Finset.prod_range_add]
    ring
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 71–87 at dfc7b5cc45a9 ↗
1. rhs_test.e have Finset.Ioc n ((2 : ℕ) * n) = Finset.Ico (n + (1 : ℕ)) ((2 : ℕ) * n + (1 : ℕ))
  Leaves (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with e : Finset.Ioc n ((2 : ℕ) * n) = Finset.Ico (n + (1 : ℕ)) ((2 : ℕ) * n + (1 : ℕ)).
  Uses library Finset.Ioc, Finset.Ico, Finset.mem_Ioc, Finset.mem_Ico.
  Lean, lines 76–77
  have e : Finset.Ioc n (2 * n) = Finset.Ico (n + 1) (2 * n + 1) := by ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 76–77 at dfc7b5cc45a9 ↗
2. rhs_test.h1 have ∀ t ∈ Finset.range n, (2 : ℚ) ^ j - (2 : ℚ) ^ (n + (1 : ℕ) + t) = (-1 : ℚ) * (2 : ℚ) ^ j * ((2 : ℚ) ^ (n - j + t + (1 : ℕ)) - (1 : ℚ))
  Leaves (∏ k ∈ Finset.range n, ((2 : ℚ) ^ j - (2 : ℚ) ^ (n + (1 : ℕ) + k))) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)), with h1 : ∀ t ∈ Finset.range n, (2 : ℚ) ^ j - (2 : ℚ) ^ (n + (1 : ℕ) + t) = (-1 : ℚ) * (2 : ℚ) ^ j * ((2 : ℚ) ^ (n - j + t + (1 : ℕ)) - (1 : ℚ)).
  Uses library Finset.range, pow_add.
  Lean, lines 79–83
  have h1 : ∀ t ∈ Finset.range n, ((2 : ℚ) ^ j - 2 ^ (n + 1 + t)) = (-1) * 2 ^ j * (2 ^ ((n - j) + t + 1) - 1) := by intro t _ rw [show n + 1 + t = j + ((n - j) + t + 1) by omega, pow_add] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 79–83 at dfc7b5cc45a9 ↗
id1_full have ∀ (n j : ℕ), j ≤ n → (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with id1_full : ∀ (n j : ℕ), j ≤ n → (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K).
Uses library Finset.range, Finset.Ioc, Finset.prod_union, mul_right_cancel₀, pow_add.
Lean, lines 89–132
```
  have id1_full : ∀ (n j : ℕ) (hj : j ≤ n),
      (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) *
            ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
          ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
      ∏ K ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ K) := by
    intro n j hj
    have hPne : ∀ m : ℕ, ∏ s ∈ Finset.range m, ((2 : ℚ) ^ (s + 1) - 1) ≠ 0 := by
      intro m
      refine Finset.prod_ne_zero_iff.mpr (fun s _ => ?_)
      have : (1 : ℚ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hG1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) =
        (∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) /
          ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + 1) - 1)) := by
      rw [Finset.prod_div_distrib, ← id1_test n j hj]
      field_simp [hPne]
    have hG2 : ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) =
        (∏ s ∈ Finset.range (2 * n - j), ((2 : ℚ) ^ (s + 1) - 1)) /
          ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) := by
      rw [Finset.prod_div_distrib, ← id1_test (2 * n - j) n (by omega), show 2 * n - j - n = n - j by omega]
      field_simp [hPne]
    have hsplit : (Finset.range (n + 1)).erase j = Finset.range j ∪ Finset.Ioc j n := by
      ext x; simp only [Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc]; omega
    have hdisj : Disjoint (Finset.range j) (Finset.Ioc j n) := by
      rw [Finset.disjoint_left]
      intro x hx hx'
      simp only [Finset.mem_range, Finset.mem_Ioc] at hx hx'
      omega
    have hR := rhs_test j n hj
    obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hj
    have hsign : (-1 : ℚ) ^ (j + m) = (-1) ^ j * (-1) ^ (j + m - j) := by
      rw [← pow_add]; congr 1; omega
    have ha : 2 * (j * (j - 1) / 2) = j * (j - 1) := Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self _)
    have hexp : j * (j + m) = j * (j - (1 : ℕ)) / (2 : ℕ) + j + j * (j - (1 : ℕ)) / (2 : ℕ) + j * (j + m - j) := by
      rw [show j + m - j = m by omega]
      rcases j with _ | i
      · simp
      · simp only [Nat.add_sub_cancel] at ha ⊢
        nlinarith [ha]
    rw [hG1, hG2, hsplit, Finset.prod_union hdisj, lower_test, upper_test j (j + m) hj]
    apply mul_right_cancel₀ (hPne (j + m - j))
    rw [hR, hsign, hexp, pow_add, pow_add, pow_add]
    field_simp [hPne]
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 89–132 at dfc7b5cc45a9 ↗
1. id1_full.hPne have ∀ (m : ℕ), ∏ s ∈ Finset.range m, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) ≠ (0 : ℚ)
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hPne : ∀ (m : ℕ), ∏ s ∈ Finset.range m, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) ≠ (0 : ℚ).
  Uses library Finset.range, Finset.prod_ne_zero_iff.
  Lean, lines 96–100
  have hPne : ∀ m : ℕ, ∏ s ∈ Finset.range m, ((2 : ℚ) ^ (s + 1) - 1) ≠ 0 := by intro m refine Finset.prod_ne_zero_iff.mpr (fun s _ => ?_) have : (1 : ℚ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega) linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 96–100 at dfc7b5cc45a9 ↗
  id1_full.hPne.this have (1 : ℚ) < (2 : ℚ) ^ (s + (1 : ℕ))
  Leaves (2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ) ≠ (0 : ℚ), with this : (1 : ℚ) < (2 : ℚ) ^ (s + (1 : ℕ)).
  Uses library one_lt_pow₀.
  Lean, line 99
  have : (1 : ℚ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 99 at dfc7b5cc45a9 ↗
2. id1_full.hG1 have ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)))
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hG1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))).
  Uses library Finset.range, Finset.prod_div_distrib.
  Lean, lines 101–105
  have hG1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + 1) - 1)) := by rw [Finset.prod_div_distrib, ← id1_test n j hj] field_simp [hPne]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 101–105 at dfc7b5cc45a9 ↗
3. id1_full.hG2 have ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)))
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hG2 : ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))).
  Uses library Finset.range, Finset.prod_div_distrib.
  Lean, lines 106–110
  have hG2 : ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range (2 * n - j), ((2 : ℚ) ^ (s + 1) - 1)) / ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) := by rw [Finset.prod_div_distrib, ← id1_test (2 * n - j) n (by omega), show 2 * n - j - n = n - j by omega] field_simp [hPne]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 106–110 at dfc7b5cc45a9 ↗
4. id1_full.hsplit have (Finset.range (n + (1 : ℕ))).erase j = Finset.range j ∪ Finset.Ioc j n
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hsplit : (Finset.range (n + (1 : ℕ))).erase j = Finset.range j ∪ Finset.Ioc j n.
  Uses library Finset.range, Finset.Ioc, Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc.
  Lean, lines 111–112
  have hsplit : (Finset.range (n + 1)).erase j = Finset.range j ∪ Finset.Ioc j n := by ext x; simp only [Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc]; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 111–112 at dfc7b5cc45a9 ↗
5. id1_full.hdisj have Disjoint (Finset.range j) (Finset.Ioc j n)
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hdisj : Disjoint (Finset.range j) (Finset.Ioc j n).
  Uses library Disjoint, Finset.range, Finset.Ioc, Finset.disjoint_left, Finset.mem_range, Finset.mem_Ioc.
  Lean, lines 113–117
  have hdisj : Disjoint (Finset.range j) (Finset.Ioc j n) := by rw [Finset.disjoint_left] intro x hx hx' simp only [Finset.mem_range, Finset.mem_Ioc] at hx hx' omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 113–117 at dfc7b5cc45a9 ↗
6. id1_full.hR have (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hR : (∏ K ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ n * (2 : ℚ) ^ (j * n) * ∏ s ∈ Finset.range ((2 : ℕ) * n - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
  Lean, line 118
  have hR := rhs_test j n hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 118 at dfc7b5cc45a9 ↗
7. id1_full.s7 obtain ∃ (k : ℕ), n = j + k
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + m - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range (j + m), ((2 : ℚ) ^ ((2 : ℕ) * (j + m) - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (j + m + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc (j + m) ((2 : ℕ) * (j + m)), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with m : ℕ, hj : j ≤ j + m, hG1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + m - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range (j + m), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (j + m - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))), hG2 : ∏ i ∈ Finset.range (j + m), ((2 : ℚ) ^ ((2 : ℕ) * (j + m) - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) = (∏ s ∈ Finset.range ((2 : ℕ) * (j + m) - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) / ((∏ s ∈ Finset.range (j + m - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))) * ∏ s ∈ Finset.range (j + m), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ))), hsplit : (Finset.range (j + m + (1 : ℕ))).erase j = Finset.range j ∪ Finset.Ioc j (j + m), hdisj : Disjoint (Finset.range j) (Finset.Ioc j (j + m)), hR : (∏ K ∈ Finset.Ioc (j + m) ((2 : ℕ) * (j + m)), ((2 : ℚ) ^ j - (2 : ℚ) ^ K)) * ∏ s ∈ Finset.range (j + m - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)) = (-1 : ℚ) ^ (j + m) * (2 : ℚ) ^ (j * (j + m)) * ∏ s ∈ Finset.range ((2 : ℕ) * (j + m) - j), ((2 : ℚ) ^ (s + (1 : ℕ)) - (1 : ℚ)).
  Uses library Nat.exists_eq_add_of_le.
  Lean, line 119
  obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 119 at dfc7b5cc45a9 ↗
8. id1_full.hsign have (-1 : ℚ) ^ (j + m) = (-1 : ℚ) ^ j * (-1 : ℚ) ^ (j + m - j)
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + m - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range (j + m), ((2 : ℚ) ^ ((2 : ℕ) * (j + m) - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (j + m + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc (j + m) ((2 : ℕ) * (j + m)), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hsign : (-1 : ℚ) ^ (j + m) = (-1 : ℚ) ^ j * (-1 : ℚ) ^ (j + m - j).
  Uses library pow_add.
  Lean, lines 120–121
  have hsign : (-1 : ℚ) ^ (j + m) = (-1) ^ j * (-1) ^ (j + m - j) := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 120–121 at dfc7b5cc45a9 ↗
9. id1_full.ha have (2 : ℕ) * (j * (j - (1 : ℕ)) / (2 : ℕ)) = j * (j - (1 : ℕ))
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + m - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range (j + m), ((2 : ℚ) ^ ((2 : ℕ) * (j + m) - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (j + m + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc (j + m) ((2 : ℕ) * (j + m)), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with ha : (2 : ℕ) * (j * (j - (1 : ℕ)) / (2 : ℕ)) = j * (j - (1 : ℕ)).
  Uses library Nat.two_mul_div_two_of_even, Nat.even_mul_pred_self.
  Lean, line 122
  have ha : 2 * (j * (j - 1) / 2) = j * (j - 1) := Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self _)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 122 at dfc7b5cc45a9 ↗
10. id1_full.hexp have j * (j + m) = j * (j - (1 : ℕ)) / (2 : ℕ) + j + j * (j - (1 : ℕ)) / (2 : ℕ) + j * (j + m - j)
  Leaves (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + m - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range (j + m), ((2 : ℚ) ^ ((2 : ℕ) * (j + m) - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (j + m + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ K ∈ Finset.Ioc (j + m) ((2 : ℕ) * (j + m)), ((2 : ℚ) ^ j - (2 : ℚ) ^ K), with hexp : j * (j + m) = j * (j - (1 : ℕ)) / (2 : ℕ) + j + j * (j - (1 : ℕ)) / (2 : ℕ) + j * (j + m - j).
  Lean, lines 123–128
  have hexp : j * (j + m) = j * (j - (1 : ℕ)) / (2 : ℕ) + j + j * (j - (1 : ℕ)) / (2 : ℕ) + j * (j + m - j) := by rw [show j + m - j = m by omega] rcases j with _ | i · simp · simp only [Nat.add_sub_cancel] at ha ⊢ nlinarith [ha]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 123–128 at dfc7b5cc45a9 ↗
  id1_full.hexp.s1 case routine: simp
  Leaves (0 : ℕ) * ((0 : ℕ) + m) = (0 : ℕ) * ((0 : ℕ) - (1 : ℕ)) / (2 : ℕ) + (0 : ℕ) + (0 : ℕ) * ((0 : ℕ) - (1 : ℕ)) / (2 : ℕ) + (0 : ℕ) * m.
  Lean, line 126
  · simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 126 at dfc7b5cc45a9 ↗
  id1_full.hexp.s2 case routine: simp, nlinarith
  Leaves (i + (1 : ℕ)) * (i + (1 : ℕ) + m) = (i + (1 : ℕ)) * (i + (1 : ℕ) - (1 : ℕ)) / (2 : ℕ) + (i + (1 : ℕ)) + (i + (1 : ℕ)) * (i + (1 : ℕ) - (1 : ℕ)) / (2 : ℕ) + (i + (1 : ℕ)) * m.
  Uses library Nat.add_sub_cancel.
  Lean, lines 127–128
  · simp only [Nat.add_sub_cancel] at ha ⊢ nlinarith [ha]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 127–128 at dfc7b5cc45a9 ↗
lagr_eval have ∀ (n : ℕ) (x : ℕ → ℚ), (∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j) → ∀ (N : Polynomial ℚ), N.degree ≤ (↑n : WithBot ℕ) → ∀ (r : ℕ → ℚ), (∀ j ≤ n, r j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (x j - x i) = Polynomial.eval (x j) N) → N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with lagr_eval : ∀ (n : ℕ) (x : ℕ → ℚ), (∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j) → ∀ (N : Polynomial ℚ), N.degree ≤ (↑n : WithBot ℕ) → ∀ (r : ℕ → ℚ), (∀ j ≤ n, r j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (x j - x i) = Polynomial.eval (x j) N) → N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)).
Uses library Finset.range, Polynomial.C, Polynomial.X, Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq.
Lean, lines 135–168
```
  have lagr_eval : ∀ (n : ℕ) (x : ℕ → ℚ) (hx : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j)
    (N : ℚ[X]) (hNdeg : N.degree ≤ n) (r : ℕ → ℚ)
    (hr : ∀ j ≤ n, r j * ∏ i ∈ (Finset.range (n + 1)).erase j, (x j - x i) = N.eval (x j)),
      N = ∑ j ∈ Finset.range (n + 1), C (r j) * ∏ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)) := by
    intro n x hx N hNdeg r hr
    have hinj : Set.InjOn x (Finset.range (n + 1) : Set ℕ) := by
      intro i hi j hj h
      simp only [Finset.coe_range, Set.mem_Iio] at hi hj
      exact hx i (by omega) j (by omega) h
    apply Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq ((Finset.range (n + 1)).image x)
    · rw [Finset.card_image_of_injOn hinj, Finset.card_range]
      refine lt_of_le_of_lt (Polynomial.degree_sub_le _ _) (max_lt ?_ ?_)
      · exact lt_of_le_of_lt hNdeg (by exact_mod_cast Nat.lt_succ_self n)
      · refine lt_of_le_of_lt (Polynomial.degree_sum_le _ _) ?_
        rw [Finset.sup_lt_iff (by exact WithBot.bot_lt_coe _)]
        intro j hj
        refine lt_of_le_of_lt (Polynomial.degree_mul_le _ _) ?_
        refine lt_of_le_of_lt (add_le_add (Polynomial.degree_C_le) (Polynomial.degree_prod_le _ _)) ?_
        have : ∑ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)).degree = ((n : ℕ) : WithBot ℕ) := by
          rw [Finset.sum_congr rfl (fun i _ => Polynomial.degree_X_sub_C (x i)), Finset.sum_const,
            Finset.card_erase_of_mem hj, Finset.card_range, Nat.add_sub_cancel]
          simp
        rw [this, zero_add]
        exact_mod_cast Nat.lt_succ_self n
    · intro y hy
      obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
      have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega
      rw [← hr k hk', Polynomial.eval_finset_sum, Finset.sum_eq_single k]
      · simp [Polynomial.eval_prod]
      · intro j hj hjk
        simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub,
          Polynomial.eval_X]
        rw [Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hjk, hk⟩) (by ring), mul_zero]
      · intro h; exact absurd hk h
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 135–168 at dfc7b5cc45a9 ↗
1. lagr_eval.hinj have Set.InjOn x (↑(Finset.range (n + (1 : ℕ))) : Set ℕ)
  Leaves N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)), with hinj : Set.InjOn x (↑(Finset.range (n + (1 : ℕ))) : Set ℕ).
  Uses library Set.InjOn, Finset.range, Set, Finset.coe_range, Set.mem_Iio.
  Lean, lines 140–143
  have hinj : Set.InjOn x (Finset.range (n + 1) : Set ℕ) := by intro i hi j hj h simp only [Finset.coe_range, Set.mem_Iio] at hi hj exact hx i (by omega) j (by omega) h
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 140–143 at dfc7b5cc45a9 ↗
2. lagr_eval.s2 case
  Leaves (N - ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))).degree < (↑(Finset.image x (Finset.range (n + (1 : ℕ)))).card : WithBot ℕ).
  Uses library Finset.card_image_of_injOn, Finset.card_range, lt_of_le_of_lt, Polynomial.degree_sub_le, max_lt.
  Lean, lines 145–158
  · rw [Finset.card_image_of_injOn hinj, Finset.card_range] refine lt_of_le_of_lt (Polynomial.degree_sub_le _ _) (max_lt ?_ ?_) · exact lt_of_le_of_lt hNdeg (by exact_mod_cast Nat.lt_succ_self n) · refine lt_of_le_of_lt (Polynomial.degree_sum_le _ _) ?_ rw [Finset.sup_lt_iff (by exact WithBot.bot_lt_coe _)] intro j hj refine lt_of_le_of_lt (Polynomial.degree_mul_le _ _) ?_ refine lt_of_le_of_lt (add_le_add (Polynomial.degree_C_le) (Polynomial.degree_prod_le _ _)) ?_ have : ∑ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)).degree = ((n : ℕ) : WithBot ℕ) := by rw [Finset.sum_congr rfl (fun i _ => Polynomial.degree_X_sub_C (x i)), Finset.sum_const, Finset.card_erase_of_mem hj, Finset.card_range, Nat.add_sub_cancel] simp rw [this, zero_add] exact_mod_cast Nat.lt_succ_self n
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 145–158 at dfc7b5cc45a9 ↗
  lagr_eval.s2.s1 case
  Leaves N.degree < (↑(n + (1 : ℕ)) : WithBot ℕ).
  Uses library lt_of_le_of_lt, Nat.lt_succ_self.
  Lean, line 147
  · exact lt_of_le_of_lt hNdeg (by exact_mod_cast Nat.lt_succ_self n)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 147 at dfc7b5cc45a9 ↗
  lagr_eval.s2.s2 case
  Leaves (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))).degree < (↑(n + (1 : ℕ)) : WithBot ℕ).
  Uses library lt_of_le_of_lt, Polynomial.degree_sum_le, Finset.sup_lt_iff, WithBot.bot_lt_coe, Polynomial.degree_mul_le, add_le_add, Polynomial.degree_C_le, Polynomial.degree_prod_le, zero_add, Nat.lt_succ_self.
  Lean, lines 148–158
  · refine lt_of_le_of_lt (Polynomial.degree_sum_le _ _) ?_ rw [Finset.sup_lt_iff (by exact WithBot.bot_lt_coe _)] intro j hj refine lt_of_le_of_lt (Polynomial.degree_mul_le _ _) ?_ refine lt_of_le_of_lt (add_le_add (Polynomial.degree_C_le) (Polynomial.degree_prod_le _ _)) ?_ have : ∑ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)).degree = ((n : ℕ) : WithBot ℕ) := by rw [Finset.sum_congr rfl (fun i _ => Polynomial.degree_X_sub_C (x i)), Finset.sum_const, Finset.card_erase_of_mem hj, Finset.card_range, Nat.add_sub_cancel] simp rw [this, zero_add] exact_mod_cast Nat.lt_succ_self n
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 148–158 at dfc7b5cc45a9 ↗
  lagr_eval.s2.s2.this have ∑ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)).degree = (↑n : WithBot ℕ)
  Leaves (0 : WithBot ℕ) + ∑ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)).degree < (↑(n + (1 : ℕ)) : WithBot ℕ), with this : ∑ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)).degree = (↑n : WithBot ℕ).
  Uses library Finset.range, Polynomial.X, Polynomial.C, WithBot, Finset.sum_congr, rfl, Polynomial.degree_X_sub_C, Finset.sum_const, Finset.card_erase_of_mem, Finset.card_range, Nat.add_sub_cancel.
  Lean, lines 153–156
  have : ∑ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)).degree = ((n : ℕ) : WithBot ℕ) := by rw [Finset.sum_congr rfl (fun i _ => Polynomial.degree_X_sub_C (x i)), Finset.sum_const, Finset.card_erase_of_mem hj, Finset.card_range, Nat.add_sub_cancel] simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 153–156 at dfc7b5cc45a9 ↗
3. lagr_eval.s3 case
  Leaves ∀ x_1 ∈ Finset.image x (Finset.range (n + (1 : ℕ))), Polynomial.eval x_1 N = Polynomial.eval x_1 (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))).
  Uses library Polynomial.eval_finset_sum, Finset.sum_eq_single.
  Lean, lines 159–168
  · intro y hy obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega rw [← hr k hk', Polynomial.eval_finset_sum, Finset.sum_eq_single k] · simp [Polynomial.eval_prod] · intro j hj hjk simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X] rw [Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hjk, hk⟩) (by ring), mul_zero] · intro h; exact absurd hk h
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 159–168 at dfc7b5cc45a9 ↗
  lagr_eval.s3.s1 obtain ∃ a ∈ Finset.range (n + (1 : ℕ)), x a = y
  Leaves Polynomial.eval (x k) N = Polynomial.eval (x k) (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))), with k : ℕ, hk : k ∈ Finset.range (n + (1 : ℕ)), hy : x k ∈ Finset.image x (Finset.range (n + (1 : ℕ))).
  Uses library Finset.mem_image.
  Lean, line 160
  obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 160 at dfc7b5cc45a9 ↗
  lagr_eval.s3.hk' have k ≤ n
  Leaves Polynomial.eval (x k) N = Polynomial.eval (x k) (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (r j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))), with hk' : k ≤ n.
  Lean, line 161
  have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 161 at dfc7b5cc45a9 ↗
  lagr_eval.s3.hk'.this have k < n + (1 : ℕ)
  Leaves k ≤ n, with this : k < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 161
  have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 161 at dfc7b5cc45a9 ↗
  lagr_eval.s3.s3 case routine: simp
  Leaves r k * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase k, (x k - x i) = Polynomial.eval (x k) ((Polynomial.C : ℚ → Polynomial ℚ) (r k) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase k, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))).
  Uses library Polynomial.eval_prod.
  Lean, line 163
  · simp [Polynomial.eval_prod]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 163 at dfc7b5cc45a9 ↗
  lagr_eval.s3.s4 case
  Leaves ∀ b ∈ Finset.range (n + (1 : ℕ)), b ≠ k → Polynomial.eval (x k) ((Polynomial.C : ℚ → Polynomial ℚ) (r b) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase b, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))) = (0 : ℚ).
  Uses library Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Finset.prod_eq_zero, Finset.mem_erase, And.intro, Ne.symm, MulZeroClass.mul_zero.
  Lean, lines 164–167
  · intro j hj hjk simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X] rw [Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hjk, hk⟩) (by ring), mul_zero]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 164–167 at dfc7b5cc45a9 ↗
  lagr_eval.s3.s5 case
  Leaves k ∉ Finset.range (n + (1 : ℕ)) → Polynomial.eval (x k) ((Polynomial.C : ℚ → Polynomial ℚ) (r k) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase k, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))) = (0 : ℚ).
  Uses library absurd.
  Lean, line 168
  · intro h; exact absurd hk h
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 168 at dfc7b5cc45a9 ↗
tail_coeff have ∀ (n K : ℕ), n < K → ∀ (Qc : ℕ → ℕ → ℚ), (∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ L)) → ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with tail_coeff : ∀ (n K : ℕ), n < K → ∀ (Qc : ℕ → ℕ → ℚ), (∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ L)) → ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i).
Uses library Finset.range, Finset.Ioc, Polynomial.X, Polynomial.C, Polynomial.eval_finset_sum, Finset.sum_div, Finset.sum_congr, rfl, Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Finset.mul_prod_erase.
Lean, lines 170–222
```
  have tail_coeff : ∀ (n K : ℕ) (hK : n < K) (Qc : ℕ → ℕ → ℚ)
    (hQc : ∀ j ≤ n, Qc n j * 2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
      ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ L)),
      ∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) =
        (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℚ) ^ K - 2 ^ i) := by
    intro n K hK Qc hQc
    set x : ℕ → ℚ := fun i => (2 : ℚ) ^ i with hxdef
    have hxinj : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j := by
      intro i _ j _ h
      have h' : (2 : ℚ) ^ i = 2 ^ j := h
      exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast h')
    have hxne : ∀ a b : ℕ, a ≠ b → (2 : ℚ) ^ a - 2 ^ b ≠ 0 := by
      intro a b hab h
      apply hab
      exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast (sub_eq_zero.mp h))
    set N : ℚ[X] := ∏ L ∈ Finset.Ioc n (2 * n), (X - C ((2 : ℚ) ^ L)) with hN
    have hNdeg : N.degree ≤ n := by
      rw [hN, Polynomial.degree_prod]
      rw [Finset.sum_congr rfl (fun L _ => Polynomial.degree_X_sub_C ((2 : ℚ) ^ L)), Finset.sum_const,
        Nat.card_Ioc]
      simp
      omega
    have hr : ∀ j ≤ n, (Qc n j * 2 ^ j) * ∏ i ∈ (Finset.range (n + 1)).erase j, (x j - x i) = N.eval (x j) := by
      intro j hj
      rw [hN, Polynomial.eval_prod]
      simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, hxdef]
      exact hQc j hj
    have hL := lagr_eval n x hxinj N hNdeg (fun j => Qc n j * 2 ^ j) hr
    have hev := congrArg (Polynomial.eval ((2 : ℚ) ^ K)) hL
    rw [Polynomial.eval_finset_sum] at hev
    have hNK : N.eval ((2 : ℚ) ^ K) = ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ K - 2 ^ L) := by
      rw [hN, Polynomial.eval_prod]
      simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C]
    have hD : ∏ i ∈ Finset.range (n + 1), ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 :=
      Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp hi; omega))
    rw [hNK] at hev
    rw [hev, Finset.sum_div]
    refine Finset.sum_congr rfl (fun j hj => ?_)
    have hjn : j < n + 1 := Finset.mem_range.mp hj
    have hne1 : (2 : ℚ) ^ K - 2 ^ j ≠ 0 := hxne K j (by omega)
    have hne2 : (2 : ℚ) ^ (K - j) - 1 ≠ 0 := by
      have : (1 : ℚ) < 2 ^ (K - j) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub,
      Polynomial.eval_X, hxdef]
    rw [← Finset.mul_prod_erase (Finset.range (n + 1)) (fun i => (2 : ℚ) ^ K - 2 ^ i) hj]
    have hE : ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 :=
      Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by
        have := Finset.mem_range.mp (Finset.mem_of_mem_erase hi); omega))
    have hpow : (2 : ℚ) ^ K = 2 ^ j * 2 ^ (K - j) := by rw [← pow_add]; congr 1; omega
    field_simp
    rw [hpow]
    ring
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 170–222 at dfc7b5cc45a9 ↗
1. tail_coeff.hxinj have ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hxinj : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j.
  Uses library Nat.pow_right_injective, le_refl.
  Lean, lines 177–180
  have hxinj : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j := by intro i _ j _ h have h' : (2 : ℚ) ^ i = 2 ^ j := h exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast h')
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 177–180 at dfc7b5cc45a9 ↗
  tail_coeff.hxinj.h' have (2 : ℚ) ^ i = (2 : ℚ) ^ j
  Leaves i = j, with h' : (2 : ℚ) ^ i = (2 : ℚ) ^ j.
  Lean, line 179
  have h' : (2 : ℚ) ^ i = 2 ^ j := h
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 179 at dfc7b5cc45a9 ↗
2. tail_coeff.hxne have ∀ (a b : ℕ), a ≠ b → (2 : ℚ) ^ a - (2 : ℚ) ^ b ≠ (0 : ℚ)
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hxne : ∀ (a b : ℕ), a ≠ b → (2 : ℚ) ^ a - (2 : ℚ) ^ b ≠ (0 : ℚ).
  Uses library Nat.pow_right_injective, le_refl, sub_eq_zero.
  Lean, lines 181–184
  have hxne : ∀ a b : ℕ, a ≠ b → (2 : ℚ) ^ a - 2 ^ b ≠ 0 := by intro a b hab h apply hab exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast (sub_eq_zero.mp h))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 181–184 at dfc7b5cc45a9 ↗
3. tail_coeff.hNdeg have N.degree ≤ (↑n : WithBot ℕ)
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hNdeg : N.degree ≤ (↑n : WithBot ℕ).
  Uses library Polynomial.degree_prod, Finset.sum_congr, rfl, Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Ioc.
  Lean, lines 186–191
  have hNdeg : N.degree ≤ n := by rw [hN, Polynomial.degree_prod] rw [Finset.sum_congr rfl (fun L _ => Polynomial.degree_X_sub_C ((2 : ℚ) ^ L)), Finset.sum_const, Nat.card_Ioc] simp omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 186–191 at dfc7b5cc45a9 ↗
4. tail_coeff.hr have ∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (x j - x i) = Polynomial.eval (x j) N
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hr : ∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (x j - x i) = Polynomial.eval (x j) N.
  Uses library Finset.range, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.
  Lean, lines 192–196
  have hr : ∀ j ≤ n, (Qc n j * 2 ^ j) * ∏ i ∈ (Finset.range (n + 1)).erase j, (x j - x i) = N.eval (x j) := by intro j hj rw [hN, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, hxdef] exact hQc j hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 192–196 at dfc7b5cc45a9 ↗
5. tail_coeff.hL have N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hL : N = ∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)).
  Lean, line 197
  have hL := lagr_eval n x hxinj N hNdeg (fun j => Qc n j * 2 ^ j) hr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 197 at dfc7b5cc45a9 ↗
6. tail_coeff.hev have Polynomial.eval ((2 : ℚ) ^ K) N = Polynomial.eval ((2 : ℚ) ^ K) (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i)))
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hev : Polynomial.eval ((2 : ℚ) ^ K) N = Polynomial.eval ((2 : ℚ) ^ K) (∑ j ∈ Finset.range (n + (1 : ℕ)), (Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))).
  Uses library congrArg, Polynomial.eval.
  Lean, line 198
  have hev := congrArg (Polynomial.eval ((2 : ℚ) ^ K)) hL
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 198 at dfc7b5cc45a9 ↗
7. tail_coeff.hNK have Polynomial.eval ((2 : ℚ) ^ K) N = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hNK : Polynomial.eval ((2 : ℚ) ^ K) N = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L).
  Uses library Finset.Ioc, Polynomial.eval_prod, Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.
  Lean, lines 200–202
  have hNK : N.eval ((2 : ℚ) ^ K) = ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ K - 2 ^ L) := by rw [hN, Polynomial.eval_prod] simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 200–202 at dfc7b5cc45a9 ↗
8. tail_coeff.hD have ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i) ≠ (0 : ℚ)
  Leaves ∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ K - (2 : ℚ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hD : ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i) ≠ (0 : ℚ).
  Uses library Finset.range, Finset.prod_ne_zero_iff.
  Lean, lines 203–204
  have hD : ∏ i ∈ Finset.range (n + 1), ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp hi; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 203–204 at dfc7b5cc45a9 ↗
  tail_coeff.hD.this have i < n + (1 : ℕ)
  Leaves K ≠ i, with this : i < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 204
  Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp hi; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 204 at dfc7b5cc45a9 ↗
9. tail_coeff.hjn have j < n + (1 : ℕ)
  Leaves Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = Polynomial.eval ((2 : ℚ) ^ K) ((Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hjn : j < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 208
  have hjn : j < n + 1 := Finset.mem_range.mp hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 208 at dfc7b5cc45a9 ↗
10. tail_coeff.hne1 have (2 : ℚ) ^ K - (2 : ℚ) ^ j ≠ (0 : ℚ)
  Leaves Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = Polynomial.eval ((2 : ℚ) ^ K) ((Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hne1 : (2 : ℚ) ^ K - (2 : ℚ) ^ j ≠ (0 : ℚ).
  Lean, line 209
  have hne1 : (2 : ℚ) ^ K - 2 ^ j ≠ 0 := hxne K j (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 209 at dfc7b5cc45a9 ↗
11. tail_coeff.hne2 have (2 : ℚ) ^ (K - j) - (1 : ℚ) ≠ (0 : ℚ)
  Leaves Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = Polynomial.eval ((2 : ℚ) ^ K) ((Polynomial.C : ℚ → Polynomial ℚ) (Qc n j * (2 : ℚ) ^ j) * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, (Polynomial.X - (Polynomial.C : ℚ → Polynomial ℚ) (x i))) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℚ) ^ K - (2 : ℚ) ^ i), with hne2 : (2 : ℚ) ^ (K - j) - (1 : ℚ) ≠ (0 : ℚ).
  Lean, lines 210–212
  have hne2 : (2 : ℚ) ^ (K - j) - 1 ≠ 0 := by have : (1 : ℚ) < 2 ^ (K - j) := one_lt_pow₀ (by norm_num) (by omega) linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 210–212 at dfc7b5cc45a9 ↗
  tail_coeff.hne2.this have (1 : ℚ) < (2 : ℚ) ^ (K - j)
  Leaves (2 : ℚ) ^ (K - j) - (1 : ℚ) ≠ (0 : ℚ), with this : (1 : ℚ) < (2 : ℚ) ^ (K - j).
  Uses library one_lt_pow₀.
  Lean, line 211
  have : (1 : ℚ) < 2 ^ (K - j) := one_lt_pow₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 211 at dfc7b5cc45a9 ↗
12. tail_coeff.hE have ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ i) ≠ (0 : ℚ)
  Leaves Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (Qc n j * (2 : ℚ) ^ j * ∏ x ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ x)) / (((2 : ℚ) ^ K - (2 : ℚ) ^ j) * ∏ x ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ x)), with hE : ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ i) ≠ (0 : ℚ).
  Uses library Finset.range, Finset.prod_ne_zero_iff.
  Lean, lines 216–218
  have hE : ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp (Finset.mem_of_mem_erase hi); omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 216–218 at dfc7b5cc45a9 ↗
  tail_coeff.hE.this have i < n + (1 : ℕ)
  Leaves K ≠ i, with this : i < n + (1 : ℕ).
  Uses library Finset.mem_range, Finset.mem_of_mem_erase.
  Lean, line 218
  have := Finset.mem_range.mp (Finset.mem_of_mem_erase hi); omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 218 at dfc7b5cc45a9 ↗
13. tail_coeff.hpow have (2 : ℚ) ^ K = (2 : ℚ) ^ j * (2 : ℚ) ^ (K - j)
  Leaves Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ)) = (Qc n j * (2 : ℚ) ^ j * ∏ x ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ x)) / (((2 : ℚ) ^ K - (2 : ℚ) ^ j) * ∏ x ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ K - (2 : ℚ) ^ x)), with hpow : (2 : ℚ) ^ K = (2 : ℚ) ^ j * (2 : ℚ) ^ (K - j).
  Uses library pow_add.
  Lean, line 219
  have hpow : (2 : ℚ) ^ K = 2 ^ j * 2 ^ (K - j) := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 219 at dfc7b5cc45a9 ↗
tail3 have ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with tail3 : ∀ (n : ℕ), (1 : ℕ) ≤ n → ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)).
Uses library tsum_congr.
Lean, lines 225–289
```
  have tail3 : ∀ (n : ℕ) (hn : 1 ≤ n),
      ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) = ∑' s : ℕ, ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) := by
    intro n hn
    set f : ℕ → ℕ → ℝ := fun m s => ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * ((1 : ℝ) / 2 ^ (s + 1)) ^ m with hf
    have key : ∀ m s, f m s = ((3 : ℝ) / 2 ^ (m + n + 1)) ^ (s + 1) := by
      intro m s
      simp only [hf]
      rw [div_pow, div_pow, div_pow, one_pow, ← pow_mul, ← pow_mul, ← pow_mul,
        show (m + n + 1) * (s + 1) = (n + 1) * (s + 1) + (s + 1) * m by ring, pow_add]
      field_simp
      exact pow_add _ _ _
    have hpos : ∀ m s, 0 ≤ f m s := fun m s => by rw [key]; positivity
    have h4 : ∀ m : ℕ, (4 : ℝ) ≤ 2 ^ (m + n + 1) := by
      intro m
      calc (4 : ℝ) = 2 ^ 2 := by norm_num
        _ ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
    have inner1 : ∀ m, HasSum (f m) ((3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by
      intro m
      have hr0 : 0 ≤ (3 : ℝ) / 2 ^ (m + n + 1) := by positivity
      have hr1 : (3 : ℝ) / 2 ^ (m + n + 1) < 1 := by
        rw [div_lt_one (by positivity)]; linarith [h4 m]
      have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left ((3 : ℝ) / 2 ^ (m + n + 1))
      have hfun : f m = fun s => (3 : ℝ) / 2 ^ (m + n + 1) * ((3 : ℝ) / 2 ^ (m + n + 1)) ^ s := by
        funext s; rw [key]; ring
      have hne : (2 : ℝ) ^ (m + n + 1) - 3 ≠ 0 := by linarith [h4 m]
      have hne2 : (2 : ℝ) ^ (m + n + 1) ≠ 0 := by positivity
      have hval : (3 : ℝ) / (2 ^ (m + n + 1) - 3) =
          3 / 2 ^ (m + n + 1) * (1 - 3 / 2 ^ (m + n + 1))⁻¹ := by
        field_simp
      rw [hfun, hval]
      exact hg
    have inner2 : ∀ s, HasSum (fun m => f m s) (((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1)) := by
      intro s
      have hr0 : 0 ≤ (1 : ℝ) / 2 ^ (s + 1) := by positivity
      have hr1 : (1 : ℝ) / 2 ^ (s + 1) < 1 := by
        rw [div_lt_one (by positivity)]
        exact one_lt_pow₀ (by norm_num) (by omega)
      have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1))
      have hne : (2 : ℝ) ^ (s + 1) - 1 ≠ 0 := by
        have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have hval : ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) =
          ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * (1 - 1 / 2 ^ (s + 1))⁻¹ := by
        rw [div_pow, div_pow, ← pow_mul, ← pow_mul, show (n + 1) * (s + 1) = n * (s + 1) + (s + 1) by ring, pow_add]
        field_simp
        exact pow_add _ _ _
      rw [hval]
      exact hg
    have hsum1 : Summable (fun m : ℕ => (3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by
      refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3)
      · have := h4 m; exact div_nonneg (by norm_num) (by linarith)
      · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
        have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
        calc (3 : ℝ) / (2 ^ (m + n + 1) - 3) ≤ 3 / 2 ^ m :=
              div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
          _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
    have hU : Summable (Function.uncurry f) := by
      refine (summable_prod_of_nonneg (fun p => hpos p.1 p.2)).mpr ⟨fun m => (inner1 m).summable, ?_⟩
      exact hsum1.congr (fun m => ((inner1 m).tsum_eq).symm)
    have hcomm := hU.tsum_comm' (fun m => (inner1 m).summable) (fun s => (inner2 s).summable)
    rw [show (∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3)) = ∑' m, ∑' s, f m s from
      tsum_congr (fun m => ((inner1 m).tsum_eq).symm)]
    rw [← hcomm]
    exact tsum_congr (fun s => (inner2 s).tsum_eq)
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 225–289 at dfc7b5cc45a9 ↗
1. tail3.key have ∀ (m s : ℕ), f m s = ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ (s + (1 : ℕ))
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with key : ∀ (m s : ℕ), f m s = ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ (s + (1 : ℕ)).
  Uses library div_pow, one_pow, pow_mul, pow_add.
  Lean, lines 229–235
  have key : ∀ m s, f m s = ((3 : ℝ) / 2 ^ (m + n + 1)) ^ (s + 1) := by intro m s simp only [hf] rw [div_pow, div_pow, div_pow, one_pow, ← pow_mul, ← pow_mul, ← pow_mul, show (m + n + 1) * (s + 1) = (n + 1) * (s + 1) + (s + 1) * m by ring, pow_add] field_simp exact pow_add _ _ _
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 229–235 at dfc7b5cc45a9 ↗
2. tail3.hpos have ∀ (m s : ℕ), (0 : ℝ) ≤ f m s
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with hpos : ∀ (m s : ℕ), (0 : ℝ) ≤ f m s.
  Lean, line 236
  have hpos : ∀ m s, 0 ≤ f m s := fun m s => by rw [key]; positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 236 at dfc7b5cc45a9 ↗
3. tail3.h4 have ∀ (m : ℕ), (4 : ℝ) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ))
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with h4 : ∀ (m : ℕ), (4 : ℝ) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ)).
  Lean, lines 237–240
  have h4 : ∀ m : ℕ, (4 : ℝ) ≤ 2 ^ (m + n + 1) := by intro m calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 237–240 at dfc7b5cc45a9 ↗
  tail3.h4.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num
  Closes the goal.
  Lean, line 239
  calc (4 : ℝ) = 2 ^ 2 := by norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 239 at dfc7b5cc45a9 ↗
  tail3.h4.s2 calc (2 : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ))
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 240
  _ ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 240 at dfc7b5cc45a9 ↗
4. tail3.inner1 have ∀ (m : ℕ), HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)))
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with inner1 : ∀ (m : ℕ), HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))).
  Uses library HasSum.
  Lean, lines 241–255
  have inner1 : ∀ m, HasSum (f m) ((3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by intro m have hr0 : 0 ≤ (3 : ℝ) / 2 ^ (m + n + 1) := by positivity have hr1 : (3 : ℝ) / 2 ^ (m + n + 1) < 1 := by rw [div_lt_one (by positivity)]; linarith [h4 m] have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left ((3 : ℝ) / 2 ^ (m + n + 1)) have hfun : f m = fun s => (3 : ℝ) / 2 ^ (m + n + 1) * ((3 : ℝ) / 2 ^ (m + n + 1)) ^ s := by funext s; rw [key]; ring have hne : (2 : ℝ) ^ (m + n + 1) - 3 ≠ 0 := by linarith [h4 m] have hne2 : (2 : ℝ) ^ (m + n + 1) ≠ 0 := by positivity have hval : (3 : ℝ) / (2 ^ (m + n + 1) - 3) = 3 / 2 ^ (m + n + 1) * (1 - 3 / 2 ^ (m + n + 1))⁻¹ := by field_simp rw [hfun, hval] exact hg
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 241–255 at dfc7b5cc45a9 ↗
  tail3.inner1.hr0 have (0 : ℝ) ≤ (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) routine: positivity
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hr0 : (0 : ℝ) ≤ (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)).
  Lean, line 243
  have hr0 : 0 ≤ (3 : ℝ) / 2 ^ (m + n + 1) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 243 at dfc7b5cc45a9 ↗
  tail3.inner1.hr1 have (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) < (1 : ℝ)
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hr1 : (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) < (1 : ℝ).
  Uses library div_lt_one.
  Lean, lines 244–245
  have hr1 : (3 : ℝ) / 2 ^ (m + n + 1) < 1 := by rw [div_lt_one (by positivity)]; linarith [h4 m]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 244–245 at dfc7b5cc45a9 ↗
  tail3.inner1.hg have HasSum (fun (i : ℕ) => (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ i) ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((1 : ℝ) - (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)))⁻¹)
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hg : HasSum (fun (i : ℕ) => (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ i) ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((1 : ℝ) - (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)))⁻¹).
  Uses library hasSum_geometric_of_lt_one.
  Lean, line 246
  have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left ((3 : ℝ) / 2 ^ (m + n + 1))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 246 at dfc7b5cc45a9 ↗
  tail3.inner1.hfun have f m = fun (s : ℕ) => (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ s
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hfun : f m = fun (s : ℕ) => (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ))) ^ s.
  Lean, lines 247–248
  have hfun : f m = fun s => (3 : ℝ) / 2 ^ (m + n + 1) * ((3 : ℝ) / 2 ^ (m + n + 1)) ^ s := by funext s; rw [key]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 247–248 at dfc7b5cc45a9 ↗
  tail3.inner1.hne have (2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ) ≠ (0 : ℝ) routine: linarith
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hne : (2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ) ≠ (0 : ℝ).
  Lean, line 249
  have hne : (2 : ℝ) ^ (m + n + 1) - 3 ≠ 0 := by linarith [h4 m]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 249 at dfc7b5cc45a9 ↗
  tail3.inner1.hne2 have (2 : ℝ) ^ (m + n + (1 : ℕ)) ≠ (0 : ℝ) routine: positivity
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hne2 : (2 : ℝ) ^ (m + n + (1 : ℕ)) ≠ (0 : ℝ).
  Lean, line 250
  have hne2 : (2 : ℝ) ^ (m + n + 1) ≠ 0 := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 250 at dfc7b5cc45a9 ↗
  tail3.inner1.hval have (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((1 : ℝ) - (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)))⁻¹ routine: field_simp
  Leaves HasSum (f m) ((3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))), with hval : (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)) * ((1 : ℝ) - (3 : ℝ) / (2 : ℝ) ^ (m + n + (1 : ℕ)))⁻¹.
  Lean, lines 251–253
  have hval : (3 : ℝ) / (2 ^ (m + n + 1) - 3) = 3 / 2 ^ (m + n + 1) * (1 - 3 / 2 ^ (m + n + 1))⁻¹ := by field_simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 251–253 at dfc7b5cc45a9 ↗
5. tail3.inner2 have ∀ (s : ℕ), HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)))
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with inner2 : ∀ (s : ℕ), HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))).
  Uses library HasSum.
  Lean, lines 256–272
  have inner2 : ∀ s, HasSum (fun m => f m s) (((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1)) := by intro s have hr0 : 0 ≤ (1 : ℝ) / 2 ^ (s + 1) := by positivity have hr1 : (1 : ℝ) / 2 ^ (s + 1) < 1 := by rw [div_lt_one (by positivity)] exact one_lt_pow₀ (by norm_num) (by omega) have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1)) have hne : (2 : ℝ) ^ (s + 1) - 1 ≠ 0 := by have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega) linarith have hval : ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) = ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * (1 - 1 / 2 ^ (s + 1))⁻¹ := by rw [div_pow, div_pow, ← pow_mul, ← pow_mul, show (n + 1) * (s + 1) = n * (s + 1) + (s + 1) by ring, pow_add] field_simp exact pow_add _ _ _ rw [hval] exact hg
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 256–272 at dfc7b5cc45a9 ↗
  tail3.inner2.hr0 have (0 : ℝ) ≤ (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)) routine: positivity
  Leaves HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))), with hr0 : (0 : ℝ) ≤ (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)).
  Lean, line 258
  have hr0 : 0 ≤ (1 : ℝ) / 2 ^ (s + 1) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 258 at dfc7b5cc45a9 ↗
  tail3.inner2.hr1 have (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)) < (1 : ℝ)
  Leaves HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))), with hr1 : (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)) < (1 : ℝ).
  Uses library div_lt_one, one_lt_pow₀.
  Lean, lines 259–261
  have hr1 : (1 : ℝ) / 2 ^ (s + 1) < 1 := by rw [div_lt_one (by positivity)] exact one_lt_pow₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 259–261 at dfc7b5cc45a9 ↗
  tail3.inner2.hg have HasSum (fun (i : ℕ) => ((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ))) ^ i) (((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) - (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)))⁻¹)
  Leaves HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))), with hg : HasSum (fun (i : ℕ) => ((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ))) ^ i) (((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) - (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)))⁻¹).
  Uses library hasSum_geometric_of_lt_one.
  Lean, line 262
  have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 262 at dfc7b5cc45a9 ↗
  tail3.inner2.hne have (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ) ≠ (0 : ℝ)
  Leaves HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))), with hne : (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ) ≠ (0 : ℝ).
  Lean, lines 263–265
  have hne : (2 : ℝ) ^ (s + 1) - 1 ≠ 0 := by have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega) linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 263–265 at dfc7b5cc45a9 ↗
  tail3.inner2.hne.this have (1 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ))
  Leaves (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ) ≠ (0 : ℝ), with this : (1 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ)).
  Uses library one_lt_pow₀.
  Lean, line 264
  have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 264 at dfc7b5cc45a9 ↗
  tail3.inner2.hval have ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)) = ((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) - (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)))⁻¹
  Leaves HasSum (fun (m : ℕ) => f m s) (((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ))), with hval : ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)) = ((3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))) ^ (s + (1 : ℕ)) * ((1 : ℝ) - (1 : ℝ) / (2 : ℝ) ^ (s + (1 : ℕ)))⁻¹.
  Uses library div_pow, pow_mul, pow_add.
  Lean, lines 266–270
  have hval : ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) = ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * (1 - 1 / 2 ^ (s + 1))⁻¹ := by rw [div_pow, div_pow, ← pow_mul, ← pow_mul, show (n + 1) * (s + 1) = n * (s + 1) + (s + 1) by ring, pow_add] field_simp exact pow_add _ _ _
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 266–270 at dfc7b5cc45a9 ↗
6. tail3.hsum1 have Summable fun (m : ℕ) => (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with hsum1 : Summable fun (m : ℕ) => (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)).
  Uses library Summable, Summable.of_nonneg_of_le, summable_geometric_two.
  Lean, lines 273–281
  have hsum1 : Summable (fun m : ℕ => (3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3) · have := h4 m; exact div_nonneg (by norm_num) (by linarith) · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega) have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring calc (3 : ℝ) / (2 ^ (m + n + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith) _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 273–281 at dfc7b5cc45a9 ↗
  tail3.hsum1.s1 case
  Leaves (0 : ℝ) ≤ (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)).
  Uses library div_nonneg.
  Lean, line 275
  · have := h4 m; exact div_nonneg (by norm_num) (by linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 275 at dfc7b5cc45a9 ↗
  tail3.hsum1.s1.this have (4 : ℝ) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ))
  Leaves (0 : ℝ) ≤ (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with this : (4 : ℝ) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ)).
  Lean, line 275
  · have := h4 m; exact div_nonneg (by norm_num) (by linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 275 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2 case
  Leaves (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m.
  Lean, lines 276–281
  · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega) have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring calc (3 : ℝ) / (2 ^ (m + n + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith) _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 276–281 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2.hpow have (2 : ℝ) ^ (m + (2 : ℕ)) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ))
  Leaves (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with hpow : (2 : ℝ) ^ (m + (2 : ℕ)) ≤ (2 : ℝ) ^ (m + n + (1 : ℕ)).
  Uses library pow_le_pow_right₀.
  Lean, line 276
  · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 276 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ m.
  Uses library one_le_pow₀.
  Lean, line 277
  have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 277 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2.h2 have (2 : ℝ) ^ (m + (2 : ℕ)) = (4 : ℝ) * (2 : ℝ) ^ m
  Leaves (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with h2 : (2 : ℝ) ^ (m + (2 : ℕ)) = (4 : ℝ) * (2 : ℝ) ^ m.
  Uses library pow_add.
  Lean, line 278
  have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 278 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2.s4 calc (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) / (2 : ℝ) ^ m
  Closes the goal.
  Uses library div_le_div_of_nonneg_left.
  Lean, lines 279–280
  calc (3 : ℝ) / (2 ^ (m + n + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 279–280 at dfc7b5cc45a9 ↗
  tail3.hsum1.s2.s5 calc
  Closes the goal.
  Uses library one_div_pow.
  Lean, line 281
  _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 281 at dfc7b5cc45a9 ↗
7. tail3.hU have Summable (Function.uncurry f)
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with hU : Summable (Function.uncurry f).
  Uses library Summable, Function.uncurry, summable_prod_of_nonneg, And.intro.
  Lean, lines 282–284
  have hU : Summable (Function.uncurry f) := by refine (summable_prod_of_nonneg (fun p => hpos p.1 p.2)).mpr ⟨fun m => (inner1 m).summable, ?_⟩ exact hsum1.congr (fun m => ((inner1 m).tsum_eq).symm)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 282–284 at dfc7b5cc45a9 ↗
8. tail3.hcomm have ∑' (c : ℕ) (b : ℕ), f b c = ∑' (b : ℕ) (c : ℕ), f b c
  Leaves ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (s : ℕ), ((3 : ℝ) / (2 : ℝ) ^ n) ^ (s + (1 : ℕ)) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)), with hcomm : ∑' (c : ℕ) (b : ℕ), f b c = ∑' (b : ℕ) (c : ℕ), f b c.
  Lean, line 285
  have hcomm := hU.tsum_comm' (fun m => (inner1 m).summable) (fun s => (inner2 s).summable)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 285 at dfc7b5cc45a9 ↗
split3T have ∀ (n : ℕ), (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with split3T : ∀ (n : ℕ), (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)).
Uses library Finset.range, zero_add.
Lean, lines 291–329
```
  have split3T : ∀ (n : ℕ),
      3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) =
        ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) + ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) := by
    intro n
    set g : ℕ → ℝ := fun m => (3 : ℝ) / (2 ^ (m + 1) - 3) with hg
    have hsum : Summable (fun m => g (m + 1)) := by
      refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3)
      · have : (4 : ℝ) ≤ 2 ^ (m + 1 + 1) := by
          calc (4 : ℝ) = 2 ^ 2 := by norm_num
            _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        simp only [hg]; exact div_nonneg (by norm_num) (by linarith)
      · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf)
        have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
        have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
        simp only [hg]
        calc (3 : ℝ) / (2 ^ (m + 1 + 1) - 3) ≤ 3 / 2 ^ m :=
              div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
          _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
    have hs : Summable g := (summable_nat_add_iff 1).mp hsum
    have e1 := hs.sum_add_tsum_nat_add n
    have e2 := hs.sum_add_tsum_nat_add 2
    have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑' m, g (m + 2) := by
      rw [← tsum_mul_left]
      refine tsum_congr (fun m => ?_)
      simp only [hg]
      rw [show m + 2 + 1 = m + 3 by ring]
      ring
    have hfirst : ∑ i ∈ Finset.range 2, g i = 0 := by
      simp only [hg, Finset.sum_range_succ, Finset.sum_range_zero]
      norm_num
    rw [h3T]
    have htail : ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) = ∑' m, g (m + n) := by
      refine tsum_congr (fun m => ?_)
      simp only [hg]
    rw [htail]
    have hrange : ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) = ∑ i ∈ Finset.range n, g i := by
      simp only [hg]
    rw [hrange, e1]
    rw [← e2, hfirst, zero_add]
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 291–329 at dfc7b5cc45a9 ↗
1. split3T.hsum have Summable fun (m : ℕ) => g (m + (1 : ℕ))
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with hsum : Summable fun (m : ℕ) => g (m + (1 : ℕ)).
  Uses library Summable, Summable.of_nonneg_of_le, summable_geometric_two.
  Lean, lines 296–308
  have hsum : Summable (fun m => g (m + 1)) := by refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3) · have : (4 : ℝ) ≤ 2 ^ (m + 1 + 1) := by calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega) simp only [hg]; exact div_nonneg (by norm_num) (by linarith) · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf) have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring simp only [hg] calc (3 : ℝ) / (2 ^ (m + 1 + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith) _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 296–308 at dfc7b5cc45a9 ↗
  split3T.hsum.s1 case
  Leaves (0 : ℝ) ≤ g (m + (1 : ℕ)).
  Uses library div_nonneg.
  Lean, lines 298–301
  · have : (4 : ℝ) ≤ 2 ^ (m + 1 + 1) := by calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega) simp only [hg]; exact div_nonneg (by norm_num) (by linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 298–301 at dfc7b5cc45a9 ↗
  split3T.hsum.s1.this have (4 : ℝ) ≤ (2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ))
  Leaves (0 : ℝ) ≤ g (m + (1 : ℕ)), with this : (4 : ℝ) ≤ (2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ)).
  Lean, lines 298–300
  · have : (4 : ℝ) ≤ 2 ^ (m + 1 + 1) := by calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 298–300 at dfc7b5cc45a9 ↗
  split3T.hsum.s1.this.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num
  Closes the goal.
  Lean, line 299
  calc (4 : ℝ) = 2 ^ 2 := by norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 299 at dfc7b5cc45a9 ↗
  split3T.hsum.s1.this.s2 calc (2 : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ))
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 300
  _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 300 at dfc7b5cc45a9 ↗
  split3T.hsum.s2 case
  Leaves g (m + (1 : ℕ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m.
  Lean, lines 302–308
  · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf) have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring simp only [hg] calc (3 : ℝ) / (2 ^ (m + 1 + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith) _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 302–308 at dfc7b5cc45a9 ↗
  split3T.hsum.s2.hpow have (2 : ℝ) ^ (m + (2 : ℕ)) ≤ (2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ))
  Leaves g (m + (1 : ℕ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with hpow : (2 : ℝ) ^ (m + (2 : ℕ)) ≤ (2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ)).
  Uses library le_of_eq.
  Lean, line 302
  · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 302 at dfc7b5cc45a9 ↗
  split3T.hsum.s2.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves g (m + (1 : ℕ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ m.
  Uses library one_le_pow₀.
  Lean, line 303
  have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 303 at dfc7b5cc45a9 ↗
  split3T.hsum.s2.h2 have (2 : ℝ) ^ (m + (2 : ℕ)) = (4 : ℝ) * (2 : ℝ) ^ m
  Leaves g (m + (1 : ℕ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m, with h2 : (2 : ℝ) ^ (m + (2 : ℕ)) = (4 : ℝ) * (2 : ℝ) ^ m.
  Uses library pow_add.
  Lean, line 304
  have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 304 at dfc7b5cc45a9 ↗
  split3T.hsum.s2.s4 calc (3 : ℝ) / ((2 : ℝ) ^ (m + (1 : ℕ) + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) / (2 : ℝ) ^ m
  Closes the goal.
  Uses library div_le_div_of_nonneg_left.
  Lean, lines 306–307
  calc (3 : ℝ) / (2 ^ (m + 1 + 1) - 3) ≤ 3 / 2 ^ m := div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 306–307 at dfc7b5cc45a9 ↗
  split3T.hsum.s2.s5 calc
  Closes the goal.
  Uses library one_div_pow.
  Lean, line 308
  _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 308 at dfc7b5cc45a9 ↗
2. split3T.hs have Summable g
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with hs : Summable g.
  Uses library Summable, summable_nat_add_iff.
  Lean, line 309
  have hs : Summable g := (summable_nat_add_iff 1).mp hsum
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 309 at dfc7b5cc45a9 ↗
3. split3T.e1 have ∑ i ∈ Finset.range n, g i + ∑' (i : ℕ), g (i + n) = ∑' (i : ℕ), g i
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with e1 : ∑ i ∈ Finset.range n, g i + ∑' (i : ℕ), g (i + n) = ∑' (i : ℕ), g i.
  Lean, line 310
  have e1 := hs.sum_add_tsum_nat_add n
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 310 at dfc7b5cc45a9 ↗
4. split3T.e2 have ∑ i ∈ Finset.range (2 : ℕ), g i + ∑' (i : ℕ), g (i + (2 : ℕ)) = ∑' (i : ℕ), g i
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with e2 : ∑ i ∈ Finset.range (2 : ℕ), g i + ∑' (i : ℕ), g (i + (2 : ℕ)) = ∑' (i : ℕ), g i.
  Lean, line 311
  have e2 := hs.sum_add_tsum_nat_add 2
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 311 at dfc7b5cc45a9 ↗
5. split3T.h3T have (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑' (m : ℕ), g (m + (2 : ℕ))
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with h3T : (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑' (m : ℕ), g (m + (2 : ℕ)).
  Uses library tsum_mul_left, tsum_congr.
  Lean, lines 312–317
  have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑' m, g (m + 2) := by rw [← tsum_mul_left] refine tsum_congr (fun m => ?_) simp only [hg] rw [show m + 2 + 1 = m + 3 by ring] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 312–317 at dfc7b5cc45a9 ↗
6. split3T.hfirst have ∑ i ∈ Finset.range (2 : ℕ), g i = (0 : ℝ) routine: simp, norm_num
  Leaves (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with hfirst : ∑ i ∈ Finset.range (2 : ℕ), g i = (0 : ℝ).
  Uses library Finset.range, Finset.sum_range_succ, Finset.sum_range_zero.
  Lean, lines 318–320
  have hfirst : ∑ i ∈ Finset.range 2, g i = 0 := by simp only [hg, Finset.sum_range_succ, Finset.sum_range_zero] norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 318–320 at dfc7b5cc45a9 ↗
7. split3T.htail have ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (m : ℕ), g (m + n)
  Leaves ∑' (m : ℕ), g (m + (2 : ℕ)) = (Finset.range n).sum g + ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)), with htail : ∑' (m : ℕ), (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) = ∑' (m : ℕ), g (m + n).
  Uses library tsum_congr.
  Lean, lines 322–324
  have htail : ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) = ∑' m, g (m + n) := by refine tsum_congr (fun m => ?_) simp only [hg]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 322–324 at dfc7b5cc45a9 ↗
8. split3T.hrange have ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) = ∑ i ∈ Finset.range n, g i routine: simp
  Leaves ∑' (m : ℕ), g (m + (2 : ℕ)) = (Finset.range n).sum g + ∑' (m : ℕ), g (m + n), with hrange : ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) = ∑ i ∈ Finset.range n, g i.
  Uses library Finset.range.
  Lean, lines 326–327
  have hrange : ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) = ∑ i ∈ Finset.range n, g i := by simp only [hg]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 326–327 at dfc7b5cc45a9 ↗
cauchy_fin have ∀ (n : ℕ) (q a : ℕ → ℝ) (x A : ℝ), HasSum (fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))) A → HasSum (fun (K : ℕ) => ∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then q j * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), q j * x ^ j) * A)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with cauchy_fin : ∀ (n : ℕ) (q a : ℕ → ℝ) (x A : ℝ), HasSum (fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))) A → HasSum (fun (K : ℕ) => ∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then q j * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), q j * x ^ j) * A).
Uses library HasSum, Finset.range, Finset.sum_mul, hasSum_sum, hasSum_nat_add_iff', sub_zero.
Lean, lines 331–351
```
  have cauchy_fin : ∀ (n : ℕ) (q a : ℕ → ℝ) (x A : ℝ) (hA : HasSum (fun s => a s * x ^ (s + 1)) A),
      HasSum (fun K => ∑ j ∈ Finset.range (n + 1), if j < K then q j * a (K - j - 1) * x ^ K else 0)
        ((∑ j ∈ Finset.range (n + 1), q j * x ^ j) * A) := by
    intro n q a x A hA
    rw [Finset.sum_mul]
    apply hasSum_sum
    intro j _
    have h := hA.mul_left (q j * x ^ j)
    rw [← hasSum_nat_add_iff' (j + 1)]
    have hz : ∑ i ∈ Finset.range (j + 1), (if j < i then q j * a (i - j - 1) * x ^ i else 0) = 0 :=
      Finset.sum_eq_zero (fun i hi => by
        have := Finset.mem_range.mp hi
        rw [if_neg (by omega)])
    rw [hz, sub_zero]
    have hfun : (fun s => if j < s + (j + 1) then q j * a (s + (j + 1) - j - 1) * x ^ (s + (j + 1)) else 0) =
        fun s => q j * x ^ j * (a s * x ^ (s + 1)) := by
      funext s
      rw [if_pos (by omega), show s + (j + 1) - j - 1 = s by omega, show s + (j + 1) = j + (s + 1) by ring, pow_add]
      ring
    rw [hfun]
    exact h
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 331–351 at dfc7b5cc45a9 ↗
1. cauchy_fin.h have HasSum (fun (i : ℕ) => q j * x ^ j * (a i * x ^ (i + (1 : ℕ)))) (q j * x ^ j * A)
  Leaves HasSum (fun (b : ℕ) => if j < b then q j * a (b - j - (1 : ℕ)) * x ^ b else (0 : ℝ)) (q j * x ^ j * A), with h : HasSum (fun (i : ℕ) => q j * x ^ j * (a i * x ^ (i + (1 : ℕ)))) (q j * x ^ j * A).
  Lean, line 338
  have h := hA.mul_left (q j * x ^ j)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 338 at dfc7b5cc45a9 ↗
2. cauchy_fin.hz have (∑ i ∈ Finset.range (j + (1 : ℕ)), if j < i then q j * a (i - j - (1 : ℕ)) * x ^ i else (0 : ℝ)) = (0 : ℝ)
  Leaves HasSum (fun (n : ℕ) => if j < n + (j + (1 : ℕ)) then q j * a (n + (j + (1 : ℕ)) - j - (1 : ℕ)) * x ^ (n + (j + (1 : ℕ))) else (0 : ℝ)) (q j * x ^ j * A - ∑ i ∈ Finset.range (j + (1 : ℕ)), if j < i then q j * a (i - j - (1 : ℕ)) * x ^ i else (0 : ℝ)), with hz : (∑ i ∈ Finset.range (j + (1 : ℕ)), if j < i then q j * a (i - j - (1 : ℕ)) * x ^ i else (0 : ℝ)) = (0 : ℝ).
  Uses library Finset.range, Finset.sum_eq_zero, if_neg.
  Lean, lines 340–343
  have hz : ∑ i ∈ Finset.range (j + 1), (if j < i then q j * a (i - j - 1) * x ^ i else 0) = 0 := Finset.sum_eq_zero (fun i hi => by have := Finset.mem_range.mp hi rw [if_neg (by omega)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 340–343 at dfc7b5cc45a9 ↗
  cauchy_fin.hz.this have i < j + (1 : ℕ)
  Leaves (if j < i then q j * a (i - j - (1 : ℕ)) * x ^ i else (0 : ℝ)) = (0 : ℝ), with this : i < j + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 342
  have := Finset.mem_range.mp hi
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 342 at dfc7b5cc45a9 ↗
3. cauchy_fin.hfun have (fun (s : ℕ) => if j < s + (j + (1 : ℕ)) then q j * a (s + (j + (1 : ℕ)) - j - (1 : ℕ)) * x ^ (s + (j + (1 : ℕ))) else (0 : ℝ)) = fun (s : ℕ) => q j * x ^ j * (a s * x ^ (s + (1 : ℕ)))
  Leaves HasSum (fun (n : ℕ) => if j < n + (j + (1 : ℕ)) then q j * a (n + (j + (1 : ℕ)) - j - (1 : ℕ)) * x ^ (n + (j + (1 : ℕ))) else (0 : ℝ)) (q j * x ^ j * A), with hfun : (fun (s : ℕ) => if j < s + (j + (1 : ℕ)) then q j * a (s + (j + (1 : ℕ)) - j - (1 : ℕ)) * x ^ (s + (j + (1 : ℕ))) else (0 : ℝ)) = fun (s : ℕ) => q j * x ^ j * (a s * x ^ (s + (1 : ℕ))).
  Uses library if_pos, pow_add.
  Lean, lines 345–349
  have hfun : (fun s => if j < s + (j + 1) then q j * a (s + (j + 1) - j - 1) * x ^ (s + (j + 1)) else 0) = fun s => q j * x ^ j * (a s * x ^ (s + 1)) := by funext s rw [if_pos (by omega), show s + (j + 1) - j - 1 = s by omega, show s + (j + 1) = j + (s + 1) by ring, pow_add] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 345–349 at dfc7b5cc45a9 ↗
h4_series have ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (1 : ℕ) ≤ n → HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with h4_series : ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (1 : ℕ) ≤ n → HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)).
Uses library Finset.range, HasSum, Finset.Ioc.
Lean, lines 353–472
```
  have h4_series : ∀ (Qc : ℕ → ℕ → ℚ) (hQc : (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)))) (Qx : ℕ → ℚ) (hQx : (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k))
    (Aq : ℕ → ℚ) (hAq : (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ)))) (n : ℕ) (hn : 1 ≤ n),
      HasSum (fun K : ℕ => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            ((3 : ℝ) / 2 ^ n) ^ K else 0)
        ((Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ)) := by
    intro Qc hQc Qx hQx Aq hAq n hn
    set x : ℝ := (3 : ℝ) / 2 ^ n with hxdef
    have hx0 : 0 ≤ x := by positivity
    have hx2 : x / 2 < 1 := by
      have : (2 : ℝ) ≤ 2 ^ n := by
        calc (2 : ℝ) = 2 ^ 1 := by norm_num
          _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
      rw [hxdef, div_div, div_lt_one (by positivity)]; linarith
    set a : ℕ → ℝ := fun s => (1 : ℝ) / (2 ^ (s + 1) - 1) with hadef
    have hapos : ∀ s, 0 < (2 : ℝ) ^ (s + 1) - 1 := fun s => by
      have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hAsum : Summable (fun s => a s * x ^ (s + 1)) := by
      refine Summable.of_nonneg_of_le (fun s => ?_) (fun s => ?_) ((summable_geometric_of_lt_one (by positivity) hx2).mul_left x)
      · simp only [hadef]; exact mul_nonneg (div_nonneg (by norm_num) (le_of_lt (hapos s))) (by positivity)
      · simp only [hadef]
        have h1 : (2 : ℝ) ^ s ≤ 2 ^ (s + 1) - 1 := by
          have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num)
          rw [pow_succ]; linarith
        have hxs : 0 ≤ x ^ (s + 1) := by positivity
        calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by
              gcongr
          _ = x * (x / 2) ^ s := by
            have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by
              intro y; rw [div_pow, pow_succ]; field_simp
            exact e x
    set A := ∑' s, a s * x ^ (s + 1) with hAdef
    have hA : HasSum (fun s => a s * x ^ (s + 1)) A := hAsum.hasSum
    -- 3T = c + A
    have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) + A := by
      rw [split3T n, tail3 n hn, hAdef]
      congr 1
      refine tsum_congr (fun s => ?_)
      simp only [hadef, hxdef]
      ring
    have hQxR : (Qx n : ℝ) = ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j := by
      rw [hQx]; push_cast; rfl
    have hAqR : (Aq n : ℝ) = ∑ k ∈ Finset.range (n + 1),
        (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * x ^ k +
        (Qx n : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) := by
      rw [hAq]; push_cast; rfl
    have hC := cauchy_fin n (fun j => (Qc n j : ℝ)) a x A hA
    set P : ℝ := ∑ k ∈ Finset.range (n + 1),
        (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * x ^ k with hPdef
    have hP : HasSum (fun K => if K < n + 1 then
        (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) P := by
      have h0 : HasSum (fun K => if K < n + 1 then
          (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0)
          (∑ K ∈ Finset.range (n + 1), if K < n + 1 then
            (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) :=
        hasSum_sum_of_ne_finset_zero (fun b hb => by rw [if_neg (by simpa using hb)])
      have h1 : (∑ K ∈ Finset.range (n + 1), if K < n + 1 then
            (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) = P := by
        rw [hPdef]
        exact Finset.sum_congr rfl (fun K hK => by rw [if_pos (Finset.mem_range.mp hK)])
      rw [h1] at h0
      exact h0
    have hD := hC.sub hP
    have hval : (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) =
        (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j) * A - P := by
      rw [h3T, hAqR, hQxR]; ring
    rw [hval]
    have hid1 : ∀ j ≤ n, Qc n j * 2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
        ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ L) := fun j hj => by subst hQc; exact id1_full n j hj
    have hfun : (fun K : ℕ => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            ((3 : ℝ) / 2 ^ n) ^ K else 0) =
        fun K => (∑ j ∈ Finset.range (n + 1), if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) -
          (if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) := by
      funext K
      symm
      by_cases hK : K < n + 1
      · rw [if_pos hK, if_neg (by omega)]
        rw [Finset.sum_range_succ (fun j => (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) K, Nat.sub_self, pow_zero, sub_self,
          div_zero, add_zero]
        have hsplit : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) =
            ∑ j ∈ Finset.range K, (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1) * x ^ K := by
          rw [← Finset.sum_range_add_sum_Ico _ (show K ≤ n + 1 by omega)]
          rw [Finset.sum_eq_zero (s := Finset.Ico K (n + 1)) (fun j hj => by
            rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero]
          refine Finset.sum_congr rfl (fun j hj => ?_)
          have hjK := Finset.mem_range.mp hj
          rw [if_pos hjK]
          simp only [hadef]
          rw [show K - j - 1 + 1 = K - j by omega]
          ring
        rw [hsplit, Finset.sum_mul, sub_self]
      · rw [if_neg hK, sub_zero]
        have hS : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) =
            ((∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) : ℚ) : ℝ) * x ^ K := by
          push_cast
          rw [Finset.sum_mul]
          refine Finset.sum_congr rfl (fun j hj => ?_)
          have hjK : j < K := by have := Finset.mem_range.mp hj; omega
          rw [if_pos hjK]
          simp only [hadef]
          rw [show K - j - 1 + 1 = K - j by omega]
          ring
        rw [hS, tail_coeff n K (by omega) Qc hid1]
        push_cast
        by_cases h2 : 2 * n < K
        · rw [if_pos h2]
        · rw [if_neg h2]
          rw [Finset.prod_eq_zero (i := K) (Finset.mem_Ioc.mpr ⟨by omega, by omega⟩) (sub_self _)]
          simp
    rw [hfun]
    exact hD
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 353–472 at dfc7b5cc45a9 ↗
1. h4_series.hx0 have (0 : ℝ) ≤ x routine: positivity
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hx0 : (0 : ℝ) ≤ x.
  Lean, line 368
  have hx0 : 0 ≤ x := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 368 at dfc7b5cc45a9 ↗
2. h4_series.hx2 have x / (2 : ℝ) < (1 : ℝ)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hx2 : x / (2 : ℝ) < (1 : ℝ).
  Uses library div_div, div_lt_one.
  Lean, lines 369–373
  have hx2 : x / 2 < 1 := by have : (2 : ℝ) ≤ 2 ^ n := by calc (2 : ℝ) = 2 ^ 1 := by norm_num _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn rw [hxdef, div_div, div_lt_one (by positivity)]; linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 369–373 at dfc7b5cc45a9 ↗
  h4_series.hx2.this have (2 : ℝ) ≤ (2 : ℝ) ^ n
  Leaves x / (2 : ℝ) < (1 : ℝ), with this : (2 : ℝ) ≤ (2 : ℝ) ^ n.
  Lean, lines 370–372
  have : (2 : ℝ) ≤ 2 ^ n := by calc (2 : ℝ) = 2 ^ 1 := by norm_num _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 370–372 at dfc7b5cc45a9 ↗
  h4_series.hx2.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ) routine: norm_num
  Closes the goal.
  Lean, line 371
  calc (2 : ℝ) = 2 ^ 1 := by norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 371 at dfc7b5cc45a9 ↗
  h4_series.hx2.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ n
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 372
  _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 372 at dfc7b5cc45a9 ↗
3. h4_series.hapos have ∀ (s : ℕ), (0 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hapos : ∀ (s : ℕ), (0 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ).
  Lean, lines 375–377
  have hapos : ∀ s, 0 < (2 : ℝ) ^ (s + 1) - 1 := fun s => by have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega) linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 375–377 at dfc7b5cc45a9 ↗
  h4_series.hapos.this have (1 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ))
  Leaves (0 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ), with this : (1 : ℝ) < (2 : ℝ) ^ (s + (1 : ℕ)).
  Uses library one_lt_pow₀.
  Lean, line 376
  have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 376 at dfc7b5cc45a9 ↗
4. h4_series.hAsum have Summable fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hAsum : Summable fun (s : ℕ) => a s * x ^ (s + (1 : ℕ)).
  Uses library Summable, Summable.of_nonneg_of_le, summable_geometric_of_lt_one.
  Lean, lines 378–391
  have hAsum : Summable (fun s => a s * x ^ (s + 1)) := by refine Summable.of_nonneg_of_le (fun s => ?_) (fun s => ?_) ((summable_geometric_of_lt_one (by positivity) hx2).mul_left x) · simp only [hadef]; exact mul_nonneg (div_nonneg (by norm_num) (le_of_lt (hapos s))) (by positivity) · simp only [hadef] have h1 : (2 : ℝ) ^ s ≤ 2 ^ (s + 1) - 1 := by have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num) rw [pow_succ]; linarith have hxs : 0 ≤ x ^ (s + 1) := by positivity calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by gcongr _ = x * (x / 2) ^ s := by have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by intro y; rw [div_pow, pow_succ]; field_simp exact e x
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 378–391 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s1 case
  Leaves (0 : ℝ) ≤ a s * x ^ (s + (1 : ℕ)).
  Uses library mul_nonneg, div_nonneg, le_of_lt.
  Lean, line 380
  · simp only [hadef]; exact mul_nonneg (div_nonneg (by norm_num) (le_of_lt (hapos s))) (by positivity)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 380 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2 case
  Leaves a s * x ^ (s + (1 : ℕ)) ≤ x * (x / (2 : ℝ)) ^ s.
  Lean, lines 381–391
  · simp only [hadef] have h1 : (2 : ℝ) ^ s ≤ 2 ^ (s + 1) - 1 := by have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num) rw [pow_succ]; linarith have hxs : 0 ≤ x ^ (s + 1) := by positivity calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by gcongr _ = x * (x / 2) ^ s := by have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by intro y; rw [div_pow, pow_succ]; field_simp exact e x
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 381–391 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.h1 have (2 : ℝ) ^ s ≤ (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)
  Leaves (1 : ℝ) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)) * x ^ (s + (1 : ℕ)) ≤ x * (x / (2 : ℝ)) ^ s, with h1 : (2 : ℝ) ^ s ≤ (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ).
  Uses library pow_succ.
  Lean, lines 382–384
  have h1 : (2 : ℝ) ^ s ≤ 2 ^ (s + 1) - 1 := by have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num) rw [pow_succ]; linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 382–384 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.h1.this have (1 : ℝ) ≤ (2 : ℝ) ^ s
  Leaves (2 : ℝ) ^ s ≤ (2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ), with this : (1 : ℝ) ≤ (2 : ℝ) ^ s.
  Uses library one_le_pow₀.
  Lean, line 383
  have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 383 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.hxs have (0 : ℝ) ≤ x ^ (s + (1 : ℕ)) routine: positivity
  Leaves (1 : ℝ) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)) * x ^ (s + (1 : ℕ)) ≤ x * (x / (2 : ℝ)) ^ s, with hxs : (0 : ℝ) ≤ x ^ (s + (1 : ℕ)).
  Lean, line 385
  have hxs : 0 ≤ x ^ (s + 1) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 385 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.s3 calc (1 : ℝ) / ((2 : ℝ) ^ (s + (1 : ℕ)) - (1 : ℝ)) * x ^ (s + (1 : ℕ)) ≤ (1 : ℝ) / (2 : ℝ) ^ s * x ^ (s + (1 : ℕ))
  Closes the goal.
  Lean, lines 386–387
  calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 386–387 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.s4 calc
  Closes the goal.
  Lean, lines 388–391
  _ = x * (x / 2) ^ s := by have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by intro y; rw [div_pow, pow_succ]; field_simp exact e x
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 388–391 at dfc7b5cc45a9 ↗
  h4_series.hAsum.s2.s4.e have ∀ (y : ℝ), (1 : ℝ) / (2 : ℝ) ^ s * y ^ (s + (1 : ℕ)) = y * (y / (2 : ℝ)) ^ s
  Leaves (1 : ℝ) / (2 : ℝ) ^ s * x ^ (s + (1 : ℕ)) = x * (x / (2 : ℝ)) ^ s, with e : ∀ (y : ℝ), (1 : ℝ) / (2 : ℝ) ^ s * y ^ (s + (1 : ℕ)) = y * (y / (2 : ℝ)) ^ s.
  Uses library div_pow, pow_succ.
  Lean, lines 389–390
  have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by intro y; rw [div_pow, pow_succ]; field_simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 389–390 at dfc7b5cc45a9 ↗
5. h4_series.hA have HasSum (fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))) A
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hA : HasSum (fun (s : ℕ) => a s * x ^ (s + (1 : ℕ))) A.
  Uses library HasSum.
  Lean, line 393
  have hA : HasSum (fun s => a s * x ^ (s + 1)) A := hAsum.hasSum
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 393 at dfc7b5cc45a9 ↗
6. h4_series.h3T have (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) + A
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with h3T : (3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) = ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)) + A.
  Uses library Finset.range, tsum_congr.
  Lean, lines 395–400
  have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) + A := by rw [split3T n, tail3 n hn, hAdef] congr 1 refine tsum_congr (fun s => ?_) simp only [hadef, hxdef] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 395–400 at dfc7b5cc45a9 ↗
7. h4_series.hQxR have (↑(Qx n) : ℝ) = ∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hQxR : (↑(Qx n) : ℝ) = ∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j.
  Uses library Finset.range.
  Lean, lines 401–402
  have hQxR : (Qx n : ℝ) = ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j := by rw [hQx]; push_cast; rfl
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 401–402 at dfc7b5cc45a9 ↗
8. h4_series.hAqR have (↑(Aq n) : ℝ) = ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (k - j) - (1 : ℝ))) * x ^ k + (↑(Qx n) : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ))
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hAqR : (↑(Aq n) : ℝ) = ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (k - j) - (1 : ℝ))) * x ^ k + (↑(Qx n) : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + (1 : ℕ)) - (3 : ℝ)).
  Uses library Finset.range.
  Lean, lines 403–406
  have hAqR : (Aq n : ℝ) = ∑ k ∈ Finset.range (n + 1), (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * x ^ k + (Qx n : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) := by rw [hAq]; push_cast; rfl
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 403–406 at dfc7b5cc45a9 ↗
9. h4_series.hC have HasSum (fun (K : ℕ) => ∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hC : HasSum (fun (K : ℕ) => ∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A).
  Lean, line 407
  have hC := cauchy_fin n (fun j => (Qc n j : ℝ)) a x A hA
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 407 at dfc7b5cc45a9 ↗
10. h4_series.hP have HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) P
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hP : HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) P.
  Uses library HasSum, Finset.range.
  Lean, lines 410–422
  have hP : HasSum (fun K => if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) P := by have h0 : HasSum (fun K => if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) (∑ K ∈ Finset.range (n + 1), if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) := hasSum_sum_of_ne_finset_zero (fun b hb => by rw [if_neg (by simpa using hb)]) have h1 : (∑ K ∈ Finset.range (n + 1), if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) = P := by rw [hPdef] exact Finset.sum_congr rfl (fun K hK => by rw [if_pos (Finset.mem_range.mp hK)]) rw [h1] at h0 exact h0
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 410–422 at dfc7b5cc45a9 ↗
  h4_series.hP.h0 have HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) (∑ K ∈ Finset.range (n + (1 : ℕ)), if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ))
  Leaves HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) P, with h0 : HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) (∑ K ∈ Finset.range (n + (1 : ℕ)), if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)).
  Uses library HasSum, Finset.range, hasSum_sum_of_ne_finset_zero, if_neg.
  Lean, lines 412–416
  have h0 : HasSum (fun K => if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) (∑ K ∈ Finset.range (n + 1), if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) := hasSum_sum_of_ne_finset_zero (fun b hb => by rw [if_neg (by simpa using hb)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 412–416 at dfc7b5cc45a9 ↗
  h4_series.hP.h1 have (∑ K ∈ Finset.range (n + (1 : ℕ)), if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) = P
  Leaves HasSum (fun (K : ℕ) => if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) P, with h1 : (∑ K ∈ Finset.range (n + (1 : ℕ)), if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) = P.
  Uses library Finset.range, Finset.sum_congr, rfl, if_pos, Finset.mem_range.
  Lean, lines 417–420
  have h1 : (∑ K ∈ Finset.range (n + 1), if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) = P := by rw [hPdef] exact Finset.sum_congr rfl (fun K hK => by rw [if_pos (Finset.mem_range.mp hK)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 417–420 at dfc7b5cc45a9 ↗
11. h4_series.hD have HasSum (fun (b : ℕ) => (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < b then (↑(Qc n j) : ℝ) * a (b - j - (1 : ℕ)) * x ^ b else (0 : ℝ)) - if b < n + (1 : ℕ) then (∑ j ∈ Finset.range (b + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (b - j) - (1 : ℝ))) * x ^ b else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hD : HasSum (fun (b : ℕ) => (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < b then (↑(Qc n j) : ℝ) * a (b - j - (1 : ℕ)) * x ^ b else (0 : ℝ)) - if b < n + (1 : ℕ) then (∑ j ∈ Finset.range (b + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (b - j) - (1 : ℝ))) * x ^ b else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P).
  Lean, line 423
  have hD := hC.sub hP
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 423 at dfc7b5cc45a9 ↗
12. h4_series.hval have (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) = (∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)), with hval : (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) = (∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P.
  Uses library Finset.range.
  Lean, lines 424–426
  have hval : (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) = (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j) * A - P := by rw [h3T, hAqR, hQxR]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 424–426 at dfc7b5cc45a9 ↗
13. h4_series.hid1 have ∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ L)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P), with hid1 : ∀ j ≤ n, Qc n j * (2 : ℚ) ^ j * ∏ i ∈ (Finset.range (n + (1 : ℕ))).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ i) = ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℚ) ^ j - (2 : ℚ) ^ L).
  Uses library Finset.range, Finset.Ioc.
  Lean, lines 428–429
  have hid1 : ∀ j ≤ n, Qc n j * 2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) = ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ L) := fun j hj => by subst hQc; exact id1_full n j hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 428–429 at dfc7b5cc45a9 ↗
14. h4_series.hfun have (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) = fun (K : ℕ) => (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) - if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)
  Leaves HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K else (0 : ℝ)) ((∑ j ∈ Finset.range (n + (1 : ℕ)), (↑(Qc n j) : ℝ) * x ^ j) * A - P), with hfun : (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) = fun (K : ℕ) => (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) - if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ).
  Uses library Finset.Ioc, Finset.range.
  Lean, lines 430–470
  have hfun : (fun K : ℕ => if 2 * n < K then ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * ((3 : ℝ) / 2 ^ n) ^ K else 0) = fun K => (∑ j ∈ Finset.range (n + 1), if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) - (if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) := by funext K symm by_cases hK : K < n + 1 · rw [if_pos hK, if_neg (by omega)] rw [Finset.sum_range_succ (fun j => (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) K, Nat.sub_self, pow_zero, sub_self, div_zero, add_zero] have hsplit : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ∑ j ∈ Finset.range K, (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1) * x ^ K := by rw [← Finset.sum_range_add_sum_Ico _ (show K ≤ n + 1 by omega)] rw [Finset.sum_eq_zero (s := Finset.Ico K (n + 1)) (fun j hj => by rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK := Finset.mem_range.mp hj rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring rw [hsplit, Finset.sum_mul, sub_self] · rw [if_neg hK, sub_zero] have hS : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ((∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) : ℚ) : ℝ) * x ^ K := by push_cast rw [Finset.sum_mul] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK : j < K := by have := Finset.mem_range.mp hj; omega rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring rw [hS, tail_coeff n K (by omega) Qc hid1] push_cast by_cases h2 : 2 * n < K · rw [if_pos h2] · rw [if_neg h2] rw [Finset.prod_eq_zero (i := K) (Finset.mem_Ioc.mpr ⟨by omega, by omega⟩) (sub_self _)] simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 430–470 at dfc7b5cc45a9 ↗
  h4_series.hfun.s1 case
  Leaves ((∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) - if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) = if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ).
  Uses library if_pos, if_neg, Finset.sum_range_succ, Nat.sub_self, pow_zero, sub_self, div_zero, add_zero, Finset.sum_mul.
  Lean, lines 438–452
  · rw [if_pos hK, if_neg (by omega)] rw [Finset.sum_range_succ (fun j => (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) K, Nat.sub_self, pow_zero, sub_self, div_zero, add_zero] have hsplit : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ∑ j ∈ Finset.range K, (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1) * x ^ K := by rw [← Finset.sum_range_add_sum_Ico _ (show K ≤ n + 1 by omega)] rw [Finset.sum_eq_zero (s := Finset.Ico K (n + 1)) (fun j hj => by rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK := Finset.mem_range.mp hj rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring rw [hsplit, Finset.sum_mul, sub_self]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 438–452 at dfc7b5cc45a9 ↗
  h4_series.hfun.s1.hsplit have (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = ∑ j ∈ Finset.range K, (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ)) * x ^ K
  Leaves (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) - (∑ x ∈ Finset.range K, (↑(Qc n x) : ℝ) / ((2 : ℝ) ^ (K - x) - (1 : ℝ))) * x ^ K = (0 : ℝ), with hsplit : (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = ∑ j ∈ Finset.range K, (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ)) * x ^ K.
  Uses library Finset.range, Finset.sum_range_add_sum_Ico, Finset.sum_eq_zero, Finset.Ico, if_neg, add_zero, Finset.sum_congr, rfl, if_pos.
  Lean, lines 441–451
  have hsplit : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ∑ j ∈ Finset.range K, (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1) * x ^ K := by rw [← Finset.sum_range_add_sum_Ico _ (show K ≤ n + 1 by omega)] rw [Finset.sum_eq_zero (s := Finset.Ico K (n + 1)) (fun j hj => by rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK := Finset.mem_range.mp hj rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 441–451 at dfc7b5cc45a9 ↗
  h4_series.hfun.s1.hsplit.this have K ≤ j ∧ j < n + (1 : ℕ)
  Leaves ¬j < K, with this : K ≤ j ∧ j < n + (1 : ℕ).
  Uses library Finset.mem_Ico.
  Lean, line 445
  rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 445 at dfc7b5cc45a9 ↗
  h4_series.hfun.s1.hsplit.hjK have j < K
  Leaves (if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ)) * x ^ K, with hjK : j < K.
  Uses library Finset.mem_range.
  Lean, line 447
  have hjK := Finset.mem_range.mp hj
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 447 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2 case
  Leaves ((∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) - if K < n + (1 : ℕ) then (∑ j ∈ Finset.range (K + (1 : ℕ)), (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ))) * x ^ K else (0 : ℝ)) = if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ).
  Uses library if_neg, sub_zero.
  Lean, lines 453–470
  · rw [if_neg hK, sub_zero] have hS : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ((∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) : ℚ) : ℝ) * x ^ K := by push_cast rw [Finset.sum_mul] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK : j < K := by have := Finset.mem_range.mp hj; omega rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring rw [hS, tail_coeff n K (by omega) Qc hid1] push_cast by_cases h2 : 2 * n < K · rw [if_pos h2] · rw [if_neg h2] rw [Finset.prod_eq_zero (i := K) (Finset.mem_Ioc.mpr ⟨by omega, by omega⟩) (sub_self _)] simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 453–470 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2.hS have (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = (↑(∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ))) : ℝ) * x ^ K
  Leaves (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ), with hS : (∑ j ∈ Finset.range (n + (1 : ℕ)), if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = (↑(∑ j ∈ Finset.range (n + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (K - j) - (1 : ℚ))) : ℝ) * x ^ K.
  Uses library Finset.range, Finset.sum_mul, Finset.sum_congr, rfl, if_pos.
  Lean, lines 454–463
  have hS : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) = ((∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) : ℚ) : ℝ) * x ^ K := by push_cast rw [Finset.sum_mul] refine Finset.sum_congr rfl (fun j hj => ?_) have hjK : j < K := by have := Finset.mem_range.mp hj; omega rw [if_pos hjK] simp only [hadef] rw [show K - j - 1 + 1 = K - j by omega] ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 454–463 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2.hS.hjK have j < K
  Leaves (if j < K then (↑(Qc n j) : ℝ) * a (K - j - (1 : ℕ)) * x ^ K else (0 : ℝ)) = (↑(Qc n j) : ℝ) / ((2 : ℝ) ^ (K - j) - (1 : ℝ)) * x ^ K, with hjK : j < K.
  Lean, line 459
  have hjK : j < K := by have := Finset.mem_range.mp hj; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 459 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2.hS.hjK.this have j < n + (1 : ℕ)
  Leaves j < K, with this : j < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 459
  have hjK : j < K := by have := Finset.mem_range.mp hj; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 459 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2.s2 case
  Leaves ((∏ x ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ x)) / ∏ x ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ x)) * x ^ K = if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ).
  Uses library if_pos.
  Lean, line 467
  · rw [if_pos h2]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 467 at dfc7b5cc45a9 ↗
  h4_series.hfun.s2.s3 case
  Leaves ((∏ x ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ x)) / ∏ x ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ x)) * x ^ K = if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ).
  Uses library if_neg, Finset.prod_eq_zero, Finset.mem_Ioc, And.intro, sub_self.
  Lean, lines 468–470
  · rw [if_neg h2] rw [Finset.prod_eq_zero (i := K) (Finset.mem_Ioc.mpr ⟨by omega, by omega⟩) (sub_self _)] simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 468–470 at dfc7b5cc45a9 ↗
prod_one_sub_ge have ∀ (s : Finset ℕ) (a : ℕ → ℝ), (∀ i ∈ s, (0 : ℝ) ≤ a i) → (∀ i ∈ s, a i ≤ (1 : ℝ)) → (1 : ℝ) - ∑ i ∈ s, a i ≤ ∏ i ∈ s, ((1 : ℝ) - a i)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with prod_one_sub_ge : ∀ (s : Finset ℕ) (a : ℕ → ℝ), (∀ i ∈ s, (0 : ℝ) ≤ a i) → (∀ i ∈ s, a i ≤ (1 : ℝ)) → (1 : ℝ) - ∑ i ∈ s, a i ≤ ∏ i ∈ s, ((1 : ℝ) - a i).
Uses library Finset, Finset.induction_on.
Lean, lines 475–490
```
  have prod_one_sub_ge : ∀ (s : Finset ℕ) (a : ℕ → ℝ) (h0 : ∀ i ∈ s, 0 ≤ a i) (h1 : ∀ i ∈ s, a i ≤ 1),
      1 - ∑ i ∈ s, a i ≤ ∏ i ∈ s, (1 - a i) := by
    intro s a h0 h1
    induction s using Finset.induction_on with
    | empty => simp
    | insert j s hj ih =>
      rw [Finset.sum_insert hj, Finset.prod_insert hj]
      have ih' := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi)) (fun i hi => h1 i (Finset.mem_insert_of_mem hi))
      have haj0 := h0 j (Finset.mem_insert_self j s)
      have haj1 := h1 j (Finset.mem_insert_self j s)
      have hP : 0 ≤ ∏ i ∈ s, (1 - a i) :=
        Finset.prod_nonneg (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)])
      have hP1 : ∏ i ∈ s, (1 - a i) ≤ 1 :=
        Finset.prod_le_one (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)])
          (fun i hi => by linarith [h0 i (Finset.mem_insert_of_mem hi)])
      nlinarith
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 475–490 at dfc7b5cc45a9 ↗
1. prod_one_sub_ge.s1 case printed form did not read back: shown as its Lean lines only
  | empty => simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 479 at dfc7b5cc45a9 ↗
2. prod_one_sub_ge.s2 case
  Leaves (1 : ℝ) - ∑ i ∈ insert j s, a i ≤ ∏ i ∈ insert j s, ((1 : ℝ) - a i), with j : ℕ, s : Finset ℕ, hj : j ∉ s, ih : (∀ i ∈ s, (0 : ℝ) ≤ a i) → (∀ i ∈ s, a i ≤ (1 : ℝ)) → (1 : ℝ) - ∑ i ∈ s, a i ≤ ∏ i ∈ s, ((1 : ℝ) - a i), h0 : ∀ i ∈ insert j s, (0 : ℝ) ≤ a i, h1 : ∀ i ∈ insert j s, a i ≤ (1 : ℝ).
  Uses library Finset.sum_insert, Finset.prod_insert.
  Lean, lines 480–490
  | insert j s hj ih => rw [Finset.sum_insert hj, Finset.prod_insert hj] have ih' := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi)) (fun i hi => h1 i (Finset.mem_insert_of_mem hi)) have haj0 := h0 j (Finset.mem_insert_self j s) have haj1 := h1 j (Finset.mem_insert_self j s) have hP : 0 ≤ ∏ i ∈ s, (1 - a i) := Finset.prod_nonneg (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)]) have hP1 : ∏ i ∈ s, (1 - a i) ≤ 1 := Finset.prod_le_one (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)]) (fun i hi => by linarith [h0 i (Finset.mem_insert_of_mem hi)]) nlinarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 480–490 at dfc7b5cc45a9 ↗
  prod_one_sub_ge.s2.ih' have (1 : ℝ) - ∑ i ∈ s, a i ≤ ∏ i ∈ s, ((1 : ℝ) - a i)
  Leaves (1 : ℝ) - (a j + ∑ x ∈ s, a x) ≤ ((1 : ℝ) - a j) * ∏ x ∈ s, ((1 : ℝ) - a x), with ih' : (1 : ℝ) - ∑ i ∈ s, a i ≤ ∏ i ∈ s, ((1 : ℝ) - a i).
  Uses library Finset.mem_insert_of_mem.
  Lean, line 482
  have ih' := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi)) (fun i hi => h1 i (Finset.mem_insert_of_mem hi))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 482 at dfc7b5cc45a9 ↗
  prod_one_sub_ge.s2.haj0 have (0 : ℝ) ≤ a j
  Leaves (1 : ℝ) - (a j + ∑ x ∈ s, a x) ≤ ((1 : ℝ) - a j) * ∏ x ∈ s, ((1 : ℝ) - a x), with haj0 : (0 : ℝ) ≤ a j.
  Uses library Finset.mem_insert_self.
  Lean, line 483
  have haj0 := h0 j (Finset.mem_insert_self j s)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 483 at dfc7b5cc45a9 ↗
  prod_one_sub_ge.s2.haj1 have a j ≤ (1 : ℝ)
  Leaves (1 : ℝ) - (a j + ∑ x ∈ s, a x) ≤ ((1 : ℝ) - a j) * ∏ x ∈ s, ((1 : ℝ) - a x), with haj1 : a j ≤ (1 : ℝ).
  Uses library Finset.mem_insert_self.
  Lean, line 484
  have haj1 := h1 j (Finset.mem_insert_self j s)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 484 at dfc7b5cc45a9 ↗
  prod_one_sub_ge.s2.hP have (0 : ℝ) ≤ ∏ i ∈ s, ((1 : ℝ) - a i)
  Leaves (1 : ℝ) - (a j + ∑ x ∈ s, a x) ≤ ((1 : ℝ) - a j) * ∏ x ∈ s, ((1 : ℝ) - a x), with hP : (0 : ℝ) ≤ ∏ i ∈ s, ((1 : ℝ) - a i).
  Uses library Finset.prod_nonneg, Finset.mem_insert_of_mem.
  Lean, lines 485–486
  have hP : 0 ≤ ∏ i ∈ s, (1 - a i) := Finset.prod_nonneg (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 485–486 at dfc7b5cc45a9 ↗
  prod_one_sub_ge.s2.hP1 have ∏ i ∈ s, ((1 : ℝ) - a i) ≤ (1 : ℝ)
  Leaves (1 : ℝ) - (a j + ∑ x ∈ s, a x) ≤ ((1 : ℝ) - a j) * ∏ x ∈ s, ((1 : ℝ) - a x), with hP1 : ∏ i ∈ s, ((1 : ℝ) - a i) ≤ (1 : ℝ).
  Uses library Finset.prod_le_one, Finset.mem_insert_of_mem.
  Lean, lines 487–489
  have hP1 : ∏ i ∈ s, (1 - a i) ≤ 1 := Finset.prod_le_one (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)]) (fun i hi => by linarith [h0 i (Finset.mem_insert_of_mem hi)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 487–489 at dfc7b5cc45a9 ↗
nd_bound have ∀ (n K : ℕ), (1 : ℕ) ≤ n → (2 : ℕ) * n + (1 : ℕ) ≤ K → (0 : ℝ) < (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ∧ (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with nd_bound : ∀ (n K : ℕ), (1 : ℕ) ≤ n → (2 : ℕ) * n + (1 : ℕ) ≤ K → (0 : ℝ) < (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ∧ (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K.
Uses library Finset.Ioc, Finset.range, And.intro, div_pos, div_le_iff₀.
Lean, lines 492–547
```
  have nd_bound : ∀ (n K : ℕ) (hn : 1 ≤ n) (hK : 2 * n + 1 ≤ K),
      0 < (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) ∧
      (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)
        ≤ 2 * (1 / 2) ^ K := by
    intro n K hn hK
    have hlt : ∀ i, i < K → (0 : ℝ) < 2 ^ K - 2 ^ i := fun i hi => by
      have : (2 : ℝ) ^ i < 2 ^ K := pow_lt_pow_right₀ (by norm_num) hi
      linarith
    have hNpos : 0 < ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) :=
      Finset.prod_pos (fun L hL => hlt L (by have := Finset.mem_Ioc.mp hL; omega))
    have hDpos : 0 < ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) :=
      Finset.prod_pos (fun i hi => hlt i (by have := Finset.mem_range.mp hi; omega))
    refine ⟨div_pos hNpos hDpos, ?_⟩
    have hN : ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := by
      have := Finset.prod_le_prod (s := Finset.Ioc n (2 * n)) (f := fun L => (2 : ℝ) ^ K - 2 ^ L) (g := fun _ => (2 : ℝ) ^ K)
        (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by
          have : (0 : ℝ) < 2 ^ L := by positivity
          linarith)
      rw [Finset.prod_const, Nat.card_Ioc, show 2 * n - n = n by omega] at this
      exact this
    have hD : (2 ^ K) ^ (n + 1) / 2 ≤ ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by
      have hfac : ∀ i ∈ Finset.range (n + 1), (2 : ℝ) ^ K - 2 ^ i = 2 ^ K * (1 - (1 / 2) ^ (K - i)) := by
        intro i hi
        have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega
        have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega
        rw [mul_sub, mul_one, one_div_pow, mul_one_div, e]
        field_simp
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range]
      have hsum : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) ≤ 1 / 2 := by
        have e : ∀ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) = (1 / 2) ^ (K - n) * (1 / 2) ^ (n - i) := by
          intro i hi
          have := Finset.mem_range.mp hi
          rw [← pow_add]; congr 1; omega
        rw [Finset.sum_congr rfl e, ← Finset.mul_sum]
        have hg : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (n - i) ≤ 2 := by
          have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1)
          simp only [Nat.add_sub_cancel] at hr
          rw [hr]
          exact sum_geometric_two_le (n + 1)
        have hk : ((1 : ℝ) / 2) ^ (K - n) ≤ (1 / 2) ^ 2 :=
          pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega)
        have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity
        nlinarith
      have hprod := prod_one_sub_ge (Finset.range (n + 1)) (fun i => ((1 : ℝ) / 2) ^ (K - i))
        (fun i _ => by positivity) (fun i _ => pow_le_one₀ (by norm_num) (by norm_num))
      have h2K : (0 : ℝ) < (2 ^ K) ^ (n + 1) := by positivity
      nlinarith
    rw [div_le_iff₀ hDpos]
    have h2K : (2 : ℝ) ^ K * (1 / 2) ^ K = 1 := by rw [← mul_pow]; norm_num
    have e2 : 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ (n + 1) / 2) = (2 ^ K) ^ n := by
      rw [pow_succ]
      calc 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ n * 2 ^ K / 2) = (2 ^ K * (1 / 2) ^ K) * (2 ^ K) ^ n := by ring
        _ = (2 ^ K) ^ n := by rw [h2K, one_mul]
    calc ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := hN
      _ = 2 * (1 / 2) ^ K * ((2 ^ K) ^ (n + 1) / 2) := e2.symm
      _ ≤ 2 * (1 / 2) ^ K * ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by gcongr
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 492–547 at dfc7b5cc45a9 ↗
1. nd_bound.hlt have ∀ i < K, (0 : ℝ) < (2 : ℝ) ^ K - (2 : ℝ) ^ i
  Leaves (0 : ℝ) < (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ∧ (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K, with hlt : ∀ i < K, (0 : ℝ) < (2 : ℝ) ^ K - (2 : ℝ) ^ i.
  Lean, lines 497–499
  have hlt : ∀ i, i < K → (0 : ℝ) < 2 ^ K - 2 ^ i := fun i hi => by have : (2 : ℝ) ^ i < 2 ^ K := pow_lt_pow_right₀ (by norm_num) hi linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 497–499 at dfc7b5cc45a9 ↗
  nd_bound.hlt.this have (2 : ℝ) ^ i < (2 : ℝ) ^ K
  Leaves (0 : ℝ) < (2 : ℝ) ^ K - (2 : ℝ) ^ i, with this : (2 : ℝ) ^ i < (2 : ℝ) ^ K.
  Uses library pow_lt_pow_right₀.
  Lean, line 498
  have : (2 : ℝ) ^ i < 2 ^ K := pow_lt_pow_right₀ (by norm_num) hi
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 498 at dfc7b5cc45a9 ↗
2. nd_bound.hNpos have (0 : ℝ) < ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)
  Leaves (0 : ℝ) < (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ∧ (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K, with hNpos : (0 : ℝ) < ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L).
  Uses library Finset.Ioc, Finset.prod_pos.
  Lean, lines 500–501
  have hNpos : 0 < ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) := Finset.prod_pos (fun L hL => hlt L (by have := Finset.mem_Ioc.mp hL; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 500–501 at dfc7b5cc45a9 ↗
  nd_bound.hNpos.this have n < L ∧ L ≤ (2 : ℕ) * n
  Leaves L < K, with this : n < L ∧ L ≤ (2 : ℕ) * n.
  Uses library Finset.mem_Ioc.
  Lean, line 501
  Finset.prod_pos (fun L hL => hlt L (by have := Finset.mem_Ioc.mp hL; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 501 at dfc7b5cc45a9 ↗
3. nd_bound.hDpos have (0 : ℝ) < ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)
  Leaves (0 : ℝ) < (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ∧ (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K, with hDpos : (0 : ℝ) < ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i).
  Uses library Finset.range, Finset.prod_pos.
  Lean, lines 502–503
  have hDpos : 0 < ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := Finset.prod_pos (fun i hi => hlt i (by have := Finset.mem_range.mp hi; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 502–503 at dfc7b5cc45a9 ↗
  nd_bound.hDpos.this have i < n + (1 : ℕ)
  Leaves i < K, with this : i < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 503
  Finset.prod_pos (fun i hi => hlt i (by have := Finset.mem_range.mp hi; omega))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 503 at dfc7b5cc45a9 ↗
4. nd_bound.hN have ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ ((2 : ℝ) ^ K) ^ n
  Leaves (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K, with hN : ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ ((2 : ℝ) ^ K) ^ n.
  Uses library Finset.Ioc, Finset.prod_const, Nat.card_Ioc.
  Lean, lines 505–511
  have hN : ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := by have := Finset.prod_le_prod (s := Finset.Ioc n (2 * n)) (f := fun L => (2 : ℝ) ^ K - 2 ^ L) (g := fun _ => (2 : ℝ) ^ K) (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by have : (0 : ℝ) < 2 ^ L := by positivity linarith) rw [Finset.prod_const, Nat.card_Ioc, show 2 * n - n = n by omega] at this exact this
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 505–511 at dfc7b5cc45a9 ↗
  nd_bound.hN.this have ∏ i ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ ∏ i ∈ Finset.Ioc n ((2 : ℕ) * n), (2 : ℝ) ^ K
  Leaves ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ ((2 : ℝ) ^ K) ^ n, with this : ∏ i ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ ∏ i ∈ Finset.Ioc n ((2 : ℕ) * n), (2 : ℝ) ^ K.
  Uses library Finset.prod_le_prod, Finset.Ioc, le_of_lt.
  Lean, lines 506–509
  have := Finset.prod_le_prod (s := Finset.Ioc n (2 * n)) (f := fun L => (2 : ℝ) ^ K - 2 ^ L) (g := fun _ => (2 : ℝ) ^ K) (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by have : (0 : ℝ) < 2 ^ L := by positivity linarith)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 506–509 at dfc7b5cc45a9 ↗
  nd_bound.hN.this.s1 have n < L ∧ L ≤ (2 : ℕ) * n
  Leaves L < K, with this : n < L ∧ L ≤ (2 : ℕ) * n.
  Uses library Finset.mem_Ioc.
  Lean, line 507
  (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 507 at dfc7b5cc45a9 ↗
  nd_bound.hN.this.s2 have (0 : ℝ) < (2 : ℝ) ^ L routine: positivity
  Leaves (2 : ℝ) ^ K - (2 : ℝ) ^ L ≤ (2 : ℝ) ^ K, with this : (0 : ℝ) < (2 : ℝ) ^ L.
  Lean, line 508
  have : (0 : ℝ) < 2 ^ L := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 508 at dfc7b5cc45a9 ↗
5. nd_bound.hD have ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)
  Leaves (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K, with hD : ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i).
  Uses library Finset.range, Finset.prod_congr, rfl, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range.
  Lean, lines 512–538
  have hD : (2 ^ K) ^ (n + 1) / 2 ≤ ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by have hfac : ∀ i ∈ Finset.range (n + 1), (2 : ℝ) ^ K - 2 ^ i = 2 ^ K * (1 - (1 / 2) ^ (K - i)) := by intro i hi have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega rw [mul_sub, mul_one, one_div_pow, mul_one_div, e] field_simp rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range] have hsum : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) ≤ 1 / 2 := by have e : ∀ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) = (1 / 2) ^ (K - n) * (1 / 2) ^ (n - i) := by intro i hi have := Finset.mem_range.mp hi rw [← pow_add]; congr 1; omega rw [Finset.sum_congr rfl e, ← Finset.mul_sum] have hg : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (n - i) ≤ 2 := by have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1) simp only [Nat.add_sub_cancel] at hr rw [hr] exact sum_geometric_two_le (n + 1) have hk : ((1 : ℝ) / 2) ^ (K - n) ≤ (1 / 2) ^ 2 := pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega) have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity nlinarith have hprod := prod_one_sub_ge (Finset.range (n + 1)) (fun i => ((1 : ℝ) / 2) ^ (K - i)) (fun i _ => by positivity) (fun i _ => pow_le_one₀ (by norm_num) (by norm_num)) have h2K : (0 : ℝ) < (2 ^ K) ^ (n + 1) := by positivity nlinarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 512–538 at dfc7b5cc45a9 ↗
  nd_bound.hD.hfac have ∀ i ∈ Finset.range (n + (1 : ℕ)), (2 : ℝ) ^ K - (2 : ℝ) ^ i = (2 : ℝ) ^ K * ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i))
  Leaves ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i), with hfac : ∀ i ∈ Finset.range (n + (1 : ℕ)), (2 : ℝ) ^ K - (2 : ℝ) ^ i = (2 : ℝ) ^ K * ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i)).
  Uses library Finset.range, mul_sub, mul_one, one_div_pow, mul_one_div.
  Lean, lines 513–518
  have hfac : ∀ i ∈ Finset.range (n + 1), (2 : ℝ) ^ K - 2 ^ i = 2 ^ K * (1 - (1 / 2) ^ (K - i)) := by intro i hi have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega rw [mul_sub, mul_one, one_div_pow, mul_one_div, e] field_simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 513–518 at dfc7b5cc45a9 ↗
  nd_bound.hD.hfac.hiK have i ≤ K
  Leaves (2 : ℝ) ^ K - (2 : ℝ) ^ i = (2 : ℝ) ^ K * ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i)), with hiK : i ≤ K.
  Lean, line 515
  have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 515 at dfc7b5cc45a9 ↗
  nd_bound.hD.hfac.hiK.this have i < n + (1 : ℕ)
  Leaves i ≤ K, with this : i < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 515
  have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 515 at dfc7b5cc45a9 ↗
  nd_bound.hD.hfac.e have (2 : ℝ) ^ K = (2 : ℝ) ^ i * (2 : ℝ) ^ (K - i)
  Leaves (2 : ℝ) ^ K - (2 : ℝ) ^ i = (2 : ℝ) ^ K * ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i)), with e : (2 : ℝ) ^ K = (2 : ℝ) ^ i * (2 : ℝ) ^ (K - i).
  Uses library pow_add.
  Lean, line 516
  have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 516 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum have ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) ≤ (1 / 2 : ℝ)
  Leaves ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) * ∏ x ∈ Finset.range (n + (1 : ℕ)), ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - x)), with hsum : ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) ≤ (1 / 2 : ℝ).
  Uses library Finset.range, Finset.sum_congr, rfl, Finset.mul_sum.
  Lean, lines 520–534
  have hsum : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) ≤ 1 / 2 := by have e : ∀ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) = (1 / 2) ^ (K - n) * (1 / 2) ^ (n - i) := by intro i hi have := Finset.mem_range.mp hi rw [← pow_add]; congr 1; omega rw [Finset.sum_congr rfl e, ← Finset.mul_sum] have hg : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (n - i) ≤ 2 := by have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1) simp only [Nat.add_sub_cancel] at hr rw [hr] exact sum_geometric_two_le (n + 1) have hk : ((1 : ℝ) / 2) ^ (K - n) ≤ (1 / 2) ^ 2 := pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega) have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity nlinarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 520–534 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.e have ∀ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) = (1 / 2 : ℝ) ^ (K - n) * (1 / 2 : ℝ) ^ (n - i)
  Leaves ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) ≤ (1 / 2 : ℝ), with e : ∀ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) = (1 / 2 : ℝ) ^ (K - n) * (1 / 2 : ℝ) ^ (n - i).
  Uses library Finset.range, pow_add.
  Lean, lines 521–524
  have e : ∀ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) = (1 / 2) ^ (K - n) * (1 / 2) ^ (n - i) := by intro i hi have := Finset.mem_range.mp hi rw [← pow_add]; congr 1; omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 521–524 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.e.this have i < n + (1 : ℕ)
  Leaves (1 / 2 : ℝ) ^ (K - i) = (1 / 2 : ℝ) ^ (K - n) * (1 / 2 : ℝ) ^ (n - i), with this : i < n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 523
  have := Finset.mem_range.mp hi
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 523 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.hg have ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (2 : ℝ)
  Leaves (1 / 2 : ℝ) ^ (K - n) * ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (1 / 2 : ℝ), with hg : ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (2 : ℝ).
  Uses library Finset.range, Nat.add_sub_cancel, sum_geometric_two_le.
  Lean, lines 526–530
  have hg : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (n - i) ≤ 2 := by have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1) simp only [Nat.add_sub_cancel] at hr rw [hr] exact sum_geometric_two_le (n + 1)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 526–530 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.hg.hr have ∑ j ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n + (1 : ℕ) - (1 : ℕ) - j) = ∑ j ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ j
  Leaves ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (2 : ℝ), with hr : ∑ j ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n + (1 : ℕ) - (1 : ℕ) - j) = ∑ j ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ j.
  Uses library Finset.sum_range_reflect.
  Lean, line 527
  have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 527 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.hk have (1 / 2 : ℝ) ^ (K - n) ≤ (1 / 2 : ℝ) ^ (2 : ℕ)
  Leaves (1 / 2 : ℝ) ^ (K - n) * ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (1 / 2 : ℝ), with hk : (1 / 2 : ℝ) ^ (K - n) ≤ (1 / 2 : ℝ) ^ (2 : ℕ).
  Uses library pow_le_pow_of_le_one.
  Lean, lines 531–532
  have hk : ((1 : ℝ) / 2) ^ (K - n) ≤ (1 / 2) ^ 2 := pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 531–532 at dfc7b5cc45a9 ↗
  nd_bound.hD.hsum.hpos have (0 : ℝ) ≤ (1 / 2 : ℝ) ^ (K - n) routine: positivity
  Leaves (1 / 2 : ℝ) ^ (K - n) * ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (n - i) ≤ (1 / 2 : ℝ), with hpos : (0 : ℝ) ≤ (1 / 2 : ℝ) ^ (K - n).
  Lean, line 533
  have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 533 at dfc7b5cc45a9 ↗
  nd_bound.hD.hprod have (1 : ℝ) - ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) ≤ ∏ i ∈ Finset.range (n + (1 : ℕ)), ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i))
  Leaves ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) * ∏ x ∈ Finset.range (n + (1 : ℕ)), ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - x)), with hprod : (1 : ℝ) - ∑ i ∈ Finset.range (n + (1 : ℕ)), (1 / 2 : ℝ) ^ (K - i) ≤ ∏ i ∈ Finset.range (n + (1 : ℕ)), ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - i)).
  Uses library Finset.range, pow_le_one₀.
  Lean, lines 535–536
  have hprod := prod_one_sub_ge (Finset.range (n + 1)) (fun i => ((1 : ℝ) / 2) ^ (K - i)) (fun i _ => by positivity) (fun i _ => pow_le_one₀ (by norm_num) (by norm_num))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 535–536 at dfc7b5cc45a9 ↗
  nd_bound.hD.h2K have (0 : ℝ) < ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) routine: positivity
  Leaves ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ) ≤ ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) * ∏ x ∈ Finset.range (n + (1 : ℕ)), ((1 : ℝ) - (1 / 2 : ℝ) ^ (K - x)), with h2K : (0 : ℝ) < ((2 : ℝ) ^ K) ^ (n + (1 : ℕ)).
  Lean, line 537
  have h2K : (0 : ℝ) < (2 ^ K) ^ (n + 1) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 537 at dfc7b5cc45a9 ↗
6. nd_bound.h2K have (2 : ℝ) ^ K * (1 / 2 : ℝ) ^ K = (1 : ℝ)
  Leaves ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K * ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i), with h2K : (2 : ℝ) ^ K * (1 / 2 : ℝ) ^ K = (1 : ℝ).
  Uses library mul_pow.
  Lean, line 540
  have h2K : (2 : ℝ) ^ K * (1 / 2) ^ K = 1 := by rw [← mul_pow]; norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 540 at dfc7b5cc45a9 ↗
7. nd_bound.e2 have (2 : ℝ) * (1 / 2 : ℝ) ^ K * (((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ)) = ((2 : ℝ) ^ K) ^ n
  Leaves ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K * ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i), with e2 : (2 : ℝ) * (1 / 2 : ℝ) ^ K * (((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ)) = ((2 : ℝ) ^ K) ^ n.
  Uses library pow_succ.
  Lean, lines 541–544
  have e2 : 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ (n + 1) / 2) = (2 ^ K) ^ n := by rw [pow_succ] calc 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ n * 2 ^ K / 2) = (2 ^ K * (1 / 2) ^ K) * (2 ^ K) ^ n := by ring _ = (2 ^ K) ^ n := by rw [h2K, one_mul]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 541–544 at dfc7b5cc45a9 ↗
  nd_bound.e2.s1 calc (2 : ℝ) * (1 / 2 : ℝ) ^ K * (((2 : ℝ) ^ K) ^ n * (2 : ℝ) ^ K / (2 : ℝ)) = (2 : ℝ) ^ K * (1 / 2 : ℝ) ^ K * ((2 : ℝ) ^ K) ^ n routine: ring
  Closes the goal.
  Lean, line 543
  calc 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ n * 2 ^ K / 2) = (2 ^ K * (1 / 2) ^ K) * (2 ^ K) ^ n := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 543 at dfc7b5cc45a9 ↗
  nd_bound.e2.s2 calc
  Closes the goal.
  Uses library one_mul.
  Lean, line 544
  _ = (2 ^ K) ^ n := by rw [h2K, one_mul]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 544 at dfc7b5cc45a9 ↗
8. nd_bound.s8 calc ∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L) ≤ ((2 : ℝ) ^ K) ^ n
  Closes the goal.
  Uses library Finset.Ioc.
  Lean, line 545
  calc ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := hN
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 545 at dfc7b5cc45a9 ↗
9. nd_bound.s9 calc ((2 : ℝ) ^ K) ^ n = (2 : ℝ) * (1 / 2 : ℝ) ^ K * (((2 : ℝ) ^ K) ^ (n + (1 : ℕ)) / (2 : ℝ))
  Closes the goal.
  Lean, line 546
  _ = 2 * (1 / 2) ^ K * ((2 ^ K) ^ (n + 1) / 2) := e2.symm
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 546 at dfc7b5cc45a9 ↗
10. nd_bound.s10 calc
  Closes the goal.
  Uses library Finset.range.
  Lean, line 547
  _ ≤ 2 * (1 / 2) ^ K * ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 547 at dfc7b5cc45a9 ↗
ineq_pow have ∀ (n : ℕ), (2 : ℕ) ≤ n → (9 : ℕ) * (81 : ℕ) ^ n ≤ (256 : ℕ) ^ n
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with ineq_pow : ∀ (n : ℕ), (2 : ℕ) ≤ n → (9 : ℕ) * (81 : ℕ) ^ n ≤ (256 : ℕ) ^ n.
Lean, lines 550–560
```
  have ineq_pow : ∀ (n : ℕ) (hn : 2 ≤ n),
      9 * 81 ^ n ≤ 256 ^ n := by
    intro n hn
    induction n with
    | zero => omega
    | succ m ih =>
      rcases (show m = 1 ∨ 2 ≤ m by omega) with rfl | hm
      · norm_num
      · have := ih hm
        rw [pow_succ, pow_succ]
        omega
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 550–560 at dfc7b5cc45a9 ↗
1. ineq_pow.s1 case routine: omega
  Leaves (9 : ℕ) * (81 : ℕ) ^ (0 : ℕ) ≤ (256 : ℕ) ^ (0 : ℕ), with hn : (2 : ℕ) ≤ (0 : ℕ).
  Uses library Nat.zero.
  Lean, line 554
  | zero => omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 554 at dfc7b5cc45a9 ↗
2. ineq_pow.s2 case
  Leaves (9 : ℕ) * (81 : ℕ) ^ (m + (1 : ℕ)) ≤ (256 : ℕ) ^ (m + (1 : ℕ)), with m : ℕ, ih : (2 : ℕ) ≤ m → (9 : ℕ) * (81 : ℕ) ^ m ≤ (256 : ℕ) ^ m, hn : (2 : ℕ) ≤ m + (1 : ℕ).
  Uses library Nat.succ.
  Lean, lines 555–560
  | succ m ih => rcases (show m = 1 ∨ 2 ≤ m by omega) with rfl | hm · norm_num · have := ih hm rw [pow_succ, pow_succ] omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 555–560 at dfc7b5cc45a9 ↗
  ineq_pow.s2.s1 case routine: norm_num
  Leaves (9 : ℕ) * (81 : ℕ) ^ ((1 : ℕ) + (1 : ℕ)) ≤ (256 : ℕ) ^ ((1 : ℕ) + (1 : ℕ)).
  Lean, line 557
  · norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 557 at dfc7b5cc45a9 ↗
  ineq_pow.s2.s2 case
  Leaves (9 : ℕ) * (81 : ℕ) ^ (m + (1 : ℕ)) ≤ (256 : ℕ) ^ (m + (1 : ℕ)).
  Uses library pow_succ.
  Lean, lines 558–560
  · have := ih hm rw [pow_succ, pow_succ] omega
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 558–560 at dfc7b5cc45a9 ↗
  ineq_pow.s2.s2.this have (9 : ℕ) * (81 : ℕ) ^ m ≤ (256 : ℕ) ^ m
  Leaves (9 : ℕ) * (81 : ℕ) ^ (m + (1 : ℕ)) ≤ (256 : ℕ) ^ (m + (1 : ℕ)), with this : (9 : ℕ) * (81 : ℕ) ^ m ≤ (256 : ℕ) ^ m.
  Lean, line 558
  · have := ih hm
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 558 at dfc7b5cc45a9 ↗
h4_big have ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (2 : ℕ) ≤ n → (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ))
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with h4_big : ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (2 : ℕ) ≤ n → (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)).
Uses library Finset.range, Finset.Ioc, And.intro.
Lean, lines 562–655
```
  have h4_big : ∀ (Qc : ℕ → ℕ → ℚ) (hQc : (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)))) (Qx : ℕ → ℚ) (hQx : (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k))
    (Aq : ℕ → ℚ) (hAq : (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ)))) (n : ℕ) (hn : 2 ≤ n),
      (0 : ℝ) < (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) ∧
      ((Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ)) ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤
        (2 : ℝ) ^ (2 * n + 4) := by
    intro Qc hQc Qx hQx Aq hAq n hn
    have hS := h4_series Qc hQc Qx hQx Aq hAq n (by omega)
    set r := (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) with hr
    set x : ℝ := (3 : ℝ) / 2 ^ n with hxdef
    have hx0 : 0 < x := by positivity
    have hx2 : x / 2 ≤ 3 / 8 := by
      have : (4 : ℝ) ≤ 2 ^ n := by
        calc (4 : ℝ) = 2 ^ 2 := by norm_num
          _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
      rw [hxdef, div_div, div_le_div_iff₀ (by positivity) (by norm_num)]; linarith
    set G : ℕ → ℝ := fun K => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            x ^ K else 0 with hG
    have hGnn : ∀ K, 0 ≤ G K := by
      intro K; simp only [hG]
      split_ifs with h
      · exact mul_nonneg (le_of_lt (nd_bound n K (by omega) (by omega)).1) (by positivity)
      · exact le_refl 0
    have hGpos : 0 < G (2 * n + 1) := by
      simp only [hG]; rw [if_pos (by omega)]
      exact mul_pos (nd_bound n (2 * n + 1) (by omega) (by omega)).1 (by positivity)
    refine ⟨?_, ?_⟩
    · exact hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1) (fun K => hGnn K) hGpos hasSum_zero hS
    · set H : ℕ → ℝ := fun K => if 2 * n < K then 2 * (x / 2) ^ K else 0 with hH
      have hGH : ∀ K, G K ≤ H K := by
        intro K; simp only [hG, hH]
        split_ifs with h
        · have hb := (nd_bound n K (by omega) (by omega)).2
          calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) /
                ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by
                gcongr
            _ = 2 * (x / 2) ^ K := by ring
        · exact le_refl 0
      have hq : x / 2 < 1 := by linarith
      have hHsum : HasSum H (2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹) := by
        rw [← hasSum_nat_add_iff' (2 * n + 1)]
        have hz : ∑ i ∈ Finset.range (2 * n + 1), H i = 0 :=
          Finset.sum_eq_zero (fun i hi => by
            simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)])
        rw [hz, sub_zero]
        have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1))
        have hfun : (fun s => H (s + (2 * n + 1))) = fun s => 2 * (x / 2) ^ (2 * n + 1) * (x / 2) ^ s := by
          funext s; simp only [hH]; rw [if_pos (by omega), pow_add]; ring
        rw [hfun]
        exact hgeo
      have hrle : r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hasSum_le hGH hS hHsum
      have hr0 : 0 ≤ r := le_of_lt (hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1)
        (fun K => hGnn K) hGpos hasSum_zero hS)
      have hr8 : r ≤ 8 * (x / 2) ^ (2 * n + 1) := by
        have hinv : (1 - x / 2)⁻¹ ≤ 4 := by
          rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith
        have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity
        calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle
          _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr
          _ = 8 * (x / 2) ^ (2 * n + 1) := by ring
      have hsq : r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := pow_le_pow_left₀ hr0 hr8 2
      have hnat : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by
        have e1 : (2 : ℕ) ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) = 2 ^ (4 * n * n) * (64 * 256 ^ n) := by
          rw [← pow_mul, ← pow_add, show (256 : ℕ) = 2 ^ 8 by norm_num, ← pow_mul,
            show (64 : ℕ) = 2 ^ 6 by norm_num, ← pow_add, ← pow_add]
          congr 1; ring
        have e2 : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) = 2 ^ (4 * n * n) * (64 * (9 * 81 ^ n)) := by
          rw [show (81 : ℕ) = 3 ^ 4 by norm_num, ← pow_mul, pow_add]; ring
        rw [e1, e2]
        have := ineq_pow n hn
        gcongr
      have hreal : (64 : ℝ) * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by
        exact_mod_cast hnat
      have hx2' : x / 2 = 3 / 2 ^ (n + 1) := by rw [hxdef, pow_succ]; field_simp
      have hsq' : r ^ 2 ≤ 64 * (x / 2) ^ (4 * n + 2) := by
        calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq
          _ = 64 * (x / 2) ^ (4 * n + 2) := by ring
      have hkey : (x / 2) ^ (4 * n + 2) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 3 ^ (4 * n + 2) := by
        rw [← mul_pow, hx2', div_mul_cancel₀ _ (by positivity)]
      have hP : (0 : ℝ) < ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := by positivity
      have hmain : 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) := by
        refine le_of_mul_le_mul_right ?_ hP
        calc 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2)
            = 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) := by rw [← hkey]; ring
          _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal
      calc r ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤ 64 * (x / 2) ^ (4 * n + 2) * 2 ^ (4 * n * n) := by gcongr
        _ ≤ 2 ^ (2 * n + 4) := hmain
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 562–655 at dfc7b5cc45a9 ↗
1. h4_big.hS have HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ))
  Leaves (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hS : HasSum (fun (K : ℕ) => if (2 : ℕ) * n < K then ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ K else (0 : ℝ)) ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)).
  Lean, line 575
  have hS := h4_series Qc hQc Qx hQx Aq hAq n (by omega)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 575 at dfc7b5cc45a9 ↗
2. h4_big.hx0 have (0 : ℝ) < x routine: positivity
  Leaves (0 : ℝ) < r ∧ r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hx0 : (0 : ℝ) < x.
  Lean, line 578
  have hx0 : 0 < x := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 578 at dfc7b5cc45a9 ↗
3. h4_big.hx2 have x / (2 : ℝ) ≤ (3 / 8 : ℝ)
  Leaves (0 : ℝ) < r ∧ r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hx2 : x / (2 : ℝ) ≤ (3 / 8 : ℝ).
  Uses library div_div, div_le_div_iff₀.
  Lean, lines 579–583
  have hx2 : x / 2 ≤ 3 / 8 := by have : (4 : ℝ) ≤ 2 ^ n := by calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn rw [hxdef, div_div, div_le_div_iff₀ (by positivity) (by norm_num)]; linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 579–583 at dfc7b5cc45a9 ↗
  h4_big.hx2.this have (4 : ℝ) ≤ (2 : ℝ) ^ n
  Leaves x / (2 : ℝ) ≤ (3 / 8 : ℝ), with this : (4 : ℝ) ≤ (2 : ℝ) ^ n.
  Lean, lines 580–582
  have : (4 : ℝ) ≤ 2 ^ n := by calc (4 : ℝ) = 2 ^ 2 := by norm_num _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 580–582 at dfc7b5cc45a9 ↗
  h4_big.hx2.this.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num
  Closes the goal.
  Lean, line 581
  calc (4 : ℝ) = 2 ^ 2 := by norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 581 at dfc7b5cc45a9 ↗
  h4_big.hx2.this.s2 calc (2 : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ n
  Closes the goal.
  Uses library pow_le_pow_right₀.
  Lean, line 582
  _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 582 at dfc7b5cc45a9 ↗
4. h4_big.hGnn have ∀ (K : ℕ), (0 : ℝ) ≤ G K
  Leaves (0 : ℝ) < r ∧ r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hGnn : ∀ (K : ℕ), (0 : ℝ) ≤ G K.
  Lean, lines 587–591
  have hGnn : ∀ K, 0 ≤ G K := by intro K; simp only [hG] split_ifs with h · exact mul_nonneg (le_of_lt (nd_bound n K (by omega) (by omega)).1) (by positivity) · exact le_refl 0
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 587–591 at dfc7b5cc45a9 ↗
  h4_big.hGnn.s1 case
  Leaves (0 : ℝ) ≤ ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ L ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) * x ^ K.
  Uses library mul_nonneg, le_of_lt.
  Lean, line 590
  · exact mul_nonneg (le_of_lt (nd_bound n K (by omega) (by omega)).1) (by positivity)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 590 at dfc7b5cc45a9 ↗
  h4_big.hGnn.s2 case
  Leaves (0 : ℝ) ≤ (0 : ℝ).
  Uses library le_refl.
  Lean, line 591
  · exact le_refl 0
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 591 at dfc7b5cc45a9 ↗
5. h4_big.hGpos have (0 : ℝ) < G ((2 : ℕ) * n + (1 : ℕ))
  Leaves (0 : ℝ) < r ∧ r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hGpos : (0 : ℝ) < G ((2 : ℕ) * n + (1 : ℕ)).
  Uses library if_pos, mul_pos.
  Lean, lines 592–594
  have hGpos : 0 < G (2 * n + 1) := by simp only [hG]; rw [if_pos (by omega)] exact mul_pos (nd_bound n (2 * n + 1) (by omega) (by omega)).1 (by positivity)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 592–594 at dfc7b5cc45a9 ↗
6. h4_big.s6 case
  Leaves (0 : ℝ) < r.
  Uses library hasSum_lt, hasSum_zero.
  Lean, line 596
  · exact hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1) (fun K => hGnn K) hGpos hasSum_zero hS
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 596 at dfc7b5cc45a9 ↗
7. h4_big.s7 case
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)).
  Lean, lines 597–655
  · set H : ℕ → ℝ := fun K => if 2 * n < K then 2 * (x / 2) ^ K else 0 with hH have hGH : ∀ K, G K ≤ H K := by intro K; simp only [hG, hH] split_ifs with h · have hb := (nd_bound n K (by omega) (by omega)).2 calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by gcongr _ = 2 * (x / 2) ^ K := by ring · exact le_refl 0 have hq : x / 2 < 1 := by linarith have hHsum : HasSum H (2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹) := by rw [← hasSum_nat_add_iff' (2 * n + 1)] have hz : ∑ i ∈ Finset.range (2 * n + 1), H i = 0 := Finset.sum_eq_zero (fun i hi => by simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)]) rw [hz, sub_zero] have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1)) have hfun : (fun s => H (s + (2 * n + 1))) = fun s => 2 * (x / 2) ^ (2 * n + 1) * (x / 2) ^ s := by funext s; simp only [hH]; rw [if_pos (by omega), pow_add]; ring rw [hfun] exact hgeo have hrle : r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hasSum_le hGH hS hHsum have hr0 : 0 ≤ r := le_of_lt (hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1) (fun K => hGnn K) hGpos hasSum_zero hS) have hr8 : r ≤ 8 * (x / 2) ^ (2 * n + 1) := by have hinv : (1 - x / 2)⁻¹ ≤ 4 := by rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr _ = 8 * (x / 2) ^ (2 * n + 1) := by ring have hsq : r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := pow_le_pow_left₀ hr0 hr8 2 have hnat : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by have e1 : (2 : ℕ) ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) = 2 ^ (4 * n * n) * (64 * 256 ^ n) := by rw [← pow_mul, ← pow_add, show (256 : ℕ) = 2 ^ 8 by norm_num, ← pow_mul, show (64 : ℕ) = 2 ^ 6 by norm_num, ← pow_add, ← pow_add] congr 1; ring have e2 : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) = 2 ^ (4 * n * n) * (64 * (9 * 81 ^ n)) := by rw [show (81 : ℕ) = 3 ^ 4 by norm_num, ← pow_mul, pow_add]; ring rw [e1, e2] have := ineq_pow n hn gcongr have hreal : (64 : ℝ) * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by exact_mod_cast hnat have hx2' : x / 2 = 3 / 2 ^ (n + 1) := by rw [hxdef, pow_succ]; field_simp have hsq' : r ^ 2 ≤ 64 * (x / 2) ^ (4 * n + 2) := by calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq _ = 64 * (x / 2) ^ (4 * n + 2) := by ring have hkey : (x / 2) ^ (4 * n + 2) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 3 ^ (4 * n + 2) := by rw [← mul_pow, hx2', div_mul_cancel₀ _ (by positivity)] have hP : (0 : ℝ) < ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := by positivity have hmain : 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) := by refine le_of_mul_le_mul_right ?_ hP calc 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) := by rw [← hkey]; ring _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal calc r ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤ 64 * (x / 2) ^ (4 * n + 2) * 2 ^ (4 * n * n) := by gcongr _ ≤ 2 ^ (2 * n + 4) := hmain
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 597–655 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH have ∀ (K : ℕ), G K ≤ H K
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hGH : ∀ (K : ℕ), G K ≤ H K.
  Lean, lines 598–606
  have hGH : ∀ K, G K ≤ H K := by intro K; simp only [hG, hH] split_ifs with h · have hb := (nd_bound n K (by omega) (by omega)).2 calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by gcongr _ = 2 * (x / 2) ^ K := by ring · exact le_refl 0
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 598–606 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH.s1 case
  Leaves ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ L ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) * x ^ K ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ K.
  Lean, lines 601–605
  · have hb := (nd_bound n K (by omega) (by omega)).2 calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by gcongr _ = 2 * (x / 2) ^ K := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 601–605 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH.s1.hb have (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K
  Leaves ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ L ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) * x ^ K ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ K, with hb : (∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i) ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K.
  Lean, line 601
  · have hb := (nd_bound n K (by omega) (by omega)).2
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 601 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH.s1.s2 calc ((∏ L ∈ Finset.Ioc n ((2 : ℕ) * n), ((2 : ℝ) ^ K - (2 : ℝ) ^ L)) / ∏ i ∈ Finset.range (n + (1 : ℕ)), ((2 : ℝ) ^ K - (2 : ℝ) ^ i)) * x ^ K ≤ (2 : ℝ) * (1 / 2 : ℝ) ^ K * x ^ K
  Closes the goal.
  Uses library Finset.Ioc, Finset.range.
  Lean, lines 602–604
  calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 602–604 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH.s1.s3 calc routine: ring
  Closes the goal.
  Lean, line 605
  _ = 2 * (x / 2) ^ K := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 605 at dfc7b5cc45a9 ↗
  h4_big.s7.hGH.s2 case
  Leaves (0 : ℝ) ≤ (0 : ℝ).
  Uses library le_refl.
  Lean, line 606
  · exact le_refl 0
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 606 at dfc7b5cc45a9 ↗
  h4_big.s7.hq have x / (2 : ℝ) < (1 : ℝ) routine: linarith
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hq : x / (2 : ℝ) < (1 : ℝ).
  Lean, line 607
  have hq : x / 2 < 1 := by linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 607 at dfc7b5cc45a9 ↗
  h4_big.s7.hHsum have HasSum H ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹)
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hHsum : HasSum H ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹).
  Uses library HasSum, hasSum_nat_add_iff', sub_zero.
  Lean, lines 608–618
  have hHsum : HasSum H (2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹) := by rw [← hasSum_nat_add_iff' (2 * n + 1)] have hz : ∑ i ∈ Finset.range (2 * n + 1), H i = 0 := Finset.sum_eq_zero (fun i hi => by simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)]) rw [hz, sub_zero] have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1)) have hfun : (fun s => H (s + (2 * n + 1))) = fun s => 2 * (x / 2) ^ (2 * n + 1) * (x / 2) ^ s := by funext s; simp only [hH]; rw [if_pos (by omega), pow_add]; ring rw [hfun] exact hgeo
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 608–618 at dfc7b5cc45a9 ↗
  h4_big.s7.hHsum.hz have ∑ i ∈ Finset.range ((2 : ℕ) * n + (1 : ℕ)), H i = (0 : ℝ)
  Leaves HasSum (fun (n_1 : ℕ) => H (n_1 + ((2 : ℕ) * n + (1 : ℕ)))) ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹ - ∑ i ∈ Finset.range ((2 : ℕ) * n + (1 : ℕ)), H i), with hz : ∑ i ∈ Finset.range ((2 : ℕ) * n + (1 : ℕ)), H i = (0 : ℝ).
  Uses library Finset.range, Finset.sum_eq_zero, if_neg.
  Lean, lines 610–612
  have hz : ∑ i ∈ Finset.range (2 * n + 1), H i = 0 := Finset.sum_eq_zero (fun i hi => by simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 610–612 at dfc7b5cc45a9 ↗
  h4_big.s7.hHsum.hz.this have i < (2 : ℕ) * n + (1 : ℕ)
  Leaves ¬(2 : ℕ) * n < i, with this : i < (2 : ℕ) * n + (1 : ℕ).
  Uses library Finset.mem_range.
  Lean, line 612
  simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)])
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 612 at dfc7b5cc45a9 ↗
  h4_big.s7.hHsum.hgeo have HasSum (fun (i : ℕ) => (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * (x / (2 : ℝ)) ^ i) ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹)
  Leaves HasSum (fun (n_1 : ℕ) => H (n_1 + ((2 : ℕ) * n + (1 : ℕ)))) ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹), with hgeo : HasSum (fun (i : ℕ) => (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * (x / (2 : ℝ)) ^ i) ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹).
  Uses library hasSum_geometric_of_lt_one.
  Lean, line 614
  have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 614 at dfc7b5cc45a9 ↗
  h4_big.s7.hHsum.hfun have (fun (s : ℕ) => H (s + ((2 : ℕ) * n + (1 : ℕ)))) = fun (s : ℕ) => (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * (x / (2 : ℝ)) ^ s
  Leaves HasSum (fun (n_1 : ℕ) => H (n_1 + ((2 : ℕ) * n + (1 : ℕ)))) ((2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹), with hfun : (fun (s : ℕ) => H (s + ((2 : ℕ) * n + (1 : ℕ)))) = fun (s : ℕ) => (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * (x / (2 : ℝ)) ^ s.
  Uses library if_pos, pow_add.
  Lean, lines 615–616
  have hfun : (fun s => H (s + (2 * n + 1))) = fun s => 2 * (x / 2) ^ (2 * n + 1) * (x / 2) ^ s := by funext s; simp only [hH]; rw [if_pos (by omega), pow_add]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 615–616 at dfc7b5cc45a9 ↗
  h4_big.s7.hrle have r ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hrle : r ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹.
  Uses library hasSum_le.
  Lean, line 619
  have hrle : r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hasSum_le hGH hS hHsum
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 619 at dfc7b5cc45a9 ↗
  h4_big.s7.hr0 have (0 : ℝ) ≤ r
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hr0 : (0 : ℝ) ≤ r.
  Uses library le_of_lt, hasSum_lt, hasSum_zero.
  Lean, lines 620–621
  have hr0 : 0 ≤ r := le_of_lt (hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1) (fun K => hGnn K) hGpos hasSum_zero hS)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 620–621 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8 have r ≤ (8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hr8 : r ≤ (8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)).
  Lean, lines 622–628
  have hr8 : r ≤ 8 * (x / 2) ^ (2 * n + 1) := by have hinv : (1 - x / 2)⁻¹ ≤ 4 := by rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr _ = 8 * (x / 2) ^ (2 * n + 1) := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 622–628 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8.hinv have ((1 : ℝ) - x / (2 : ℝ))⁻¹ ≤ (4 : ℝ)
  Leaves r ≤ (8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)), with hinv : ((1 : ℝ) - x / (2 : ℝ))⁻¹ ≤ (4 : ℝ).
  Uses library inv_le_comm₀.
  Lean, lines 623–624
  have hinv : (1 - x / 2)⁻¹ ≤ 4 := by rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 623–624 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8.hp have (0 : ℝ) ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) routine: positivity
  Leaves r ≤ (8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)), with hp : (0 : ℝ) ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)).
  Lean, line 625
  have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 625 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8.s3 calc r ≤ (2 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ)) * ((1 : ℝ) - x / (2 : ℝ))⁻¹
  Closes the goal.
  Lean, line 626
  calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 626 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8.s4 calc
  Closes the goal.
  Lean, line 627
  _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 627 at dfc7b5cc45a9 ↗
  h4_big.s7.hr8.s5 calc routine: ring
  Closes the goal.
  Lean, line 628
  _ = 8 * (x / 2) ^ (2 * n + 1) := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 628 at dfc7b5cc45a9 ↗
  h4_big.s7.hsq have r ^ (2 : ℕ) ≤ ((8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ))) ^ (2 : ℕ)
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hsq : r ^ (2 : ℕ) ≤ ((8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ))) ^ (2 : ℕ).
  Uses library pow_le_pow_left₀.
  Lean, line 629
  have hsq : r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := pow_le_pow_left₀ hr0 hr8 2
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 629 at dfc7b5cc45a9 ↗
  h4_big.s7.hnat have (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hnat : (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)).
  Lean, lines 630–639
  have hnat : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by have e1 : (2 : ℕ) ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) = 2 ^ (4 * n * n) * (64 * 256 ^ n) := by rw [← pow_mul, ← pow_add, show (256 : ℕ) = 2 ^ 8 by norm_num, ← pow_mul, show (64 : ℕ) = 2 ^ 6 by norm_num, ← pow_add, ← pow_add] congr 1; ring have e2 : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) = 2 ^ (4 * n * n) * (64 * (9 * 81 ^ n)) := by rw [show (81 : ℕ) = 3 ^ 4 by norm_num, ← pow_mul, pow_add]; ring rw [e1, e2] have := ineq_pow n hn gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 630–639 at dfc7b5cc45a9 ↗
  h4_big.s7.hnat.e1 have (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) = (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * (256 : ℕ) ^ n)
  Leaves (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)), with e1 : (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) = (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * (256 : ℕ) ^ n).
  Uses library pow_mul, pow_add.
  Lean, lines 631–634
  have e1 : (2 : ℕ) ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) = 2 ^ (4 * n * n) * (64 * 256 ^ n) := by rw [← pow_mul, ← pow_add, show (256 : ℕ) = 2 ^ 8 by norm_num, ← pow_mul, show (64 : ℕ) = 2 ^ 6 by norm_num, ← pow_add, ← pow_add] congr 1; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 631–634 at dfc7b5cc45a9 ↗
  h4_big.s7.hnat.e2 have (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) = (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * ((9 : ℕ) * (81 : ℕ) ^ n))
  Leaves (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℕ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℕ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)), with e2 : (64 : ℕ) * (3 : ℕ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℕ) ^ ((4 : ℕ) * n * n) = (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * ((9 : ℕ) * (81 : ℕ) ^ n)).
  Uses library pow_mul, pow_add.
  Lean, lines 635–636
  have e2 : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) = 2 ^ (4 * n * n) * (64 * (9 * 81 ^ n)) := by rw [show (81 : ℕ) = 3 ^ 4 by norm_num, ← pow_mul, pow_add]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 635–636 at dfc7b5cc45a9 ↗
  h4_big.s7.hnat.this have (9 : ℕ) * (81 : ℕ) ^ n ≤ (256 : ℕ) ^ n
  Leaves (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * ((9 : ℕ) * (81 : ℕ) ^ n)) ≤ (2 : ℕ) ^ ((4 : ℕ) * n * n) * ((64 : ℕ) * (256 : ℕ) ^ n), with this : (9 : ℕ) * (81 : ℕ) ^ n ≤ (256 : ℕ) ^ n.
  Lean, line 638
  have := ineq_pow n hn
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 638 at dfc7b5cc45a9 ↗
  h4_big.s7.hreal have (64 : ℝ) * (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hreal : (64 : ℝ) * (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)).
  Lean, lines 640–641
  have hreal : (64 : ℝ) * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by exact_mod_cast hnat
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 640–641 at dfc7b5cc45a9 ↗
  h4_big.s7.hx2' have x / (2 : ℝ) = (3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hx2' : x / (2 : ℝ) = (3 : ℝ) / (2 : ℝ) ^ (n + (1 : ℕ)).
  Uses library pow_succ.
  Lean, line 642
  have hx2' : x / 2 = 3 / 2 ^ (n + 1) := by rw [hxdef, pow_succ]; field_simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 642 at dfc7b5cc45a9 ↗
  h4_big.s7.hsq' have r ^ (2 : ℕ) ≤ (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hsq' : r ^ (2 : ℕ) ≤ (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)).
  Lean, lines 643–645
  have hsq' : r ^ 2 ≤ 64 * (x / 2) ^ (4 * n + 2) := by calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq _ = 64 * (x / 2) ^ (4 * n + 2) := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 643–645 at dfc7b5cc45a9 ↗
  h4_big.s7.hsq'.s1 calc r ^ (2 : ℕ) ≤ ((8 : ℝ) * (x / (2 : ℝ)) ^ ((2 : ℕ) * n + (1 : ℕ))) ^ (2 : ℕ)
  Closes the goal.
  Lean, line 644
  calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 644 at dfc7b5cc45a9 ↗
  h4_big.s7.hsq'.s2 calc routine: ring
  Closes the goal.
  Lean, line 645
  _ = 64 * (x / 2) ^ (4 * n + 2) := by ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 645 at dfc7b5cc45a9 ↗
  h4_big.s7.hkey have (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) = (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hkey : (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) = (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ)).
  Uses library mul_pow, div_mul_cancel₀.
  Lean, lines 646–647
  have hkey : (x / 2) ^ (4 * n + 2) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 3 ^ (4 * n + 2) := by rw [← mul_pow, hx2', div_mul_cancel₀ _ (by positivity)]
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 646–647 at dfc7b5cc45a9 ↗
  h4_big.s7.hP have (0 : ℝ) < ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) routine: positivity
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hP : (0 : ℝ) < ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)).
  Lean, line 648
  have hP : (0 : ℝ) < ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 648 at dfc7b5cc45a9 ↗
  h4_big.s7.hmain have (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ))
  Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with hmain : (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)).
  Uses library le_of_mul_le_mul_right.
  Lean, lines 649–653
  have hmain : 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) := by refine le_of_mul_le_mul_right ?_ hP calc 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) := by rw [← hkey]; ring _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 649–653 at dfc7b5cc45a9 ↗
  h4_big.s7.hmain.s1 calc (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ)) = (64 : ℝ) * (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n)
  Closes the goal.
  Lean, lines 651–652
  calc 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) := by rw [← hkey]; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 651–652 at dfc7b5cc45a9 ↗
  h4_big.s7.hmain.s2 calc (64 : ℝ) * (3 : ℝ) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) * ((2 : ℝ) ^ (n + (1 : ℕ))) ^ ((4 : ℕ) * n + (2 : ℕ))
  Closes the goal.
  Lean, line 653
  _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 653 at dfc7b5cc45a9 ↗
  h4_big.s7.s15 calc r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n)
  Closes the goal.
  Lean, line 654
  calc r ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤ 64 * (x / 2) ^ (4 * n + 2) * 2 ^ (4 * n * n) := by gcongr
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 654 at dfc7b5cc45a9 ↗
  h4_big.s7.s16 calc (64 : ℝ) * (x / (2 : ℝ)) ^ ((4 : ℕ) * n + (2 : ℕ)) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ))
  Closes the goal.
  Lean, line 655
  _ ≤ 2 ^ (2 * n + 4) := hmain
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 655 at dfc7b5cc45a9 ↗
T_facts have (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with T_facts : (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ).
Uses library Summable, And.intro.
Lean, lines 658–681
```
  have T_facts : Summable (fun m : ℕ => (1 : ℝ) / (2 ^ (m + 3) - 3)) ∧
      0 < ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ∧ ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ 2 / 5 := by
    have h5 : ∀ m : ℕ, 5 * (2 : ℝ) ^ m ≤ 2 ^ (m + 3) - 3 := by
      intro m
      have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have e : (2 : ℝ) ^ (m + 3) = 8 * 2 ^ m := by rw [pow_add]; norm_num; ring
      rw [e]; linarith
    have hpos : ∀ m : ℕ, 0 < (2 : ℝ) ^ (m + 3) - 3 := fun m => by
      have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity
      linarith
    have hle : ∀ m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ (1 / 5) * (1 / 2) ^ m := by
      intro m
      have e : (1 / 5 : ℝ) * (1 / 2) ^ m = 1 / (5 * 2 ^ m) := by
        rw [one_div_pow]; field_simp
      rw [e]
      exact one_div_le_one_div_of_le (by positivity) (h5 m)
    have hs : Summable (fun m : ℕ => (1 : ℝ) / (2 ^ (m + 3) - 3)) :=
      Summable.of_nonneg_of_le (fun m => le_of_lt (one_div_pos.mpr (hpos m))) hle
        (summable_geometric_two.mul_left (1 / 5))
    refine ⟨hs, ?_, ?_⟩
    · exact hs.tsum_pos (fun m => le_of_lt (one_div_pos.mpr (hpos m))) 0 (one_div_pos.mpr (hpos 0))
    · calc ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2) ^ m :=
            hs.tsum_le_tsum hle (summable_geometric_two.mul_left (1 / 5))
        _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_two]; norm_num
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 658–681 at dfc7b5cc45a9 ↗
1. T_facts.h5 have ∀ (m : ℕ), (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)
  Leaves (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ), with h5 : ∀ (m : ℕ), (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ).
  Lean, lines 660–664
  have h5 : ∀ m : ℕ, 5 * (2 : ℝ) ^ m ≤ 2 ^ (m + 3) - 3 := by intro m have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num) have e : (2 : ℝ) ^ (m + 3) = 8 * 2 ^ m := by rw [pow_add]; norm_num; ring rw [e]; linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 660–664 at dfc7b5cc45a9 ↗
  T_facts.h5.h1 have (1 : ℝ) ≤ (2 : ℝ) ^ m
  Leaves (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ), with h1 : (1 : ℝ) ≤ (2 : ℝ) ^ m.
  Uses library one_le_pow₀.
  Lean, line 662
  have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 662 at dfc7b5cc45a9 ↗
  T_facts.h5.e have (2 : ℝ) ^ (m + (3 : ℕ)) = (8 : ℝ) * (2 : ℝ) ^ m
  Leaves (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ), with e : (2 : ℝ) ^ (m + (3 : ℕ)) = (8 : ℝ) * (2 : ℝ) ^ m.
  Uses library pow_add.
  Lean, line 663
  have e : (2 : ℝ) ^ (m + 3) = 8 * 2 ^ m := by rw [pow_add]; norm_num; ring
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 663 at dfc7b5cc45a9 ↗
2. T_facts.hpos have ∀ (m : ℕ), (0 : ℝ) < (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)
  Leaves (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ), with hpos : ∀ (m : ℕ), (0 : ℝ) < (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ).
  Lean, lines 665–667
  have hpos : ∀ m : ℕ, 0 < (2 : ℝ) ^ (m + 3) - 3 := fun m => by have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity linarith
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 665–667 at dfc7b5cc45a9 ↗
  T_facts.hpos.s1 have (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)
  Leaves (0 : ℝ) < (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ), with this : (5 : ℝ) * (2 : ℝ) ^ m ≤ (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ).
  Lean, line 666
  have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 666 at dfc7b5cc45a9 ↗
  T_facts.hpos.s2 have (0 : ℝ) < (2 : ℝ) ^ m routine: positivity
  Leaves (0 : ℝ) < (2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ), with this : (0 : ℝ) < (2 : ℝ) ^ m.
  Lean, line 666
  have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 666 at dfc7b5cc45a9 ↗
3. T_facts.hle have ∀ (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m
  Leaves (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ), with hle : ∀ (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m.
  Uses library one_div_le_one_div_of_le.
  Lean, lines 668–673
  have hle : ∀ m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ (1 / 5) * (1 / 2) ^ m := by intro m have e : (1 / 5 : ℝ) * (1 / 2) ^ m = 1 / (5 * 2 ^ m) := by rw [one_div_pow]; field_simp rw [e] exact one_div_le_one_div_of_le (by positivity) (h5 m)
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 668–673 at dfc7b5cc45a9 ↗
  T_facts.hle.e have (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m = (1 : ℝ) / ((5 : ℝ) * (2 : ℝ) ^ m)
  Leaves (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m, with e : (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m = (1 : ℝ) / ((5 : ℝ) * (2 : ℝ) ^ m).
  Uses library one_div_pow.
  Lean, lines 670–671
  have e : (1 / 5 : ℝ) * (1 / 2) ^ m = 1 / (5 * 2 ^ m) := by rw [one_div_pow]; field_simp
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 670–671 at dfc7b5cc45a9 ↗
4. T_facts.hs have Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))
  Leaves (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ), with hs : Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)).
  Uses library Summable, Summable.of_nonneg_of_le, le_of_lt, one_div_pos, summable_geometric_two.
  Lean, lines 674–676
  have hs : Summable (fun m : ℕ => (1 : ℝ) / (2 ^ (m + 3) - 3)) := Summable.of_nonneg_of_le (fun m => le_of_lt (one_div_pos.mpr (hpos m))) hle (summable_geometric_two.mul_left (1 / 5))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 674–676 at dfc7b5cc45a9 ↗
5. T_facts.s5 case
  Leaves (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)).
  Uses library le_of_lt, one_div_pos.
  Lean, line 678
  · exact hs.tsum_pos (fun m => le_of_lt (one_div_pos.mpr (hpos m))) 0 (one_div_pos.mpr (hpos 0))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 678 at dfc7b5cc45a9 ↗
6. T_facts.s6 case
  Leaves ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ).
  Lean, lines 679–681
  · calc ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2) ^ m := hs.tsum_le_tsum hle (summable_geometric_two.mul_left (1 / 5)) _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_two]; norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 679–681 at dfc7b5cc45a9 ↗
  T_facts.s6.s1 calc ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ ∑' (m : ℕ), (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m
  Closes the goal.
  Uses library summable_geometric_two.
  Lean, lines 679–680
  · calc ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2) ^ m := hs.tsum_le_tsum hle (summable_geometric_two.mul_left (1 / 5))
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 679–680 at dfc7b5cc45a9 ↗
  T_facts.s6.s2 calc
  Closes the goal.
  Uses library tsum_mul_left, tsum_geometric_two.
  Lean, line 681
  _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_two]; norm_num
  targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 681 at dfc7b5cc45a9 ↗
s17 obtain (Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) ∧ (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ∧ ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ)
Leaves ∀ (Qc : ℕ → ℕ → ℚ), (Qc = fun (n k : ℕ) => ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) * ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) * ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) → ∀ (Qx : ℕ → ℚ), (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) → ∀ (Aq : ℕ → ℚ), (Aq = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k + Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) → ∀ (n : ℕ), (0 : ℝ) < (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ) ∧ ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq n) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)), with left✝ : Summable fun (m : ℕ) => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)), hTpos : (0 : ℝ) < ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)), hTle : ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ).
Lean, line 682
```
  obtain ⟨_, hTpos, hTle⟩ := T_facts
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 682 at dfc7b5cc45a9 ↗
hQx0 have Qx (0 : ℕ) = (1 : ℚ)
Leaves (0 : ℝ) < (↑(Qx (0 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (0 : ℕ)) : ℝ) ∧ ((↑(Qx (0 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (0 : ℕ)) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (0 : ℕ) * (0 : ℕ)) ≤ (2 : ℝ) ^ ((2 : ℕ) * (0 : ℕ) + (4 : ℕ)), with hQx0 : Qx (0 : ℕ) = (1 : ℚ).
Lean, line 686
```
    have hQx0 : Qx 0 = 1 := by rw [hQx, hQc]; simp
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 686 at dfc7b5cc45a9 ↗
hAq0 have Aq (0 : ℕ) = (0 : ℚ)
Leaves (0 : ℝ) < (↑(Qx (0 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (0 : ℕ)) : ℝ) ∧ ((↑(Qx (0 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (0 : ℕ)) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (0 : ℕ) * (0 : ℕ)) ≤ (2 : ℝ) ^ ((2 : ℕ) * (0 : ℕ) + (4 : ℕ)), with hAq0 : Aq (0 : ℕ) = (0 : ℚ).
Lean, line 687
```
    have hAq0 : Aq 0 = 0 := by rw [hAq]; simp
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 687 at dfc7b5cc45a9 ↗
e have ((1 : ℝ) * ((3 : ℝ) * T) - (0 : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (0 : ℕ) * (0 : ℕ)) = (9 : ℝ) * (T * T) routine: ring
Leaves ((1 : ℝ) * ((3 : ℝ) * T) - (0 : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ (0 : ℕ) ≤ (2 : ℝ) ^ (4 : ℕ), with e : ((1 : ℝ) * ((3 : ℝ) * T) - (0 : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (0 : ℕ) * (0 : ℕ)) = (9 : ℝ) * (T * T).
Lean, line 692
```
    have e : ((1 : ℝ) * (3 * T) - 0) ^ 2 * (2 : ℝ) ^ (4 * 0 * 0) = 9 * (T * T) := by ring
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, line 692 at dfc7b5cc45a9 ↗
hQx1 have Qx (1 : ℕ) = (3 / 2 : ℚ)
Leaves (0 : ℝ) < (↑(Qx (1 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (1 : ℕ)) : ℝ) ∧ ((↑(Qx (1 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (1 : ℕ)) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (1 : ℕ) * (1 : ℕ)) ≤ (2 : ℝ) ^ ((2 : ℕ) * (1 : ℕ) + (4 : ℕ)), with hQx1 : Qx (1 : ℕ) = (3 / 2 : ℚ).
Uses library Finset.sum_range_succ, Finset.prod_range_succ.
Lean, lines 698–699
```
    have hQx1 : Qx 1 = 3 / 2 := by
      rw [hQx, hQc]; norm_num [Finset.sum_range_succ, Finset.prod_range_succ]
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 698–699 at dfc7b5cc45a9 ↗
hAq1 have Aq (1 : ℕ) = (0 : ℚ)
Leaves (0 : ℝ) < (↑(Qx (1 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (1 : ℕ)) : ℝ) ∧ ((↑(Qx (1 : ℕ)) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) - (↑(Aq (1 : ℕ)) : ℝ)) ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * (1 : ℕ) * (1 : ℕ)) ≤ (2 : ℝ) ^ ((2 : ℕ) * (1 : ℕ) + (4 : ℕ)), with hAq1 : Aq (1 : ℕ) = (0 : ℚ).
Uses library Finset.sum_range_succ, Finset.prod_range_succ.
Lean, lines 700–702
```
    have hAq1 : Aq 1 = 0 := by
      rw [hAq]; simp only [hQx1]; rw [hQc]
      norm_num [Finset.sum_range_succ, Finset.prod_range_succ]
```
targets/erdos-1050/nodes/spec-180d8b72/Proof.lean, lines 700–702 at dfc7b5cc45a9 ↗

The proof in Lean (Proof.lean)

Checked by the kernel — these bytes are the ones its attestation names: attestations/000327.json, merge commit dfc7b5cc45a9.

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/spec-180d8b72/Proof.lean, sha256 3ff2a2cfeae6.

import Mathlib
import Nodes.«spec-180d8b72».Context

/-- erdos-1050, Borwein's route: the remainder bound `hrem` of the skeleton on erdos-1050--h1-v2 (hole erdos-1050--h1-v2--h4), without the denominator bound `hden` that the hole carries as an unused hypothesis (revision request, graph PR #229). With T = sum 1/(2^(n+3) - 3) and x_n = 3/2^n, the Pade remainder R_n = Q_n(x_n) * 3T - A_n is positive and R_n^2 * 2^(4n^2) <= 2^(2n+4). Numerically checked for n <= 30 (tightest at n = 1). -/
theorem erdos_1050_pade_remainder_bound : ∀ (Qc : ℕ → ℕ → ℚ),
  (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) →
    ∀ (Qx : ℕ → ℚ),
      (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) →
        ∀ (Aq : ℕ → ℚ),
          (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) →
              ∀ (n : ℕ),
                (0 : ℝ) <
                    (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                      (↑(Aq n) : ℝ) ∧
                  ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                          (↑(Aq n) : ℝ)) ^
                        (2 : ℕ) *
                      (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤
                    (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) := by
  open Polynomial in
  have id1_test : ∀ (n j : ℕ) (hj : j ≤ n),
      (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) =
        ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1) := by
    intro n j hj
    induction j with
    | zero => simp
    | succ j ih =>
      have ih' := ih (by omega)
      rw [Finset.prod_range_succ, ← ih']
      obtain ⟨m, hm⟩ : ∃ m, n - j = m + 1 := ⟨n - j - 1, by omega⟩
      rw [hm, Finset.prod_range_succ, show n - (j + 1) = m by omega]
      ring
  open Polynomial in
  have lower_test : ∀ (j : ℕ),
      ∏ i ∈ Finset.range j, ((2 : ℚ) ^ j - 2 ^ i) =
        2 ^ (j * (j - 1) / 2) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j
    have h1 : ∀ i ∈ Finset.range j, ((2 : ℚ) ^ j - 2 ^ i) = 2 ^ i * (2 ^ (j - 1 - i + 1) - 1) := by
      intro i hi
      have hi' := Finset.mem_range.mp hi
      rw [mul_sub, ← pow_add, show i + (j - 1 - i + 1) = j by omega]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum,
      Finset.prod_range_reflect (fun s => ((2 : ℚ) ^ (s + 1) - 1)) j]
    congr 1
    congr 1
    have := Finset.sum_range_id_mul_two j
    omega
  open Polynomial in
  have upper_test : ∀ (j n : ℕ) (hj : j ≤ n),
      ∏ i ∈ Finset.Ioc j n, ((2 : ℚ) ^ j - 2 ^ i) =
        (-1) ^ (n - j) * 2 ^ (j * (n - j)) * ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j n hj
    have e : Finset.Ioc j n = Finset.Ico (j + 1) (n + 1) := by
      ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
    rw [e, Finset.prod_Ico_eq_prod_range, show n + 1 - (j + 1) = n - j by omega]
    have h1 : ∀ t ∈ Finset.range (n - j), ((2 : ℚ) ^ j - 2 ^ (j + 1 + t)) = (-1) * 2 ^ j * (2 ^ (t + 1) - 1) := by
      intro t _
      rw [show j + 1 + t = j + (t + 1) by ring, pow_add]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
      Finset.prod_const, Finset.card_range, ← pow_mul]
  open Polynomial in
  have rhs_test : ∀ (j n : ℕ) (hj : j ≤ n),
      (∏ K ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ K)) *
        ∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1) =
        (-1) ^ n * 2 ^ (j * n) * ∏ s ∈ Finset.range (2 * n - j), ((2 : ℚ) ^ (s + 1) - 1) := by
    intro j n hj
    have e : Finset.Ioc n (2 * n) = Finset.Ico (n + 1) (2 * n + 1) := by
      ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
    rw [e, Finset.prod_Ico_eq_prod_range, show 2 * n + 1 - (n + 1) = n by omega]
    have h1 : ∀ t ∈ Finset.range n, ((2 : ℚ) ^ j - 2 ^ (n + 1 + t)) =
        (-1) * 2 ^ j * (2 ^ ((n - j) + t + 1) - 1) := by
      intro t _
      rw [show n + 1 + t = j + ((n - j) + t + 1) by omega, pow_add]
      ring
    rw [Finset.prod_congr rfl h1, Finset.prod_mul_distrib, Finset.prod_mul_distrib, Finset.prod_const,
      Finset.prod_const, Finset.card_range, ← pow_mul,
      show 2 * n - j = (n - j) + n by omega, Finset.prod_range_add]
    ring
  open Polynomial in
  have id1_full : ∀ (n j : ℕ) (hj : j ≤ n),
      (((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - (1 : ℕ)) / (2 : ℕ)) *
            ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
          ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
      ∏ K ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ K) := by
    intro n j hj
    have hPne : ∀ m : ℕ, ∏ s ∈ Finset.range m, ((2 : ℚ) ^ (s + 1) - 1) ≠ 0 := by
      intro m
      refine Finset.prod_ne_zero_iff.mpr (fun s _ => ?_)
      have : (1 : ℚ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hG1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) =
        (∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) /
          ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range j, ((2 : ℚ) ^ (s + 1) - 1)) := by
      rw [Finset.prod_div_distrib, ← id1_test n j hj]
      field_simp [hPne]
    have hG2 : ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - j - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)) =
        (∏ s ∈ Finset.range (2 * n - j), ((2 : ℚ) ^ (s + 1) - 1)) /
          ((∏ s ∈ Finset.range (n - j), ((2 : ℚ) ^ (s + 1) - 1)) * ∏ s ∈ Finset.range n, ((2 : ℚ) ^ (s + 1) - 1)) := by
      rw [Finset.prod_div_distrib, ← id1_test (2 * n - j) n (by omega), show 2 * n - j - n = n - j by omega]
      field_simp [hPne]
    have hsplit : (Finset.range (n + 1)).erase j = Finset.range j ∪ Finset.Ioc j n := by
      ext x; simp only [Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc]; omega
    have hdisj : Disjoint (Finset.range j) (Finset.Ioc j n) := by
      rw [Finset.disjoint_left]
      intro x hx hx'
      simp only [Finset.mem_range, Finset.mem_Ioc] at hx hx'
      omega
    have hR := rhs_test j n hj
    obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hj
    have hsign : (-1 : ℚ) ^ (j + m) = (-1) ^ j * (-1) ^ (j + m - j) := by
      rw [← pow_add]; congr 1; omega
    have ha : 2 * (j * (j - 1) / 2) = j * (j - 1) := Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self _)
    have hexp : j * (j + m) = j * (j - (1 : ℕ)) / (2 : ℕ) + j + j * (j - (1 : ℕ)) / (2 : ℕ) + j * (j + m - j) := by
      rw [show j + m - j = m by omega]
      rcases j with _ | i
      · simp
      · simp only [Nat.add_sub_cancel] at ha ⊢
        nlinarith [ha]
    rw [hG1, hG2, hsplit, Finset.prod_union hdisj, lower_test, upper_test j (j + m) hj]
    apply mul_right_cancel₀ (hPne (j + m - j))
    rw [hR, hsign, hexp, pow_add, pow_add, pow_add]
    field_simp [hPne]
  clear id1_test lower_test upper_test rhs_test
  open Polynomial in
  have lagr_eval : ∀ (n : ℕ) (x : ℕ → ℚ) (hx : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j)
    (N : ℚ[X]) (hNdeg : N.degree ≤ n) (r : ℕ → ℚ)
    (hr : ∀ j ≤ n, r j * ∏ i ∈ (Finset.range (n + 1)).erase j, (x j - x i) = N.eval (x j)),
      N = ∑ j ∈ Finset.range (n + 1), C (r j) * ∏ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)) := by
    intro n x hx N hNdeg r hr
    have hinj : Set.InjOn x (Finset.range (n + 1) : Set ℕ) := by
      intro i hi j hj h
      simp only [Finset.coe_range, Set.mem_Iio] at hi hj
      exact hx i (by omega) j (by omega) h
    apply Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq ((Finset.range (n + 1)).image x)
    · rw [Finset.card_image_of_injOn hinj, Finset.card_range]
      refine lt_of_le_of_lt (Polynomial.degree_sub_le _ _) (max_lt ?_ ?_)
      · exact lt_of_le_of_lt hNdeg (by exact_mod_cast Nat.lt_succ_self n)
      · refine lt_of_le_of_lt (Polynomial.degree_sum_le _ _) ?_
        rw [Finset.sup_lt_iff (by exact WithBot.bot_lt_coe _)]
        intro j hj
        refine lt_of_le_of_lt (Polynomial.degree_mul_le _ _) ?_
        refine lt_of_le_of_lt (add_le_add (Polynomial.degree_C_le) (Polynomial.degree_prod_le _ _)) ?_
        have : ∑ i ∈ (Finset.range (n + 1)).erase j, (X - C (x i)).degree = ((n : ℕ) : WithBot ℕ) := by
          rw [Finset.sum_congr rfl (fun i _ => Polynomial.degree_X_sub_C (x i)), Finset.sum_const,
            Finset.card_erase_of_mem hj, Finset.card_range, Nat.add_sub_cancel]
          simp
        rw [this, zero_add]
        exact_mod_cast Nat.lt_succ_self n
    · intro y hy
      obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy
      have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega
      rw [← hr k hk', Polynomial.eval_finset_sum, Finset.sum_eq_single k]
      · simp [Polynomial.eval_prod]
      · intro j hj hjk
        simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub,
          Polynomial.eval_X]
        rw [Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hjk, hk⟩) (by ring), mul_zero]
      · intro h; exact absurd hk h
  open Polynomial in
  have tail_coeff : ∀ (n K : ℕ) (hK : n < K) (Qc : ℕ → ℕ → ℚ)
    (hQc : ∀ j ≤ n, Qc n j * 2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
      ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ L)),
      ∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) =
        (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℚ) ^ K - 2 ^ i) := by
    intro n K hK Qc hQc
    set x : ℕ → ℚ := fun i => (2 : ℚ) ^ i with hxdef
    have hxinj : ∀ i ≤ n, ∀ j ≤ n, x i = x j → i = j := by
      intro i _ j _ h
      have h' : (2 : ℚ) ^ i = 2 ^ j := h
      exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast h')
    have hxne : ∀ a b : ℕ, a ≠ b → (2 : ℚ) ^ a - 2 ^ b ≠ 0 := by
      intro a b hab h
      apply hab
      exact Nat.pow_right_injective (le_refl 2) (by exact_mod_cast (sub_eq_zero.mp h))
    set N : ℚ[X] := ∏ L ∈ Finset.Ioc n (2 * n), (X - C ((2 : ℚ) ^ L)) with hN
    have hNdeg : N.degree ≤ n := by
      rw [hN, Polynomial.degree_prod]
      rw [Finset.sum_congr rfl (fun L _ => Polynomial.degree_X_sub_C ((2 : ℚ) ^ L)), Finset.sum_const,
        Nat.card_Ioc]
      simp
      omega
    have hr : ∀ j ≤ n, (Qc n j * 2 ^ j) * ∏ i ∈ (Finset.range (n + 1)).erase j, (x j - x i) = N.eval (x j) := by
      intro j hj
      rw [hN, Polynomial.eval_prod]
      simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C, hxdef]
      exact hQc j hj
    have hL := lagr_eval n x hxinj N hNdeg (fun j => Qc n j * 2 ^ j) hr
    have hev := congrArg (Polynomial.eval ((2 : ℚ) ^ K)) hL
    rw [Polynomial.eval_finset_sum] at hev
    have hNK : N.eval ((2 : ℚ) ^ K) = ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ K - 2 ^ L) := by
      rw [hN, Polynomial.eval_prod]
      simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C]
    have hD : ∏ i ∈ Finset.range (n + 1), ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 :=
      Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp hi; omega))
    rw [hNK] at hev
    rw [hev, Finset.sum_div]
    refine Finset.sum_congr rfl (fun j hj => ?_)
    have hjn : j < n + 1 := Finset.mem_range.mp hj
    have hne1 : (2 : ℚ) ^ K - 2 ^ j ≠ 0 := hxne K j (by omega)
    have hne2 : (2 : ℚ) ^ (K - j) - 1 ≠ 0 := by
      have : (1 : ℚ) < 2 ^ (K - j) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    simp only [Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod, Polynomial.eval_sub,
      Polynomial.eval_X, hxdef]
    rw [← Finset.mul_prod_erase (Finset.range (n + 1)) (fun i => (2 : ℚ) ^ K - 2 ^ i) hj]
    have hE : ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ K - 2 ^ i) ≠ 0 :=
      Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by
        have := Finset.mem_range.mp (Finset.mem_of_mem_erase hi); omega))
    have hpow : (2 : ℚ) ^ K = 2 ^ j * 2 ^ (K - j) := by rw [← pow_add]; congr 1; omega
    field_simp
    rw [hpow]
    ring
  clear lagr_eval
  open Polynomial in
  have tail3 : ∀ (n : ℕ) (hn : 1 ≤ n),
      ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) = ∑' s : ℕ, ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) := by
    intro n hn
    set f : ℕ → ℕ → ℝ := fun m s => ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * ((1 : ℝ) / 2 ^ (s + 1)) ^ m with hf
    have key : ∀ m s, f m s = ((3 : ℝ) / 2 ^ (m + n + 1)) ^ (s + 1) := by
      intro m s
      simp only [hf]
      rw [div_pow, div_pow, div_pow, one_pow, ← pow_mul, ← pow_mul, ← pow_mul,
        show (m + n + 1) * (s + 1) = (n + 1) * (s + 1) + (s + 1) * m by ring, pow_add]
      field_simp
      exact pow_add _ _ _
    have hpos : ∀ m s, 0 ≤ f m s := fun m s => by rw [key]; positivity
    have h4 : ∀ m : ℕ, (4 : ℝ) ≤ 2 ^ (m + n + 1) := by
      intro m
      calc (4 : ℝ) = 2 ^ 2 := by norm_num
        _ ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
    have inner1 : ∀ m, HasSum (f m) ((3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by
      intro m
      have hr0 : 0 ≤ (3 : ℝ) / 2 ^ (m + n + 1) := by positivity
      have hr1 : (3 : ℝ) / 2 ^ (m + n + 1) < 1 := by
        rw [div_lt_one (by positivity)]; linarith [h4 m]
      have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left ((3 : ℝ) / 2 ^ (m + n + 1))
      have hfun : f m = fun s => (3 : ℝ) / 2 ^ (m + n + 1) * ((3 : ℝ) / 2 ^ (m + n + 1)) ^ s := by
        funext s; rw [key]; ring
      have hne : (2 : ℝ) ^ (m + n + 1) - 3 ≠ 0 := by linarith [h4 m]
      have hne2 : (2 : ℝ) ^ (m + n + 1) ≠ 0 := by positivity
      have hval : (3 : ℝ) / (2 ^ (m + n + 1) - 3) =
          3 / 2 ^ (m + n + 1) * (1 - 3 / 2 ^ (m + n + 1))⁻¹ := by
        field_simp
      rw [hfun, hval]
      exact hg
    have inner2 : ∀ s, HasSum (fun m => f m s) (((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1)) := by
      intro s
      have hr0 : 0 ≤ (1 : ℝ) / 2 ^ (s + 1) := by positivity
      have hr1 : (1 : ℝ) / 2 ^ (s + 1) < 1 := by
        rw [div_lt_one (by positivity)]
        exact one_lt_pow₀ (by norm_num) (by omega)
      have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1))
      have hne : (2 : ℝ) ^ (s + 1) - 1 ≠ 0 := by
        have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have hval : ((3 : ℝ) / 2 ^ n) ^ (s + 1) / (2 ^ (s + 1) - 1) =
          ((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1) * (1 - 1 / 2 ^ (s + 1))⁻¹ := by
        rw [div_pow, div_pow, ← pow_mul, ← pow_mul, show (n + 1) * (s + 1) = n * (s + 1) + (s + 1) by ring, pow_add]
        field_simp
        exact pow_add _ _ _
      rw [hval]
      exact hg
    have hsum1 : Summable (fun m : ℕ => (3 : ℝ) / (2 ^ (m + n + 1) - 3)) := by
      refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3)
      · have := h4 m; exact div_nonneg (by norm_num) (by linarith)
      · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
        have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
        calc (3 : ℝ) / (2 ^ (m + n + 1) - 3) ≤ 3 / 2 ^ m :=
              div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
          _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
    have hU : Summable (Function.uncurry f) := by
      refine (summable_prod_of_nonneg (fun p => hpos p.1 p.2)).mpr ⟨fun m => (inner1 m).summable, ?_⟩
      exact hsum1.congr (fun m => ((inner1 m).tsum_eq).symm)
    have hcomm := hU.tsum_comm' (fun m => (inner1 m).summable) (fun s => (inner2 s).summable)
    rw [show (∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3)) = ∑' m, ∑' s, f m s from
      tsum_congr (fun m => ((inner1 m).tsum_eq).symm)]
    rw [← hcomm]
    exact tsum_congr (fun s => (inner2 s).tsum_eq)
  open Polynomial in
  have split3T : ∀ (n : ℕ),
      3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) =
        ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) + ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) := by
    intro n
    set g : ℕ → ℝ := fun m => (3 : ℝ) / (2 ^ (m + 1) - 3) with hg
    have hsum : Summable (fun m => g (m + 1)) := by
      refine Summable.of_nonneg_of_le (fun m => ?_) (fun m => ?_) ((summable_geometric_two).mul_left 3)
      · have : (4 : ℝ) ≤ 2 ^ (m + 1 + 1) := by
          calc (4 : ℝ) = 2 ^ 2 := by norm_num
            _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        simp only [hg]; exact div_nonneg (by norm_num) (by linarith)
      · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf)
        have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
        have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring
        simp only [hg]
        calc (3 : ℝ) / (2 ^ (m + 1 + 1) - 3) ≤ 3 / 2 ^ m :=
              div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
          _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring
    have hs : Summable g := (summable_nat_add_iff 1).mp hsum
    have e1 := hs.sum_add_tsum_nat_add n
    have e2 := hs.sum_add_tsum_nat_add 2
    have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑' m, g (m + 2) := by
      rw [← tsum_mul_left]
      refine tsum_congr (fun m => ?_)
      simp only [hg]
      rw [show m + 2 + 1 = m + 3 by ring]
      ring
    have hfirst : ∑ i ∈ Finset.range 2, g i = 0 := by
      simp only [hg, Finset.sum_range_succ, Finset.sum_range_zero]
      norm_num
    rw [h3T]
    have htail : ∑' m : ℕ, (3 : ℝ) / (2 ^ (m + n + 1) - 3) = ∑' m, g (m + n) := by
      refine tsum_congr (fun m => ?_)
      simp only [hg]
    rw [htail]
    have hrange : ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) = ∑ i ∈ Finset.range n, g i := by
      simp only [hg]
    rw [hrange, e1]
    rw [← e2, hfirst, zero_add]
  open Polynomial in
  have cauchy_fin : ∀ (n : ℕ) (q a : ℕ → ℝ) (x A : ℝ) (hA : HasSum (fun s => a s * x ^ (s + 1)) A),
      HasSum (fun K => ∑ j ∈ Finset.range (n + 1), if j < K then q j * a (K - j - 1) * x ^ K else 0)
        ((∑ j ∈ Finset.range (n + 1), q j * x ^ j) * A) := by
    intro n q a x A hA
    rw [Finset.sum_mul]
    apply hasSum_sum
    intro j _
    have h := hA.mul_left (q j * x ^ j)
    rw [← hasSum_nat_add_iff' (j + 1)]
    have hz : ∑ i ∈ Finset.range (j + 1), (if j < i then q j * a (i - j - 1) * x ^ i else 0) = 0 :=
      Finset.sum_eq_zero (fun i hi => by
        have := Finset.mem_range.mp hi
        rw [if_neg (by omega)])
    rw [hz, sub_zero]
    have hfun : (fun s => if j < s + (j + 1) then q j * a (s + (j + 1) - j - 1) * x ^ (s + (j + 1)) else 0) =
        fun s => q j * x ^ j * (a s * x ^ (s + 1)) := by
      funext s
      rw [if_pos (by omega), show s + (j + 1) - j - 1 = s by omega, show s + (j + 1) = j + (s + 1) by ring, pow_add]
      ring
    rw [hfun]
    exact h
  open Polynomial in
  have h4_series : ∀ (Qc : ℕ → ℕ → ℚ) (hQc : (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)))) (Qx : ℕ → ℚ) (hQx : (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k))
    (Aq : ℕ → ℚ) (hAq : (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ)))) (n : ℕ) (hn : 1 ≤ n),
      HasSum (fun K : ℕ => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            ((3 : ℝ) / 2 ^ n) ^ K else 0)
        ((Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ)) := by
    intro Qc hQc Qx hQx Aq hAq n hn
    set x : ℝ := (3 : ℝ) / 2 ^ n with hxdef
    have hx0 : 0 ≤ x := by positivity
    have hx2 : x / 2 < 1 := by
      have : (2 : ℝ) ≤ 2 ^ n := by
        calc (2 : ℝ) = 2 ^ 1 := by norm_num
          _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
      rw [hxdef, div_div, div_lt_one (by positivity)]; linarith
    set a : ℕ → ℝ := fun s => (1 : ℝ) / (2 ^ (s + 1) - 1) with hadef
    have hapos : ∀ s, 0 < (2 : ℝ) ^ (s + 1) - 1 := fun s => by
      have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hAsum : Summable (fun s => a s * x ^ (s + 1)) := by
      refine Summable.of_nonneg_of_le (fun s => ?_) (fun s => ?_) ((summable_geometric_of_lt_one (by positivity) hx2).mul_left x)
      · simp only [hadef]; exact mul_nonneg (div_nonneg (by norm_num) (le_of_lt (hapos s))) (by positivity)
      · simp only [hadef]
        have h1 : (2 : ℝ) ^ s ≤ 2 ^ (s + 1) - 1 := by
          have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num)
          rw [pow_succ]; linarith
        have hxs : 0 ≤ x ^ (s + 1) := by positivity
        calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by
              gcongr
          _ = x * (x / 2) ^ s := by
            have e : ∀ y : ℝ, 1 / 2 ^ s * y ^ (s + 1) = y * (y / 2) ^ s := by
              intro y; rw [div_pow, pow_succ]; field_simp
            exact e x
    set A := ∑' s, a s * x ^ (s + 1) with hAdef
    have hA : HasSum (fun s => a s * x ^ (s + 1)) A := hAsum.hasSum
    -- 3T = c + A
    have h3T : 3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) = ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) + A := by
      rw [split3T n, tail3 n hn, hAdef]
      congr 1
      refine tsum_congr (fun s => ?_)
      simp only [hadef, hxdef]
      ring
    have hQxR : (Qx n : ℝ) = ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j := by
      rw [hQx]; push_cast; rfl
    have hAqR : (Aq n : ℝ) = ∑ k ∈ Finset.range (n + 1),
        (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * x ^ k +
        (Qx n : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / (2 ^ (j + 1) - 3) := by
      rw [hAq]; push_cast; rfl
    have hC := cauchy_fin n (fun j => (Qc n j : ℝ)) a x A hA
    set P : ℝ := ∑ k ∈ Finset.range (n + 1),
        (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * x ^ k with hPdef
    have hP : HasSum (fun K => if K < n + 1 then
        (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) P := by
      have h0 : HasSum (fun K => if K < n + 1 then
          (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0)
          (∑ K ∈ Finset.range (n + 1), if K < n + 1 then
            (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) :=
        hasSum_sum_of_ne_finset_zero (fun b hb => by rw [if_neg (by simpa using hb)])
      have h1 : (∑ K ∈ Finset.range (n + 1), if K < n + 1 then
            (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) = P := by
        rw [hPdef]
        exact Finset.sum_congr rfl (fun K hK => by rw [if_pos (Finset.mem_range.mp hK)])
      rw [h1] at h0
      exact h0
    have hD := hC.sub hP
    have hval : (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) =
        (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * x ^ j) * A - P := by
      rw [h3T, hAqR, hQxR]; ring
    rw [hval]
    have hid1 : ∀ j ≤ n, Qc n j * 2 ^ j * ∏ i ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ i) =
        ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℚ) ^ j - 2 ^ L) := fun j hj => by subst hQc; exact id1_full n j hj
    have hfun : (fun K : ℕ => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            ((3 : ℝ) / 2 ^ n) ^ K else 0) =
        fun K => (∑ j ∈ Finset.range (n + 1), if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) -
          (if K < n + 1 then (∑ j ∈ Finset.range (K + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) * x ^ K else 0) := by
      funext K
      symm
      by_cases hK : K < n + 1
      · rw [if_pos hK, if_neg (by omega)]
        rw [Finset.sum_range_succ (fun j => (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1)) K, Nat.sub_self, pow_zero, sub_self,
          div_zero, add_zero]
        have hsplit : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) =
            ∑ j ∈ Finset.range K, (Qc n j : ℝ) / ((2 : ℝ) ^ (K - j) - 1) * x ^ K := by
          rw [← Finset.sum_range_add_sum_Ico _ (show K ≤ n + 1 by omega)]
          rw [Finset.sum_eq_zero (s := Finset.Ico K (n + 1)) (fun j hj => by
            rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero]
          refine Finset.sum_congr rfl (fun j hj => ?_)
          have hjK := Finset.mem_range.mp hj
          rw [if_pos hjK]
          simp only [hadef]
          rw [show K - j - 1 + 1 = K - j by omega]
          ring
        rw [hsplit, Finset.sum_mul, sub_self]
      · rw [if_neg hK, sub_zero]
        have hS : ∑ j ∈ Finset.range (n + 1), (if j < K then (Qc n j : ℝ) * a (K - j - 1) * x ^ K else 0) =
            ((∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (K - j) - 1) : ℚ) : ℝ) * x ^ K := by
          push_cast
          rw [Finset.sum_mul]
          refine Finset.sum_congr rfl (fun j hj => ?_)
          have hjK : j < K := by have := Finset.mem_range.mp hj; omega
          rw [if_pos hjK]
          simp only [hadef]
          rw [show K - j - 1 + 1 = K - j by omega]
          ring
        rw [hS, tail_coeff n K (by omega) Qc hid1]
        push_cast
        by_cases h2 : 2 * n < K
        · rw [if_pos h2]
        · rw [if_neg h2]
          rw [Finset.prod_eq_zero (i := K) (Finset.mem_Ioc.mpr ⟨by omega, by omega⟩) (sub_self _)]
          simp
    rw [hfun]
    exact hD
  clear split3T tail3 cauchy_fin tail_coeff id1_full
  open Polynomial in
  have prod_one_sub_ge : ∀ (s : Finset ℕ) (a : ℕ → ℝ) (h0 : ∀ i ∈ s, 0 ≤ a i) (h1 : ∀ i ∈ s, a i ≤ 1),
      1 - ∑ i ∈ s, a i ≤ ∏ i ∈ s, (1 - a i) := by
    intro s a h0 h1
    induction s using Finset.induction_on with
    | empty => simp
    | insert j s hj ih =>
      rw [Finset.sum_insert hj, Finset.prod_insert hj]
      have ih' := ih (fun i hi => h0 i (Finset.mem_insert_of_mem hi)) (fun i hi => h1 i (Finset.mem_insert_of_mem hi))
      have haj0 := h0 j (Finset.mem_insert_self j s)
      have haj1 := h1 j (Finset.mem_insert_self j s)
      have hP : 0 ≤ ∏ i ∈ s, (1 - a i) :=
        Finset.prod_nonneg (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)])
      have hP1 : ∏ i ∈ s, (1 - a i) ≤ 1 :=
        Finset.prod_le_one (fun i hi => by linarith [h1 i (Finset.mem_insert_of_mem hi)])
          (fun i hi => by linarith [h0 i (Finset.mem_insert_of_mem hi)])
      nlinarith
  open Polynomial in
  have nd_bound : ∀ (n K : ℕ) (hn : 1 ≤ n) (hK : 2 * n + 1 ≤ K),
      0 < (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) ∧
      (∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)
        ≤ 2 * (1 / 2) ^ K := by
    intro n K hn hK
    have hlt : ∀ i, i < K → (0 : ℝ) < 2 ^ K - 2 ^ i := fun i hi => by
      have : (2 : ℝ) ^ i < 2 ^ K := pow_lt_pow_right₀ (by norm_num) hi
      linarith
    have hNpos : 0 < ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) :=
      Finset.prod_pos (fun L hL => hlt L (by have := Finset.mem_Ioc.mp hL; omega))
    have hDpos : 0 < ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) :=
      Finset.prod_pos (fun i hi => hlt i (by have := Finset.mem_range.mp hi; omega))
    refine ⟨div_pos hNpos hDpos, ?_⟩
    have hN : ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := by
      have := Finset.prod_le_prod (s := Finset.Ioc n (2 * n)) (f := fun L => (2 : ℝ) ^ K - 2 ^ L) (g := fun _ => (2 : ℝ) ^ K)
        (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by
          have : (0 : ℝ) < 2 ^ L := by positivity
          linarith)
      rw [Finset.prod_const, Nat.card_Ioc, show 2 * n - n = n by omega] at this
      exact this
    have hD : (2 ^ K) ^ (n + 1) / 2 ≤ ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by
      have hfac : ∀ i ∈ Finset.range (n + 1), (2 : ℝ) ^ K - 2 ^ i = 2 ^ K * (1 - (1 / 2) ^ (K - i)) := by
        intro i hi
        have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega
        have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega
        rw [mul_sub, mul_one, one_div_pow, mul_one_div, e]
        field_simp
      rw [Finset.prod_congr rfl hfac, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range]
      have hsum : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) ≤ 1 / 2 := by
        have e : ∀ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (K - i) = (1 / 2) ^ (K - n) * (1 / 2) ^ (n - i) := by
          intro i hi
          have := Finset.mem_range.mp hi
          rw [← pow_add]; congr 1; omega
        rw [Finset.sum_congr rfl e, ← Finset.mul_sum]
        have hg : ∑ i ∈ Finset.range (n + 1), ((1 : ℝ) / 2) ^ (n - i) ≤ 2 := by
          have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1)
          simp only [Nat.add_sub_cancel] at hr
          rw [hr]
          exact sum_geometric_two_le (n + 1)
        have hk : ((1 : ℝ) / 2) ^ (K - n) ≤ (1 / 2) ^ 2 :=
          pow_le_pow_of_le_one (by norm_num) (by norm_num) (by omega)
        have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity
        nlinarith
      have hprod := prod_one_sub_ge (Finset.range (n + 1)) (fun i => ((1 : ℝ) / 2) ^ (K - i))
        (fun i _ => by positivity) (fun i _ => pow_le_one₀ (by norm_num) (by norm_num))
      have h2K : (0 : ℝ) < (2 ^ K) ^ (n + 1) := by positivity
      nlinarith
    rw [div_le_iff₀ hDpos]
    have h2K : (2 : ℝ) ^ K * (1 / 2) ^ K = 1 := by rw [← mul_pow]; norm_num
    have e2 : 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ (n + 1) / 2) = (2 ^ K) ^ n := by
      rw [pow_succ]
      calc 2 * (1 / 2 : ℝ) ^ K * ((2 ^ K) ^ n * 2 ^ K / 2) = (2 ^ K * (1 / 2) ^ K) * (2 ^ K) ^ n := by ring
        _ = (2 ^ K) ^ n := by rw [h2K, one_mul]
    calc ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := hN
      _ = 2 * (1 / 2) ^ K * ((2 ^ K) ^ (n + 1) / 2) := e2.symm
      _ ≤ 2 * (1 / 2) ^ K * ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by gcongr
  clear prod_one_sub_ge
  open Polynomial in
  have ineq_pow : ∀ (n : ℕ) (hn : 2 ≤ n),
      9 * 81 ^ n ≤ 256 ^ n := by
    intro n hn
    induction n with
    | zero => omega
    | succ m ih =>
      rcases (show m = 1 ∨ 2 ≤ m by omega) with rfl | hm
      · norm_num
      · have := ih hm
        rw [pow_succ, pow_succ]
        omega
  open Polynomial in
  have h4_big : ∀ (Qc : ℕ → ℕ → ℚ) (hQc : (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ)))) (Qx : ℕ → ℚ) (hQx : (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k))
    (Aq : ℕ → ℚ) (hAq : (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ)))) (n : ℕ) (hn : 2 ≤ n),
      (0 : ℝ) < (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) ∧
      ((Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ)) ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤
        (2 : ℝ) ^ (2 * n + 4) := by
    intro Qc hQc Qx hQx Aq hAq n hn
    have hS := h4_series Qc hQc Qx hQx Aq hAq n (by omega)
    set r := (Qx n : ℝ) * (3 * ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3)) - (Aq n : ℝ) with hr
    set x : ℝ := (3 : ℝ) / 2 ^ n with hxdef
    have hx0 : 0 < x := by positivity
    have hx2 : x / 2 ≤ 3 / 8 := by
      have : (4 : ℝ) ≤ 2 ^ n := by
        calc (4 : ℝ) = 2 ^ 2 := by norm_num
          _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
      rw [hxdef, div_div, div_le_div_iff₀ (by positivity) (by norm_num)]; linarith
    set G : ℕ → ℝ := fun K => if 2 * n < K then
          ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) / ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) *
            x ^ K else 0 with hG
    have hGnn : ∀ K, 0 ≤ G K := by
      intro K; simp only [hG]
      split_ifs with h
      · exact mul_nonneg (le_of_lt (nd_bound n K (by omega) (by omega)).1) (by positivity)
      · exact le_refl 0
    have hGpos : 0 < G (2 * n + 1) := by
      simp only [hG]; rw [if_pos (by omega)]
      exact mul_pos (nd_bound n (2 * n + 1) (by omega) (by omega)).1 (by positivity)
    refine ⟨?_, ?_⟩
    · exact hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1) (fun K => hGnn K) hGpos hasSum_zero hS
    · set H : ℕ → ℝ := fun K => if 2 * n < K then 2 * (x / 2) ^ K else 0 with hH
      have hGH : ∀ K, G K ≤ H K := by
        intro K; simp only [hG, hH]
        split_ifs with h
        · have hb := (nd_bound n K (by omega) (by omega)).2
          calc ((∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L)) /
                ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i)) * x ^ K ≤ 2 * (1 / 2) ^ K * x ^ K := by
                gcongr
            _ = 2 * (x / 2) ^ K := by ring
        · exact le_refl 0
      have hq : x / 2 < 1 := by linarith
      have hHsum : HasSum H (2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹) := by
        rw [← hasSum_nat_add_iff' (2 * n + 1)]
        have hz : ∑ i ∈ Finset.range (2 * n + 1), H i = 0 :=
          Finset.sum_eq_zero (fun i hi => by
            simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)])
        rw [hz, sub_zero]
        have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1))
        have hfun : (fun s => H (s + (2 * n + 1))) = fun s => 2 * (x / 2) ^ (2 * n + 1) * (x / 2) ^ s := by
          funext s; simp only [hH]; rw [if_pos (by omega), pow_add]; ring
        rw [hfun]
        exact hgeo
      have hrle : r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hasSum_le hGH hS hHsum
      have hr0 : 0 ≤ r := le_of_lt (hasSum_lt (f := fun _ => (0 : ℝ)) (g := G) (i := 2 * n + 1)
        (fun K => hGnn K) hGpos hasSum_zero hS)
      have hr8 : r ≤ 8 * (x / 2) ^ (2 * n + 1) := by
        have hinv : (1 - x / 2)⁻¹ ≤ 4 := by
          rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith
        have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity
        calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle
          _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr
          _ = 8 * (x / 2) ^ (2 * n + 1) := by ring
      have hsq : r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := pow_le_pow_left₀ hr0 hr8 2
      have hnat : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by
        have e1 : (2 : ℕ) ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) = 2 ^ (4 * n * n) * (64 * 256 ^ n) := by
          rw [← pow_mul, ← pow_add, show (256 : ℕ) = 2 ^ 8 by norm_num, ← pow_mul,
            show (64 : ℕ) = 2 ^ 6 by norm_num, ← pow_add, ← pow_add]
          congr 1; ring
        have e2 : 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) = 2 ^ (4 * n * n) * (64 * (9 * 81 ^ n)) := by
          rw [show (81 : ℕ) = 3 ^ 4 by norm_num, ← pow_mul, pow_add]; ring
        rw [e1, e2]
        have := ineq_pow n hn
        gcongr
      have hreal : (64 : ℝ) * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) * (2 ^ (n + 1)) ^ (4 * n + 2) := by
        exact_mod_cast hnat
      have hx2' : x / 2 = 3 / 2 ^ (n + 1) := by rw [hxdef, pow_succ]; field_simp
      have hsq' : r ^ 2 ≤ 64 * (x / 2) ^ (4 * n + 2) := by
        calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq
          _ = 64 * (x / 2) ^ (4 * n + 2) := by ring
      have hkey : (x / 2) ^ (4 * n + 2) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) = 3 ^ (4 * n + 2) := by
        rw [← mul_pow, hx2', div_mul_cancel₀ _ (by positivity)]
      have hP : (0 : ℝ) < ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := by positivity
      have hmain : 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) ≤ 2 ^ (2 * n + 4) := by
        refine le_of_mul_le_mul_right ?_ hP
        calc 64 * (x / 2) ^ (4 * n + 2) * (2 : ℝ) ^ (4 * n * n) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2)
            = 64 * 3 ^ (4 * n + 2) * 2 ^ (4 * n * n) := by rw [← hkey]; ring
          _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal
      calc r ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤ 64 * (x / 2) ^ (4 * n + 2) * 2 ^ (4 * n * n) := by gcongr
        _ ≤ 2 ^ (2 * n + 4) := hmain
  open Polynomial in
  clear h4_series nd_bound ineq_pow
  have T_facts : Summable (fun m : ℕ => (1 : ℝ) / (2 ^ (m + 3) - 3)) ∧
      0 < ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ∧ ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ 2 / 5 := by
    have h5 : ∀ m : ℕ, 5 * (2 : ℝ) ^ m ≤ 2 ^ (m + 3) - 3 := by
      intro m
      have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)
      have e : (2 : ℝ) ^ (m + 3) = 8 * 2 ^ m := by rw [pow_add]; norm_num; ring
      rw [e]; linarith
    have hpos : ∀ m : ℕ, 0 < (2 : ℝ) ^ (m + 3) - 3 := fun m => by
      have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity
      linarith
    have hle : ∀ m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ (1 / 5) * (1 / 2) ^ m := by
      intro m
      have e : (1 / 5 : ℝ) * (1 / 2) ^ m = 1 / (5 * 2 ^ m) := by
        rw [one_div_pow]; field_simp
      rw [e]
      exact one_div_le_one_div_of_le (by positivity) (h5 m)
    have hs : Summable (fun m : ℕ => (1 : ℝ) / (2 ^ (m + 3) - 3)) :=
      Summable.of_nonneg_of_le (fun m => le_of_lt (one_div_pos.mpr (hpos m))) hle
        (summable_geometric_two.mul_left (1 / 5))
    refine ⟨hs, ?_, ?_⟩
    · exact hs.tsum_pos (fun m => le_of_lt (one_div_pos.mpr (hpos m))) 0 (one_div_pos.mpr (hpos 0))
    · calc ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2) ^ m :=
            hs.tsum_le_tsum hle (summable_geometric_two.mul_left (1 / 5))
        _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_two]; norm_num
  obtain ⟨_, hTpos, hTle⟩ := T_facts
  intro Qc hQc Qx hQx Aq hAq n
  match n with
  | 0 =>
    have hQx0 : Qx 0 = 1 := by rw [hQx, hQc]; simp
    have hAq0 : Aq 0 = 0 := by rw [hAq]; simp
    rw [hQx0, hAq0]
    push_cast
    set T := ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) with hT
    refine ⟨by linarith, ?_⟩
    have e : ((1 : ℝ) * (3 * T) - 0) ^ 2 * (2 : ℝ) ^ (4 * 0 * 0) = 9 * (T * T) := by ring
    rw [e]
    norm_num
    nlinarith
  | 1 =>
    -- Qx 1 = 3 - 3/2 = 3/2 and Aq 1 = 3 * (3/2) + (3/2) * (3/(2-3)) = 0, so the remainder is (9/2) T.
    have hQx1 : Qx 1 = 3 / 2 := by
      rw [hQx, hQc]; norm_num [Finset.sum_range_succ, Finset.prod_range_succ]
    have hAq1 : Aq 1 = 0 := by
      rw [hAq]; simp only [hQx1]; rw [hQc]
      norm_num [Finset.sum_range_succ, Finset.prod_range_succ]
    rw [hQx1, hAq1]
    push_cast
    set T := ∑' m : ℕ, (1 : ℝ) / (2 ^ (m + 3) - 3) with hT
    refine ⟨by nlinarith, ?_⟩
    norm_num
    nlinarith
  | n + 2 => exact h4_big Qc hQc Qx hQx Aq hAq (n + 2) (by omega)

What is known

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Record

Statement: the Lean file and its words (2 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.«spec-180d8b72».Context

/-- erdos-1050, Borwein's route: the remainder bound `hrem` of the skeleton on erdos-1050--h1-v2 (hole erdos-1050--h1-v2--h4), without the denominator bound `hden` that the hole carries as an unused hypothesis (revision request, graph PR #229). With T = sum 1/(2^(n+3) - 3) and x_n = 3/2^n, the Pade remainder R_n = Q_n(x_n) * 3T - A_n is positive and R_n^2 * 2^(4n^2) <= 2^(2n+4). Numerically checked for n <= 30 (tightest at n = 1). -/
theorem erdos_1050_pade_remainder_bound : ∀ (Qc : ℕ → ℕ → ℚ),
  (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) →
    ∀ (Qx : ℕ → ℚ),
      (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) →
        ∀ (Aq : ℕ → ℚ),
          (Aq = fun (n : ℕ) =>
              ∑ k ∈ Finset.range (n + (1 : ℕ)),
                  (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
                Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) →
              ∀ (n : ℕ),
                (0 : ℝ) <
                    (↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                      (↑(Aq n) : ℝ) ∧
                  ((↑(Qx n) : ℝ) * ((3 : ℝ) * ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))) -
                          (↑(Aq n) : ℝ)) ^
                        (2 : ℕ) *
                      (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤
                    (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)) := 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

$\displaystyle \forall n\in\mathbb{N}:\qquad R_n>0\quad\wedge\quad R_n^{2}\,2^{4n^2}\le 2^{2n+4},\qquad R_n=3T\,Q_x(n)-A_q(n),\quad T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$

Rendered from gloss/e4490fe8efbd….md · Improve these words →

History: 2 versions on record

2026-10-06 · written by t1006-w3, drafted with Claude Opus 5.5 (claude-opus-5-5) in a Claude Code agent session · not signed · gloss/624faabaa55d….md · the first version of its chain
Read this version
This file states the theorem erdos_1050_pade_remainder_bound. Its doc comment calls it the remainder bound hrem of the skeleton on erdos-1050--h1-v2 (the hole erdos-1050--h1-v2--h4), without the hypothesis hden that the hole carries. The statement below is the hole's statement with that one hypothesis removed.

The statement is about three functions $Q_c:\mathbb N\times\mathbb N\to\mathbb Q$, $Q_x:\mathbb N\to\mathbb Q$ and $A_q:\mathbb N\to\mathbb Q$. It takes as hypotheses that they are given by the following formulas, so these are the only functions it speaks about:
- $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\displaystyle\prod_{i=0}^{k-1}\frac{2^{n-i}-1}{2^{i+1}-1}\;\prod_{i=0}^{n-1}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$;
- $Q_x(n)=\displaystyle\sum_{k=0}^{n}Q_c(n,k)\,\Big(\frac{3}{2^n}\Big)^k$;
- $A_q(n)=\displaystyle\sum_{k=0}^{n}\Big(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\Big)\Big(\frac{3}{2^n}\Big)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$.
Exponents are natural numbers: subtraction in them is truncated at $0$, and $k(k-1)/2$ is natural-number division (with $k-1=0$ when $k=0$). Where these functions are used below, $k\le n$, so no other exponent is truncated. Division is Lean's division of rationals, in which $x/0=0$: the term $j=k$ of the inner sum in $A_q$ has denominator $2^0-1=0$ and so contributes $0$.

Let $T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$, a real number (Lean's tsum over all $m\in\mathbb N$, which would be $0$ for a series that does not converge; this one converges). With $x_n=3/2^n$, $Q_x(n)$ is the polynomial $\sum_k Q_c(n,k)X^k$ evaluated at $x_n$. For each $n$ write, in the real numbers, $$R_n=3T\,Q_x(n)-A_q(n).$$

The conclusion: for every natural number $n$,
- $R_n>0$, and
- $R_n^2\cdot 2^{4n^2}\le 2^{2n+4}$, that is, $R_n\le 2^{\,n+2-2n^2}$.
There are no other hypotheses. The doc comment also says the bound was checked numerically for $n\le 30$ and is tightest at $n=1$; that remark is a comment, not part of what Lean checks.
2026-10-18 · written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1) · not signed · current · gloss/e4490fe8efbd….md · diff to its predecessor
Read this version
$\displaystyle \forall n\in\mathbb{N}:\qquad R_n>0\quad\wedge\quad R_n^{2}\,2^{4n^2}\le 2^{2n+4},\qquad R_n=3T\,Q_x(n)-A_q(n),\quad T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$

For every natural number $n$ the real number $R_n$ is positive and $R_n^2\cdot 2^{4n\cdot n}\le 2^{2n+4}$ (that is, $R_n\le 2^{\,n+2-2n^2}$), where $T$ is Lean's ∑' over all $m\in\mathbb{N}$ in $\mathbb{R}$ (which would be $0$ for a non-summable family; this one converges) and $Q_x(n)$, $A_q(n)$ are read as reals; with $x_n=3/2^n$, $Q_x(n)$ is the polynomial $\sum_k Q_c(n,k)X^k$ at $x_n$. The functions $Q_c:\mathbb{N}\times\mathbb{N}\to\mathbb{Q}$, $Q_x:\mathbb{N}\to\mathbb{Q}$ and $A_q:\mathbb{N}\to\mathbb{Q}$ are fixed by hypotheses as $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\prod_{i=0}^{k-1}\frac{2^{n-i}-1}{2^{i+1}-1}\prod_{i=0}^{n-1}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$, $Q_x(n)=\sum_{k=0}^{n}Q_c(n,k)\,(3/2^n)^k$ and $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. Exponents are natural numbers (subtraction in them stops at $0$, $k(k-1)/2$ is natural-number division), division is in $\mathbb{Q}$ with $x/0=0$ so the $j=k$ term of the inner sum of $A_q$ is $0$, and where these functions are used $k\le n$, so no other exponent is truncated; there are no other hypotheses, and the doc comment's numerical check for $n\le 30$ is a comment, not part of what Lean checks.

Diff of e4490fe8efbd against 624faabaa55d

@@ -1,19 +1,3 @@
-This file states the theorem `erdos_1050_pade_remainder_bound`. Its doc comment calls it the remainder bound `hrem` of the skeleton on `erdos-1050--h1-v2` (the hole `erdos-1050--h1-v2--h4`), without the hypothesis `hden` that the hole carries. The statement below is the hole's statement with that one hypothesis removed.
+$$\forall n\in\mathbb{N}:\qquad R_n>0\quad\wedge\quad R_n^{2}\,2^{4n^2}\le 2^{2n+4},\qquad R_n=3T\,Q_x(n)-A_q(n),\quad T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$$
 
-The statement is about three functions $Q_c:\mathbb N\times\mathbb N\to\mathbb Q$, $Q_x:\mathbb N\to\mathbb Q$ and $A_q:\mathbb N\to\mathbb Q$. It takes as hypotheses that they are given by the following formulas, so these are the only functions it speaks about:
-
-- $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\displaystyle\prod_{i=0}^{k-1}\frac{2^{n-i}-1}{2^{i+1}-1}\;\prod_{i=0}^{n-1}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$;
-- $Q_x(n)=\displaystyle\sum_{k=0}^{n}Q_c(n,k)\,\Big(\frac{3}{2^n}\Big)^k$;
-- $A_q(n)=\displaystyle\sum_{k=0}^{n}\Big(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\Big)\Big(\frac{3}{2^n}\Big)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$.
-
-Exponents are natural numbers: subtraction in them is truncated at $0$, and $k(k-1)/2$ is natural-number division (with $k-1=0$ when $k=0$). Where these functions are used below, $k\le n$, so no other exponent is truncated. Division is Lean's division of rationals, in which $x/0=0$: the term $j=k$ of the inner sum in $A_q$ has denominator $2^0-1=0$ and so contributes $0$.
-
-Let $T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$, a real number (Lean's `tsum` over all $m\in\mathbb N$, which would be $0$ for a series that does not converge; this one converges). With $x_n=3/2^n$, $Q_x(n)$ is the polynomial $\sum_k Q_c(n,k)X^k$ evaluated at $x_n$. For each $n$ write, in the real numbers,
-$$R_n=3T\,Q_x(n)-A_q(n).$$
-
-The conclusion: for every natural number $n$,
-
-- $R_n>0$, and
-- $R_n^2\cdot 2^{4n^2}\le 2^{2n+4}$, that is, $R_n\le 2^{\,n+2-2n^2}$.
-
-There are no other hypotheses. The doc comment also says the bound was checked numerically for $n\le 30$ and is tightest at $n=1$; that remark is a comment, not part of what Lean checks.
+For every natural number $n$ the real number $R_n$ is positive and $R_n^2\cdot 2^{4n\cdot n}\le 2^{2n+4}$ (that is, $R_n\le 2^{\,n+2-2n^2}$), where $T$ is Lean's `∑'` over all $m\in\mathbb{N}$ in $\mathbb{R}$ (which would be $0$ for a non-summable family; this one converges) and $Q_x(n)$, $A_q(n)$ are read as reals; with $x_n=3/2^n$, $Q_x(n)$ is the polynomial $\sum_k Q_c(n,k)X^k$ at $x_n$. The functions $Q_c:\mathbb{N}\times\mathbb{N}\to\mathbb{Q}$, $Q_x:\mathbb{N}\to\mathbb{Q}$ and $A_q:\mathbb{N}\to\mathbb{Q}$ are fixed by hypotheses as $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\prod_{i=0}^{k-1}\frac{2^{n-i}-1}{2^{i+1}-1}\prod_{i=0}^{n-1}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$, $Q_x(n)=\sum_{k=0}^{n}Q_c(n,k)\,(3/2^n)^k$ and $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. Exponents are natural numbers (subtraction in them stops at $0$, $k(k-1)/2$ is natural-number division), division is in $\mathbb{Q}$ with $x/0=0$ so the $j=k$ term of the inner sum of $A_q$ is $0$, and where these functions are used $k\le n$, so no other exponent is truncated; there are no other hypotheses, and the doc comment's numerical check for $n\le 30$ is a comment, not part of what Lean checks.

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/spec-180d8b72/Statement.lean. Origin: authored.

Attestation: pass, 7 of 7 checks

Checked by the kernel — the gate’s record of the run: attestations/000327.json, merge commit dfc7b5cc45a9.

Verdict pass, from attestations/000327.json.

Submitter: t1001-1050a
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/000327.json, merge commit dfc7b5cc45a9.

Witness.lean — checked at step 7 of the run recorded below: pass. Rendered from targets/erdos-1050/nodes/spec-180d8b72/Witness.lean, sha256 b97324b1a56e.

import Mathlib

theorem witness : ∃ (Qc : ℕ → ℕ → ℚ) (Qx : ℕ → ℚ) (Aq : ℕ → ℚ) (n : ℕ),
  (Qc = fun (n k : ℕ) =>
      ((-1 : ℚ) ^ k * (2 : ℚ) ^ (k * (k - (1 : ℕ)) / (2 : ℕ)) *
          ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) *
        ∏ i ∈ Finset.range n, ((2 : ℚ) ^ ((2 : ℕ) * n - k - i) - (1 : ℚ)) / ((2 : ℚ) ^ (i + (1 : ℕ)) - (1 : ℚ))) ∧
    (Qx = fun (n : ℕ) => ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) ∧
      (Aq = fun (n : ℕ) =>
          ∑ k ∈ Finset.range (n + (1 : ℕ)),
              (∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k +
            Qx n * ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + (1 : ℕ)) - (3 : ℚ))) :=
  ⟨_, _, _, 0, rfl, rfl, rfl⟩

In words, unverified — what this witness says; written by t1006-w3, drafted with Claude Opus 5.5 (claude-opus-5-5) in a Claude Code agent session, 2026-10-06; no one has read it against the Lean and signed it.

Draft · drafted, by the model named above

This file imports only Mathlib. It states and proves a theorem named witness: there exist functions $Q_c:\mathbb N\times\mathbb N\to\mathbb Q$, $Q_x:\mathbb N\to\mathbb Q$, $A_q:\mathbb N\to\mathbb Q$ and a natural number $n$ such that

$Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\prod_{i<k}\frac{2^{n-i}-1}{2^{i+1}-1}\prod_{i<n}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$ for all $n,k$,
$Q_x(n)=\sum_{k=0}^{n}Q_c(n,k)(3/2^n)^k$ for all $n$, and
$A_q(n)=\sum_{k=0}^{n}\big(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\big)(3/2^n)^k+Q_x(n)\sum_{j<n}\frac{3}{2^{j+1}-3}$ for all $n$.

These are the three hypotheses of the statement of this node (erdos_1050_pade_remainder_bound), written exactly as there (as equations between functions), so the theorem says those hypotheses can all hold at once. The existentially quantified $n$ appears in none of the three equations; the statement's own $n$ is quantified after the hypotheses and has no counterpart here.

The proof takes for $Q_c$, $Q_x$ and $A_q$ the functions the formulas define, and $n=0$; each equation then holds by definition (rfl).

Rendered from gloss/817faba6618a….md · Improve these words →

Acknowledged findings (13)