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Problems / erdos-1050 / erdos-1050--h1-v2--h4

Borwein's remainder bound (with the denominator hypothesis)

erdos-1050--h1-v2--h4

proved

Hole hrem of erdos-1050--h1-v2 (split by 20260924T123718Z-h0924-1050-http-partial.lean).

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

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. 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 $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. The statement carries one further hypothesis, which the conclusion does not use: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. 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.

unverified · history

Lean
theorem erdos_1050__h1_v2__h4 : ∀ (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),
                ∃ (d : ℕ),
                  (0 : ℕ) < d ∧
                    (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                      (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) →
              ∀ (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/d4f1735e26c2….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 proof never uses the statement's hypothesis about the denominators $d$: it introduces it and discards it (the _ in intro Qc hQc Qx hQx Aq hAq _ n).

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)

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 $2^0,2^1,\dots,2^n$.

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).

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.

  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

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.

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 $\mathbb R$.

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

  1. 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/d4f1735e26c2….md · the first version of its chain

    Read this version

    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 proof never uses the statement's hypothesis about the denominators $d$: it introduces it and discards it (the _ in intro Qc hQc Qx hQx Aq hAq _ n).

    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, 000320, records the gate’s run.

The proof step by step (outline)

Checked by the kernel — an outline of the attested proof below: attestations/000320.json, merge commit df09144aaebf.

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.

  1. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. .

    Lean, lines 35–46
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 35–46 at df09144aaebf ↗

    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 Zero, the smallest natural number. .

      Lean, line 40
          | zero => simp

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 40 at df09144aaebf ↗

    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 The successor of a natural number n. , Finset.prod_range_succ.

      Lean, lines 41–46
          | 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/erdos-1050--h1-v2--h4/Proof.lean, lines 41–46 at df09144aaebf ↗

      1. 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 42
              have ih' := ih (by omega)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 42 at df09144aaebf ↗

      2. 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 Existential introduction. .

        Lean, line 44
              obtain ⟨m, hm⟩ : ∃ m, n - j = m + 1 := ⟨n - j - 1, by omega⟩

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 44 at df09144aaebf ↗

  2. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , Finset.prod_congr, rfl rfl : a = a is the unique constructor of the equality type. , Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, Finset.prod_range_reflect.

    Lean, lines 48–62
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 48–62 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , mul_sub Alias of mul_sub_left_distrib. , pow_add.

      Lean, lines 52–56
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 52–56 at df09144aaebf ↗

      1. 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 54
              have hi' := Finset.mem_range.mp hi

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 54 at df09144aaebf ↗

    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 Gauss' summation formula .

      Lean, line 61
          have := Finset.sum_range_id_mul_two j

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 61 at df09144aaebf ↗

  3. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.range range n is the set of natural numbers less than n. , Finset.prod_Ico_eq_prod_range, Finset.prod_congr, rfl rfl : a = a is the unique constructor of the equality type. , Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, pow_mul.

    Lean, lines 64–76
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 64–76 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.Ico The finset $[a, b)$ of elements x such that a ≤ x and x < b. , Finset.mem_Ioc, Finset.mem_Ico.

      Lean, lines 68–69
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 68–69 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , pow_add.

      Lean, lines 71–74
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 71–74 at df09144aaebf ↗

  4. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.range range n is the set of natural numbers less than n. , Finset.prod_Ico_eq_prod_range, Finset.prod_congr, rfl rfl : a = a is the unique constructor of the equality type. , Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, pow_mul, Finset.prod_range_add.

    Lean, lines 78–94
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 78–94 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.Ico The finset $[a, b)$ of elements x such that a ≤ x and x < b. , Finset.mem_Ioc, Finset.mem_Ico.

      Lean, lines 83–84
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 83–84 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , pow_add.

      Lean, lines 86–90
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 86–90 at df09144aaebf ↗

  5. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.prod_union, mul_right_cancel₀, pow_add.

    Lean, lines 96–139
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 96–139 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_ne_zero_iff.

      Lean, lines 103–107
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 103–107 at df09144aaebf ↗

      1. 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 106
              have : (1 : ℚ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 106 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_div_distrib.

      Lean, lines 108–112
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 108–112 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_div_distrib.

      Lean, lines 113–117
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 113–117 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc.

      Lean, lines 118–119
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 118–119 at df09144aaebf ↗

    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 Two elements of a lattice are disjoint if their inf is the bottom element. , Finset.range range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.disjoint_left, Finset.mem_range, Finset.mem_Ioc.

      Lean, lines 120–124
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 120–124 at df09144aaebf ↗

    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 125
          have hR := rhs_test j n hj

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 125 at df09144aaebf ↗

    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 126
          obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hj

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 126 at df09144aaebf ↗

    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 127–128
          have hsign : (-1 : ℚ) ^ (j + m) = (-1) ^ j * (-1) ^ (j + m - j) := by
      rw [← pow_add]; congr 1; omega

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 127–128 at df09144aaebf ↗

    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 129
          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/erdos-1050--h1-v2--h4/Proof.lean, line 129 at df09144aaebf ↗

    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 130–135
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 130–135 at df09144aaebf ↗

      1. 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 133
              · simp

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 133 at df09144aaebf ↗

      2. 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 134–135
              · simp only [Nat.add_sub_cancel] at ha ⊢
        nlinarith [ha]

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 134–135 at df09144aaebf ↗

  6. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , Polynomial.C C a is the constant polynomial a. , Polynomial.X X is the polynomial variable (aka indeterminate). , Polynomial.eq_of_degree_sub_lt_of_eval_finset_eq.

    Lean, lines 142–175
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 142–175 at df09144aaebf ↗

    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 f is injective on s if the restriction of f to s is injective. , Finset.range range n is the set of natural numbers less than n. , Set A set is a collection of elements of some type α. , Finset.coe_range, Set.mem_Iio.

      Lean, lines 147–150
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 147–150 at df09144aaebf ↗

    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 152–165
          · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 152–165 at df09144aaebf ↗

      1. 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 154
              · exact lt_of_le_of_lt hNdeg (by exact_mod_cast Nat.lt_succ_self n)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 154 at df09144aaebf ↗

      2. 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 155–165
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 155–165 at df09144aaebf ↗

        1. 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 range n is the set of natural numbers less than n. , Polynomial.X X is the polynomial variable (aka indeterminate). , Polynomial.C C a is the constant polynomial a. , WithBot Attach ⊥ to a type. , Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , Polynomial.degree_X_sub_C, Finset.sum_const, Finset.card_erase_of_mem $\#(s \setminus \{a\}) = \#s - 1$ if $a \in s$. , Finset.card_range, Nat.add_sub_cancel.

          Lean, lines 160–163
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 160–163 at df09144aaebf ↗

    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 Alias of Polynomial.eval_finsetSum. , Finset.sum_eq_single.

      Lean, lines 166–175
          · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 166–175 at df09144aaebf ↗

      1. 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 167
              obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hy

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 167 at df09144aaebf ↗

      2. 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 168
              have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 168 at df09144aaebf ↗

        1. 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 168
                have hk' : k ≤ n := by have := Finset.mem_range.mp hk; omega

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 168 at df09144aaebf ↗

      3. 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 Polynomial evaluation commutes with Finset.prod .

        Lean, line 170
              · simp [Polynomial.eval_prod]

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 170 at df09144aaebf ↗

      4. 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 evaluation commutes with Finset.prod , Polynomial.eval_sub, Polynomial.eval_X, Finset.prod_eq_zero, Finset.mem_erase, And.intro And.intro : a → b → a ∧ b is the constructor for the And operation. , Ne.symm, MulZeroClass.mul_zero Zero is a right absorbing element for multiplication .

        Lean, lines 171–174
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 171–174 at df09144aaebf ↗

      5. 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 Anything follows from two contradictory hypotheses. .

        Lean, line 175
              · intro h; exact absurd hk h

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 175 at df09144aaebf ↗

  7. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Polynomial.X X is the polynomial variable (aka indeterminate). , Polynomial.C C a is the constant polynomial a. , Polynomial.eval_finset_sum Alias of Polynomial.eval_finsetSum. , Finset.sum_div, Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , Polynomial.eval_mul, Polynomial.eval_C, Polynomial.eval_prod Polynomial evaluation commutes with Finset.prod , Polynomial.eval_sub, Polynomial.eval_X, Finset.mul_prod_erase Taking a product over s : Finset ι is the same as multiplying the value on a single element f a by the product of s.erase a. .

    Lean, lines 177–229
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 177–229 at df09144aaebf ↗

    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 The relation ≤ on a preorder is reflexive. .

      Lean, lines 184–187
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 184–187 at df09144aaebf ↗

      1. tail_coeff.hxinj.h' have (2 : ℚ) ^ i = (2 : ℚ) ^ j

        Leaves i = j, with h' : (2 : ℚ) ^ i = (2 : ℚ) ^ j.

        Lean, line 186
              have h' : (2 : ℚ) ^ i = 2 ^ j := h

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 186 at df09144aaebf ↗

    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 The relation ≤ on a preorder is reflexive. , sub_eq_zero.

      Lean, lines 188–191
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 188–191 at df09144aaebf ↗

    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 The degree of a product of polynomials is equal to the sum of the degrees, where the degree of the zero polynomial is ⊥. , Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , Polynomial.degree_X_sub_C, Finset.sum_const, Nat.card_Ioc.

      Lean, lines 193–198
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 193–198 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Polynomial.eval_prod Polynomial evaluation commutes with Finset.prod , Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.

      Lean, lines 199–203
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 199–203 at df09144aaebf ↗

    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 204
          have hL := lagr_eval n x hxinj N hNdeg (fun j => Qc n j * 2 ^ j) hr

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 204 at df09144aaebf ↗

    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 Congruence in the function argument: if a₁ = a₂ then f a₁ = f a₂ for any (nondependent) function f. , Polynomial.eval eval x p is the evaluation of the polynomial p at x .

      Lean, line 205
          have hev := congrArg (Polynomial.eval ((2 : ℚ) ^ K)) hL

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 205 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Polynomial.eval_prod Polynomial evaluation commutes with Finset.prod , Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C.

      Lean, lines 207–209
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 207–209 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_ne_zero_iff.

      Lean, lines 210–211
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 210–211 at df09144aaebf ↗

      1. 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 211
              Finset.prod_ne_zero_iff.mpr (fun i hi => hxne K i (by have := Finset.mem_range.mp hi; omega))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 211 at df09144aaebf ↗

    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 215
          have hjn : j < n + 1 := Finset.mem_range.mp hj

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 215 at df09144aaebf ↗

    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 216
          have hne1 : (2 : ℚ) ^ K - 2 ^ j ≠ 0 := hxne K j (by omega)

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 216 at df09144aaebf ↗

    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 217–219
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 217–219 at df09144aaebf ↗

      1. 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 218
              have : (1 : ℚ) < 2 ^ (K - j) := one_lt_pow₀ (by norm_num) (by omega)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 218 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_ne_zero_iff.

      Lean, lines 223–225
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 223–225 at df09144aaebf ↗

      1. 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 225
                have := Finset.mem_range.mp (Finset.mem_of_mem_erase hi); omega))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 225 at df09144aaebf ↗

    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 226
          have hpow : (2 : ℚ) ^ K = 2 ^ j * 2 ^ (K - j) := by rw [← pow_add]; congr 1; omega

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 226 at df09144aaebf ↗

  8. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 232–296
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 232–296 at df09144aaebf ↗

    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 236–242
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 236–242 at df09144aaebf ↗

    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 243
          have hpos : ∀ m s, 0 ≤ f m s := fun m s => by rw [key]; positivity

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 243 at df09144aaebf ↗

    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 244–247
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 244–247 at df09144aaebf ↗

      1. tail3.h4.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num

        Closes the goal.

        Lean, line 246
              calc (4 : ℝ) = 2 ^ 2 := by norm_num

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 246 at df09144aaebf ↗

      2. 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 247
                _ ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 247 at df09144aaebf ↗

    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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. .

      Lean, lines 248–262
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 248–262 at df09144aaebf ↗

      1. 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 250
              have hr0 : 0 ≤ (3 : ℝ) / 2 ^ (m + n + 1) := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 250 at df09144aaebf ↗

      2. 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 251–252
              have hr1 : (3 : ℝ) / 2 ^ (m + n + 1) < 1 := by
        rw [div_lt_one (by positivity)]; linarith [h4 m]

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 251–252 at df09144aaebf ↗

      3. 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 253
              have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left ((3 : ℝ) / 2 ^ (m + n + 1))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 253 at df09144aaebf ↗

      4. 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 254–255
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 254–255 at df09144aaebf ↗

      5. 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 256
              have hne : (2 : ℝ) ^ (m + n + 1) - 3 ≠ 0 := by linarith [h4 m]

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 256 at df09144aaebf ↗

      6. 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 257
              have hne2 : (2 : ℝ) ^ (m + n + 1) ≠ 0 := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 257 at df09144aaebf ↗

      7. 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 258–260
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 258–260 at df09144aaebf ↗

    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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. .

      Lean, lines 263–279
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 263–279 at df09144aaebf ↗

      1. 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 265
              have hr0 : 0 ≤ (1 : ℝ) / 2 ^ (s + 1) := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 265 at df09144aaebf ↗

      2. 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 266–268
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 266–268 at df09144aaebf ↗

      3. 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 269
              have hg := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (((3 : ℝ) / 2 ^ (n + 1)) ^ (s + 1))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 269 at df09144aaebf ↗

      4. 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 270–272
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 270–272 at df09144aaebf ↗

        1. 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 271
                  have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 271 at df09144aaebf ↗

      5. 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 273–277
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 273–277 at df09144aaebf ↗

    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 f means that f has some (infinite) sum with respect to L. , Summable.of_nonneg_of_le Comparison test of convergence of series of non-negative real numbers. , summable_geometric_two.

      Lean, lines 280–288
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 280–288 at df09144aaebf ↗

      1. tail3.hsum1.s1 case

        Leaves (0 : ℝ) ≤ (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)).

        Uses library div_nonneg.

        Lean, line 282
              · have := h4 m; exact div_nonneg (by norm_num) (by linarith)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 282 at df09144aaebf ↗

        1. 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 282
                · have := h4 m; exact div_nonneg (by norm_num) (by linarith)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 282 at df09144aaebf ↗

      2. tail3.hsum1.s2 case

        Leaves (3 : ℝ) / ((2 : ℝ) ^ (m + n + (1 : ℕ)) - (3 : ℝ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m.

        Lean, lines 283–288
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 283–288 at df09144aaebf ↗

        1. 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 283
                · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 283 at df09144aaebf ↗

        2. 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 284
                  have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 284 at df09144aaebf ↗

        3. 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 285
                  have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 285 at df09144aaebf ↗

        4. 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 286–287
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 286–287 at df09144aaebf ↗

        5. tail3.hsum1.s2.s5 calc

          Closes the goal.

          Uses library one_div_pow.

          Lean, line 288
                    _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 288 at df09144aaebf ↗

    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 Summable f means that f has some (infinite) sum with respect to L. , Function.uncurry Transforms a two-parameter function into an equivalent function from pairs. , summable_prod_of_nonneg, And.intro And.intro : a → b → a ∧ b is the constructor for the And operation. .

      Lean, lines 289–291
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 289–291 at df09144aaebf ↗

    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 292
          have hcomm := hU.tsum_comm' (fun m => (inner1 m).summable) (fun s => (inner2 s).summable)

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 292 at df09144aaebf ↗

  9. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , zero_add.

    Lean, lines 298–336
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 298–336 at df09144aaebf ↗

    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 f means that f has some (infinite) sum with respect to L. , Summable.of_nonneg_of_le Comparison test of convergence of series of non-negative real numbers. , summable_geometric_two.

      Lean, lines 303–315
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 303–315 at df09144aaebf ↗

      1. split3T.hsum.s1 case

        Leaves (0 : ℝ) ≤ g (m + (1 : ℕ)).

        Uses library div_nonneg.

        Lean, lines 305–308
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 305–308 at df09144aaebf ↗

        1. 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 305–307
                · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 305–307 at df09144aaebf ↗

          1. split3T.hsum.s1.this.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num

            Closes the goal.

            Lean, line 306
                      calc (4 : ℝ) = 2 ^ 2 := by norm_num

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 306 at df09144aaebf ↗

          2. 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 307
                        _ ≤ 2 ^ (m + 1 + 1) := pow_le_pow_right₀ (by norm_num) (by omega)

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 307 at df09144aaebf ↗

      2. split3T.hsum.s2 case

        Leaves g (m + (1 : ℕ)) ≤ (3 : ℝ) * (1 / 2 : ℝ) ^ m.

        Lean, lines 309–315
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 309–315 at df09144aaebf ↗

        1. 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 309
                · have hpow : (2 : ℝ) ^ (m + 2) ≤ 2 ^ (m + 1 + 1) := le_of_eq (by ring_nf)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 309 at df09144aaebf ↗

        2. 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 310
                  have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 310 at df09144aaebf ↗

        3. 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 311
                  have h2 : (2 : ℝ) ^ (m + 2) = 4 * 2 ^ m := by rw [pow_add]; ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 311 at df09144aaebf ↗

        4. 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 313–314
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 313–314 at df09144aaebf ↗

        5. split3T.hsum.s2.s5 calc

          Closes the goal.

          Uses library one_div_pow.

          Lean, line 315
                    _ = 3 * (1 / 2) ^ m := by rw [one_div_pow]; ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 315 at df09144aaebf ↗

    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 f means that f has some (infinite) sum with respect to L. , summable_nat_add_iff.

      Lean, line 316
          have hs : Summable g := (summable_nat_add_iff 1).mp hsum

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 316 at df09144aaebf ↗

    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 317
          have e1 := hs.sum_add_tsum_nat_add n

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 317 at df09144aaebf ↗

    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 318
          have e2 := hs.sum_add_tsum_nat_add 2

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 318 at df09144aaebf ↗

    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 319–324
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 319–324 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.sum_range_succ, Finset.sum_range_zero.

      Lean, lines 325–327
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 325–327 at df09144aaebf ↗

    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 329–331
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 329–331 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. .

      Lean, lines 333–334
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 333–334 at df09144aaebf ↗

  10. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. , Finset.range range n is the set of natural numbers less than n. , Finset.sum_mul, hasSum_sum, hasSum_nat_add_iff', sub_zero.

    Lean, lines 338–358
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 338–358 at df09144aaebf ↗

    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 345
          have h := hA.mul_left (q j * x ^ j)

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 345 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.sum_eq_zero, if_neg.

      Lean, lines 347–350
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 347–350 at df09144aaebf ↗

      1. 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 349
                have := Finset.mem_range.mp hi

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 349 at df09144aaebf ↗

    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 352–356
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 352–356 at df09144aaebf ↗

  11. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , HasSum HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. .

    Lean, lines 360–479
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 360–479 at df09144aaebf ↗

    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 375
          have hx0 : 0 ≤ x := by positivity

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 375 at df09144aaebf ↗

    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 376–380
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 376–380 at df09144aaebf ↗

      1. 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 377–379
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 377–379 at df09144aaebf ↗

        1. h4_series.hx2.this.s1 calc (2 : ℝ) = (2 : ℝ) ^ (1 : ℕ) routine: norm_num

          Closes the goal.

          Lean, line 378
                  calc (2 : ℝ) = 2 ^ 1 := by norm_num

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 378 at df09144aaebf ↗

        2. h4_series.hx2.this.s2 calc (2 : ℝ) ^ (1 : ℕ) ≤ (2 : ℝ) ^ n

          Closes the goal.

          Uses library pow_le_pow_right₀.

          Lean, line 379
                    _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 379 at df09144aaebf ↗

    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 382–384
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 382–384 at df09144aaebf ↗

      1. 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 383
              have : (1 : ℝ) < 2 ^ (s + 1) := one_lt_pow₀ (by norm_num) (by omega)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 383 at df09144aaebf ↗

    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 f means that f has some (infinite) sum with respect to L. , Summable.of_nonneg_of_le Comparison test of convergence of series of non-negative real numbers. , summable_geometric_of_lt_one.

      Lean, lines 385–398
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 385–398 at df09144aaebf ↗

      1. h4_series.hAsum.s1 case

        Leaves (0 : ℝ) ≤ a s * x ^ (s + (1 : ℕ)).

        Uses library mul_nonneg Alias of Left.mul_nonneg. , div_nonneg, le_of_lt.

        Lean, line 387
              · simp only [hadef]; exact mul_nonneg (div_nonneg (by norm_num) (le_of_lt (hapos s))) (by positivity)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 387 at df09144aaebf ↗

      2. h4_series.hAsum.s2 case

        Leaves a s * x ^ (s + (1 : ℕ)) ≤ x * (x / (2 : ℝ)) ^ s.

        Lean, lines 388–398
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 388–398 at df09144aaebf ↗

        1. 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 389–391
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 389–391 at df09144aaebf ↗

          1. 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 390
                      have : (1 : ℝ) ≤ 2 ^ s := one_le_pow₀ (by norm_num)

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 390 at df09144aaebf ↗

        2. 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 392
                  have hxs : 0 ≤ x ^ (s + 1) := by positivity

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 392 at df09144aaebf ↗

        3. 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 393–394
                  calc 1 / (2 ^ (s + 1) - 1) * x ^ (s + 1) ≤ 1 / 2 ^ s * x ^ (s + 1) := by
              gcongr

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 393–394 at df09144aaebf ↗

        4. h4_series.hAsum.s2.s4 calc

          Closes the goal.

          Lean, lines 395–398
                    _ = 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/erdos-1050--h1-v2--h4/Proof.lean, lines 395–398 at df09144aaebf ↗

          1. 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 396–397
                        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/erdos-1050--h1-v2--h4/Proof.lean, lines 396–397 at df09144aaebf ↗

    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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. .

      Lean, line 400
          have hA : HasSum (fun s => a s * x ^ (s + 1)) A := hAsum.hasSum

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 400 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , tsum_congr.

      Lean, lines 402–407
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 402–407 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. .

      Lean, lines 408–409
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 408–409 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. .

      Lean, lines 410–413
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 410–413 at df09144aaebf ↗

    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 414
          have hC := cauchy_fin n (fun j => (Qc n j : ℝ)) a x A hA

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 414 at df09144aaebf ↗

    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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. , Finset.range range n is the set of natural numbers less than n. .

      Lean, lines 417–429
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 417–429 at df09144aaebf ↗

      1. 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 HasSum f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. , Finset.range range n is the set of natural numbers less than n. , hasSum_sum_of_ne_finset_zero If a function f vanishes outside of a finite set s, then it HasSum ∑ b ∈ s, f b. , if_neg.

        Lean, lines 419–423
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 419–423 at df09144aaebf ↗

      2. 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 range n is the set of natural numbers less than n. , Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , if_pos, Finset.mem_range.

        Lean, lines 424–427
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 424–427 at df09144aaebf ↗

    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 430
          have hD := hC.sub hP

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 430 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. .

      Lean, lines 431–433
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 431–433 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. .

      Lean, lines 435–436
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 435–436 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.range range n is the set of natural numbers less than n. .

      Lean, lines 437–477
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 437–477 at df09144aaebf ↗

      1. 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 445–459
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 445–459 at df09144aaebf ↗

        1. 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 range n is the set of natural numbers less than n. , Finset.sum_range_add_sum_Ico, Finset.sum_eq_zero, Finset.Ico The finset $[a, b)$ of elements x such that a ≤ x and x < b. , if_neg, add_zero, Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , if_pos.

          Lean, lines 448–458
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 448–458 at df09144aaebf ↗

          1. 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 452
                        rw [if_neg (by have := Finset.mem_Ico.mp hj; omega)]), add_zero]

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 452 at df09144aaebf ↗

          2. 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 454
                      have hjK := Finset.mem_range.mp hj

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 454 at df09144aaebf ↗

      2. 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 460–477
              · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 460–477 at df09144aaebf ↗

        1. 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 range n is the set of natural numbers less than n. , Finset.sum_mul, Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , if_pos.

          Lean, lines 461–470
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 461–470 at df09144aaebf ↗

          1. 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 466
                      have hjK : j < K := by have := Finset.mem_range.mp hj; omega

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 466 at df09144aaebf ↗

            1. 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 466
                        have hjK : j < K := by have := Finset.mem_range.mp hj; omega

              targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 466 at df09144aaebf ↗

        2. 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 474
                  · rw [if_pos h2]

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 474 at df09144aaebf ↗

        3. 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 And.intro : a → b → a ∧ b is the constructor for the And operation. , sub_self.

          Lean, lines 475–477
                  · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 475–477 at df09144aaebf ↗

  12. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 α is the type of finite sets of elements of α. , Finset.induction_on To prove a proposition about an arbitrary Finset α, it suffices to prove it for the empty Finset, and to show that if it holds for some Finset α, then it holds for the Finset obtained by inserting a new element. .

    Lean, lines 482–497
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 482–497 at df09144aaebf ↗

    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/erdos-1050--h1-v2--h4/Proof.lean, line 486 at df09144aaebf ↗

    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 487–497
          | 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/erdos-1050--h1-v2--h4/Proof.lean, lines 487–497 at df09144aaebf ↗

      1. 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 489
              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/erdos-1050--h1-v2--h4/Proof.lean, line 489 at df09144aaebf ↗

      2. 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 490
              have haj0 := h0 j (Finset.mem_insert_self j s)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 490 at df09144aaebf ↗

      3. 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 491
              have haj1 := h1 j (Finset.mem_insert_self j s)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 491 at df09144aaebf ↗

      4. 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 492–493
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 492–493 at df09144aaebf ↗

      5. 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 If each f i, i ∈ s belongs to [0, 1], then their product is less than or equal to one. , Finset.mem_insert_of_mem.

        Lean, lines 494–496
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 494–496 at df09144aaebf ↗

  13. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.range range n is the set of natural numbers less than n. , And.intro And.intro : a → b → a ∧ b is the constructor for the And operation. , div_pos, div_le_iff₀ See div_le_iff₀' for a version with multiplication on the other side. .

    Lean, lines 499–554
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 499–554 at df09144aaebf ↗

    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 504–506
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 504–506 at df09144aaebf ↗

      1. 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 505
              have : (2 : ℝ) ^ i < 2 ^ K := pow_lt_pow_right₀ (by norm_num) hi

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 505 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.prod_pos.

      Lean, lines 507–508
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 507–508 at df09144aaebf ↗

      1. 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 508
              Finset.prod_pos (fun L hL => hlt L (by have := Finset.mem_Ioc.mp hL; omega))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 508 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_pos.

      Lean, lines 509–510
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 509–510 at df09144aaebf ↗

      1. 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 510
              Finset.prod_pos (fun i hi => hlt i (by have := Finset.mem_range.mp hi; omega))

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 510 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.prod_const, Nat.card_Ioc.

      Lean, lines 512–518
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 512–518 at df09144aaebf ↗

      1. 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 If all f i, i ∈ s, are nonnegative and each f i is less than or equal to g i, then the product of f i is less than or equal to the product of g i. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , le_of_lt.

        Lean, lines 513–516
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 513–516 at df09144aaebf ↗

        1. 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 514
                  (fun L hL => le_of_lt (hlt L (by have := Finset.mem_Ioc.mp hL; omega))) (fun L _ => by

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 514 at df09144aaebf ↗

        2. 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 515
                    have : (0 : ℝ) < 2 ^ L := by positivity

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 515 at df09144aaebf ↗

    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 range n is the set of natural numbers less than n. , Finset.prod_congr, rfl rfl : a = a is the unique constructor of the equality type. , Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range.

      Lean, lines 519–545
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 519–545 at df09144aaebf ↗

      1. 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 range n is the set of natural numbers less than n. , mul_sub Alias of mul_sub_left_distrib. , mul_one, one_div_pow, mul_one_div.

        Lean, lines 520–525
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 520–525 at df09144aaebf ↗

        1. 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 522
                  have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 522 at df09144aaebf ↗

          1. 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 522
                    have hiK : i ≤ K := by have := Finset.mem_range.mp hi; omega

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 522 at df09144aaebf ↗

        2. 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 523
                  have e : (2 : ℝ) ^ K = 2 ^ i * 2 ^ (K - i) := by rw [← pow_add]; congr 1; omega

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 523 at df09144aaebf ↗

      2. 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 range n is the set of natural numbers less than n. , Finset.sum_congr, rfl rfl : a = a is the unique constructor of the equality type. , Finset.mul_sum.

        Lean, lines 527–541
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 527–541 at df09144aaebf ↗

        1. 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 range n is the set of natural numbers less than n. , pow_add.

          Lean, lines 528–531
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 528–531 at df09144aaebf ↗

          1. 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 530
                      have := Finset.mem_range.mp hi

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 530 at df09144aaebf ↗

        2. 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 range n is the set of natural numbers less than n. , Nat.add_sub_cancel, sum_geometric_two_le.

          Lean, lines 533–537
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 533–537 at df09144aaebf ↗

          1. 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 534
                      have hr := Finset.sum_range_reflect (fun i => ((1 : ℝ) / 2) ^ i) (n + 1)

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 534 at df09144aaebf ↗

        3. 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 538–539
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 538–539 at df09144aaebf ↗

        4. 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 540
                  have hpos : 0 ≤ ((1 : ℝ) / 2) ^ (K - n) := by positivity

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 540 at df09144aaebf ↗

      3. 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 range n is the set of natural numbers less than n. , pow_le_one₀.

        Lean, lines 542–543
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 542–543 at df09144aaebf ↗

      4. 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 544
              have h2K : (0 : ℝ) < (2 ^ K) ^ (n + 1) := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 544 at df09144aaebf ↗

    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 547
          have h2K : (2 : ℝ) ^ K * (1 / 2) ^ K = 1 := by rw [← mul_pow]; norm_num

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 547 at df09144aaebf ↗

    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 548–551
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 548–551 at df09144aaebf ↗

      1. 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 550
              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/erdos-1050--h1-v2--h4/Proof.lean, line 550 at df09144aaebf ↗

      2. nd_bound.e2.s2 calc

        Closes the goal.

        Uses library one_mul.

        Lean, line 551
                _ = (2 ^ K) ^ n := by rw [h2K, one_mul]

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 551 at df09144aaebf ↗

    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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. .

      Lean, line 552
          calc ∏ L ∈ Finset.Ioc n (2 * n), ((2 : ℝ) ^ K - 2 ^ L) ≤ (2 ^ K) ^ n := hN

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 552 at df09144aaebf ↗

    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 553
            _ = 2 * (1 / 2) ^ K * ((2 ^ K) ^ (n + 1) / 2) := e2.symm

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 553 at df09144aaebf ↗

    10. nd_bound.s10 calc

      Closes the goal.

      Uses library Finset.range range n is the set of natural numbers less than n. .

      Lean, line 554
            _ ≤ 2 * (1 / 2) ^ K * ∏ i ∈ Finset.range (n + 1), ((2 : ℝ) ^ K - 2 ^ i) := by gcongr

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 554 at df09144aaebf ↗

  14. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 557–567
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 557–567 at df09144aaebf ↗

    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 Zero, the smallest natural number. .

      Lean, line 561
          | zero => omega

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 561 at df09144aaebf ↗

    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 The successor of a natural number n. .

      Lean, lines 562–567
          | 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/erdos-1050--h1-v2--h4/Proof.lean, lines 562–567 at df09144aaebf ↗

      1. 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 564
              · norm_num

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 564 at df09144aaebf ↗

      2. ineq_pow.s2.s2 case

        Leaves (9 : ℕ) * (81 : ℕ) ^ (m + (1 : ℕ)) ≤ (256 : ℕ) ^ (m + (1 : ℕ)).

        Uses library pow_succ.

        Lean, lines 565–567
              · have := ih hm
        rw [pow_succ, pow_succ]
        omega

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 565–567 at df09144aaebf ↗

        1. 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 565
                · have := ih hm

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 565 at df09144aaebf ↗

  15. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 range n is the set of natural numbers less than n. , Finset.Ioc The finset $(a, b]$ of elements x such that a < x and x ≤ b. , And.intro And.intro : a → b → a ∧ b is the constructor for the And operation. .

    Lean, lines 569–662
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 569–662 at df09144aaebf ↗

    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 582
          have hS := h4_series Qc hQc Qx hQx Aq hAq n (by omega)

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 582 at df09144aaebf ↗

    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 585
          have hx0 : 0 < x := by positivity

      targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 585 at df09144aaebf ↗

    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 586–590
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 586–590 at df09144aaebf ↗

      1. 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 587–589
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 587–589 at df09144aaebf ↗

        1. h4_big.hx2.this.s1 calc (4 : ℝ) = (2 : ℝ) ^ (2 : ℕ) routine: norm_num

          Closes the goal.

          Lean, line 588
                  calc (4 : ℝ) = 2 ^ 2 := by norm_num

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 588 at df09144aaebf ↗

        2. h4_big.hx2.this.s2 calc (2 : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ n

          Closes the goal.

          Uses library pow_le_pow_right₀.

          Lean, line 589
                    _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 589 at df09144aaebf ↗

    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 594–598
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 594–598 at df09144aaebf ↗

      1. 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 Alias of Left.mul_nonneg. , le_of_lt.

        Lean, line 597
              · exact mul_nonneg (le_of_lt (nd_bound n K (by omega) (by omega)).1) (by positivity)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 597 at df09144aaebf ↗

      2. h4_big.hGnn.s2 case

        Leaves (0 : ℝ) ≤ (0 : ℝ).

        Uses library le_refl The relation ≤ on a preorder is reflexive. .

        Lean, line 598
              · exact le_refl 0

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 598 at df09144aaebf ↗

    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 Alias of Left.mul_pos. .

      Lean, lines 599–601
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 599–601 at df09144aaebf ↗

    6. h4_big.s6 case

      Leaves (0 : ℝ) < r.

      Uses library hasSum_lt, hasSum_zero Constant zero function has sum 0 .

      Lean, line 603
          · 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/erdos-1050--h1-v2--h4/Proof.lean, line 603 at df09144aaebf ↗

    7. h4_big.s7 case

      Leaves r ^ (2 : ℕ) * (2 : ℝ) ^ ((4 : ℕ) * n * n) ≤ (2 : ℝ) ^ ((2 : ℕ) * n + (4 : ℕ)).

      Lean, lines 604–662
          · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 604–662 at df09144aaebf ↗

      1. 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 605–613
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 605–613 at df09144aaebf ↗

        1. 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 608–612
                  · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 608–612 at df09144aaebf ↗

          1. 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 608
                    · have hb := (nd_bound n K (by omega) (by omega)).2

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 608 at df09144aaebf ↗

          2. 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 The finset $(a, b]$ of elements x such that a < x and x ≤ b. , Finset.range range n is the set of natural numbers less than n. .

            Lean, lines 609–611
                      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/erdos-1050--h1-v2--h4/Proof.lean, lines 609–611 at df09144aaebf ↗

          3. h4_big.s7.hGH.s1.s3 calc routine: ring

            Closes the goal.

            Lean, line 612
                        _ = 2 * (x / 2) ^ K := by ring

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 612 at df09144aaebf ↗

        2. h4_big.s7.hGH.s2 case

          Leaves (0 : ℝ) ≤ (0 : ℝ).

          Uses library le_refl The relation ≤ on a preorder is reflexive. .

          Lean, line 613
                  · exact le_refl 0

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 613 at df09144aaebf ↗

      2. 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 614
              have hq : x / 2 < 1 := by linarith

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 614 at df09144aaebf ↗

      3. 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 f a L means that the (potentially infinite) sum of the f b for b : β converges to a along the SummationFilter L. , hasSum_nat_add_iff', sub_zero.

        Lean, lines 615–625
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 615–625 at df09144aaebf ↗

        1. 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 range n is the set of natural numbers less than n. , Finset.sum_eq_zero, if_neg.

          Lean, lines 617–619
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 617–619 at df09144aaebf ↗

          1. 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 619
                        simp only [hH]; rw [if_neg (by have := Finset.mem_range.mp hi; omega)])

            targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 619 at df09144aaebf ↗

        2. 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 621
                  have hgeo := (hasSum_geometric_of_lt_one (by positivity) hq).mul_left (2 * (x / 2) ^ (2 * n + 1))

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 621 at df09144aaebf ↗

        3. 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 622–623
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 622–623 at df09144aaebf ↗

      4. 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 626
              have hrle : r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hasSum_le hGH hS hHsum

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 626 at df09144aaebf ↗

      5. 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 Constant zero function has sum 0 .

        Lean, lines 627–628
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 627–628 at df09144aaebf ↗

      6. 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 629–635
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 629–635 at df09144aaebf ↗

        1. 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 See also inv_le_of_inv_le₀ for a one-sided implication with one fewer assumption. .

          Lean, lines 630–631
                  have hinv : (1 - x / 2)⁻¹ ≤ 4 := by
          rw [inv_le_comm₀ (by linarith) (by norm_num)]; linarith

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 630–631 at df09144aaebf ↗

        2. 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 632
                  have hp : 0 ≤ 2 * (x / 2) ^ (2 * n + 1) := by positivity

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 632 at df09144aaebf ↗

        3. 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 633
                  calc r ≤ 2 * (x / 2) ^ (2 * n + 1) * (1 - x / 2)⁻¹ := hrle

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 633 at df09144aaebf ↗

        4. h4_big.s7.hr8.s4 calc

          Closes the goal.

          Lean, line 634
                    _ ≤ 2 * (x / 2) ^ (2 * n + 1) * 4 := by gcongr

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 634 at df09144aaebf ↗

        5. h4_big.s7.hr8.s5 calc routine: ring

          Closes the goal.

          Lean, line 635
                    _ = 8 * (x / 2) ^ (2 * n + 1) := by ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 635 at df09144aaebf ↗

      7. 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 636
              have hsq : r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := pow_le_pow_left₀ hr0 hr8 2

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 636 at df09144aaebf ↗

      8. 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 637–646
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 637–646 at df09144aaebf ↗

        1. 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 638–641
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 638–641 at df09144aaebf ↗

        2. 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 642–643
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 642–643 at df09144aaebf ↗

        3. 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 645
                  have := ineq_pow n hn

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 645 at df09144aaebf ↗

      9. 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 647–648
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 647–648 at df09144aaebf ↗

      10. 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 649
              have hx2' : x / 2 = 3 / 2 ^ (n + 1) := by rw [hxdef, pow_succ]; field_simp

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 649 at df09144aaebf ↗

      11. 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 650–652
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 650–652 at df09144aaebf ↗

        1. 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 651
                  calc r ^ 2 ≤ (8 * (x / 2) ^ (2 * n + 1)) ^ 2 := hsq

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 651 at df09144aaebf ↗

        2. h4_big.s7.hsq'.s2 calc routine: ring

          Closes the goal.

          Lean, line 652
                    _ = 64 * (x / 2) ^ (4 * n + 2) := by ring

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 652 at df09144aaebf ↗

      12. 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 653–654
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 653–654 at df09144aaebf ↗

      13. 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 655
              have hP : (0 : ℝ) < ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 655 at df09144aaebf ↗

      14. 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 656–660
              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/erdos-1050--h1-v2--h4/Proof.lean, lines 656–660 at df09144aaebf ↗

        1. 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 658–659
                  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/erdos-1050--h1-v2--h4/Proof.lean, lines 658–659 at df09144aaebf ↗

        2. 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 660
                    _ ≤ 2 ^ (2 * n + 4) * ((2 : ℝ) ^ (n + 1)) ^ (4 * n + 2) := hreal

          targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 660 at df09144aaebf ↗

      15. 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 661
              calc r ^ 2 * (2 : ℝ) ^ (4 * n * n) ≤ 64 * (x / 2) ^ (4 * n + 2) * 2 ^ (4 * n * n) := by gcongr

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 661 at df09144aaebf ↗

      16. 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 662
                _ ≤ 2 ^ (2 * n + 4) := hmain

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 662 at df09144aaebf ↗

  16. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 Summable f means that f has some (infinite) sum with respect to L. , And.intro And.intro : a → b → a ∧ b is the constructor for the And operation. .

    Lean, lines 665–688
      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/erdos-1050--h1-v2--h4/Proof.lean, lines 665–688 at df09144aaebf ↗

    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 667–671
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 667–671 at df09144aaebf ↗

      1. 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 669
              have h1 : (1 : ℝ) ≤ 2 ^ m := one_le_pow₀ (by norm_num)

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 669 at df09144aaebf ↗

      2. 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 670
              have e : (2 : ℝ) ^ (m + 3) = 8 * 2 ^ m := by rw [pow_add]; norm_num; ring

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 670 at df09144aaebf ↗

    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 672–674
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 672–674 at df09144aaebf ↗

      1. 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 673
              have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 673 at df09144aaebf ↗

      2. 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 673
              have := h5 m; have : (0 : ℝ) < 2 ^ m := by positivity

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 673 at df09144aaebf ↗

    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 675–680
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 675–680 at df09144aaebf ↗

      1. 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 677–678
              have e : (1 / 5 : ℝ) * (1 / 2) ^ m = 1 / (5 * 2 ^ m) := by
        rw [one_div_pow]; field_simp

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 677–678 at df09144aaebf ↗

    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 f means that f has some (infinite) sum with respect to L. , Summable.of_nonneg_of_le Comparison test of convergence of series of non-negative real numbers. , le_of_lt, one_div_pos, summable_geometric_two.

      Lean, lines 681–683
          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/erdos-1050--h1-v2--h4/Proof.lean, lines 681–683 at df09144aaebf ↗

    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 685
          · 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/erdos-1050--h1-v2--h4/Proof.lean, line 685 at df09144aaebf ↗

    6. T_facts.s6 case

      Leaves ∑' (m : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ)) ≤ (2 / 5 : ℝ).

      Lean, lines 686–688
          · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 686–688 at df09144aaebf ↗

      1. 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 686–687
            · 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/erdos-1050--h1-v2--h4/Proof.lean, lines 686–687 at df09144aaebf ↗

      2. T_facts.s6.s2 calc

        Closes the goal.

        Uses library tsum_mul_left, tsum_geometric_two.

        Lean, line 688
                _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_two]; norm_num

        targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 688 at df09144aaebf ↗

  17. 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), ∃ (d : ℕ), (0 : ℕ) < d ∧ (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧ (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) → ∀ (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 689
      obtain ⟨_, hTpos, hTle⟩ := T_facts

    targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 689 at df09144aaebf ↗

  18. 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 693
        have hQx0 : Qx 0 = 1 := by rw [hQx, hQc]; simp

    targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 693 at df09144aaebf ↗

  19. 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 694
        have hAq0 : Aq 0 = 0 := by rw [hAq]; simp

    targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 694 at df09144aaebf ↗

  20. 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 699
        have e : ((1 : ℝ) * (3 * T) - 0) ^ 2 * (2 : ℝ) ^ (4 * 0 * 0) = 9 * (T * T) := by ring

    targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, line 699 at df09144aaebf ↗

  21. 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 705–706
        have hQx1 : Qx 1 = 3 / 2 := by
      rw [hQx, hQc]; norm_num [Finset.sum_range_succ, Finset.prod_range_succ]

    targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, lines 705–706 at df09144aaebf ↗

  22. 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 707–709
        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/erdos-1050--h1-v2--h4/Proof.lean, lines 707–709 at df09144aaebf ↗

The proof in Lean (Proof.lean)

Checked by the kernel — these bytes are the ones its attestation names: attestations/000320.json, merge commit df09144aaebf.

Proof.lean — its sha256 is the artifact_hash of the attestation below, so these are the bytes the gate checked. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Proof.lean, sha256 88ecdb43be99.
import Mathlib
import Nodes.«erdos-1050--h1-v2--h4».Context

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

theorem erdos_1050__h1_v2__h4 : ∀ (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),
                ∃ (d : ℕ),
                  (0 : ℕ) < d ∧
                    (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                      (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) →
              ∀ (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

Notes

Work on it

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

Record

Statement: the Lean file and its words (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.«erdos-1050--h1-v2--h4».Context

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

theorem erdos_1050__h1_v2__h4 : ∀ (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),
                ∃ (d : ℕ),
                  (0 : ℕ) < d ∧
                    (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                      (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) →
              ∀ (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

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 →

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

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

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. 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 $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. The statement carries one further hypothesis, which the conclusion does not use: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. 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.

Rendered from gloss/979179f29af3….md · Improve these words →

History: 2 versions on record

  1. 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/5da96e2ecab5….md · the first version of its chain

    Read this version

    This file states the theorem erdos_1050__h1_v2__h4, which the gate wrote from the hole hrem of a merged partial proof.

    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 $0$.

    The statement has one further hypothesis: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. This is word for word the conclusion of the node erdos-1050--h1-v2--h3. The conclusion below does not mention $d$.

    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). 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, $R_n\le 2^{\,n+2-2n^2}$.

  2. 2026-10-18 · written by f29-words, drafted with Claude Fable 5.1 (claude-fable-5-1) · not signed · current · gloss/979179f29af3….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. 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 $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. The statement carries one further hypothesis, which the conclusion does not use: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. 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.

Diff of 979179f29af3 against 5da96e2ecab5
@@ -1,19 +1,3 @@
-This file states the theorem `erdos_1050__h1_v2__h4`, which the gate wrote from the hole `hrem` of a merged partial proof.
+$$\forall n\in\mathbb{N}:\qquad R_n>0\quad\wedge\quad R_n^{2}\,2^{4n^2}\le 2^{2n+4},\qquad R_n=3T\,Q_x(n)-A_q(n),\quad T=\sum_{m=0}^{\infty}\frac{1}{2^{m+3}-3}$$
 
-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 $0$.
-
-The statement has one further hypothesis: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. This is word for word the conclusion of the node `erdos-1050--h1-v2--h3`. The conclusion below does not mention $d$.
-
-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). 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, $R_n\le 2^{\,n+2-2n^2}$.
+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. 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 $A_q(n)=\sum_{k=0}^{n}\bigl(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\bigr)(3/2^n)^k+Q_x(n)\sum_{j=0}^{n-1}\frac{3}{2^{j+1}-3}$. The statement carries one further hypothesis, which the conclusion does not use: for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers. 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.

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

Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Statement.lean. Origin: compiler-derived.

Attestation: pass, 7 of 7 checks

Checked by the kernel — the gate’s record of the run: attestations/000320.json, merge commit df09144aaebf.

Verdict pass, from attestations/000320.json.

Submitter
t0929-1
Toolchain
leanprover/lean4:v4.33.1 (hash f54d0bc916bf76a7fae460ce9d078ccd1f74aeb4fd12c4b6546bca28b6aa1c63)
Mathlib
0df444a360eaa60ab8c11dca51a86af692955474
Reviewer
none needed (calibration)
StepCheckResult
1toolchainpass
2pathspass
4kernel-replaypass
5axiomspass
6hazardspass
7witnesspass
8depspass
Witness

Checked by the kernel — step 7 passed in the run its attestation records: attestations/000320.json, merge commit df09144aaebf.

Witness.lean — checked at step 7 of the run recorded below: pass. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/Witness.lean, sha256 5a1e7eba2640.
import Mathlib
import Nodes.«erdos-1050--h1-v2--h4».Context

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))) ∧
        ∀ (n : ℕ),
          ∃ (d : ℕ),
            (0 : ℕ) < d ∧
              (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n := by
  have H : ∀ (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),
                    ∃ (d : ℕ),
                      (0 : ℕ) < d ∧
                        (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                          (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n := by
    -- annex: 4670684d2f0db4568cdaf74d8147b2ae959ae6b3136c3eb5460dbfc117de376f
    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
    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
    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]
    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
    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]
    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 deriv_prod_eval : ∀ (S : Finset ℕ) (x : ℕ → ℚ) (z : ℚ),
        (derivative (∏ i ∈ S, (X - C (x i)))).eval z = ∑ l ∈ S, ∏ i ∈ S.erase l, (z - x i) := by
      intro S x z
      rw [Polynomial.derivative_prod_finset, Polynomial.eval_finset_sum]
      refine Finset.sum_congr rfl (fun l _ => ?_)
      simp [Polynomial.eval_prod]
    open Polynomial in
    have deriv_prod_eval_node : ∀ (S : Finset ℕ) (x : ℕ → ℚ) (k : ℕ) (hk : k ∈ S),
        (derivative (∏ i ∈ S, (X - C (x i)))).eval (x k) = ∏ i ∈ S.erase k, (x k - x i) := by
      intro S x k hk
      rw [deriv_prod_eval, Finset.sum_eq_single k]
      · intro l hl hlk
        exact Finset.prod_eq_zero (i := k) (Finset.mem_erase.mpr ⟨Ne.symm hlk, hk⟩) (by ring)
      · intro h; exact absurd hk h
    open Polynomial in
    have lagr_deriv : ∀ (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))
      (k : ℕ) (hk : k ≤ n),
        (derivative N).eval (x k) =
          ∑ j ∈ (Finset.range (n + 1)).erase k, r j * ∏ i ∈ ((Finset.range (n + 1)).erase j).erase k, (x k - x i) +
          r k * ∑ l ∈ (Finset.range (n + 1)).erase k, ∏ i ∈ ((Finset.range (n + 1)).erase k).erase l, (x k - x i) := by
      intro n x hx N hNdeg r hr k hk
      have hkmem : k ∈ Finset.range (n + 1) := Finset.mem_range.mpr (by omega)
      conv_lhs => rw [lagr_eval n x hx N hNdeg r hr]
      rw [Polynomial.derivative_sum, Polynomial.eval_finset_sum, ← Finset.add_sum_erase _ _ hkmem, add_comm]
      congr 1
      · refine Finset.sum_congr rfl (fun j hj => ?_)
        have hjk : j ≠ k := Finset.ne_of_mem_erase hj
        rw [Polynomial.derivative_C_mul, Polynomial.eval_mul, Polynomial.eval_C,
          deriv_prod_eval_node _ x k (Finset.mem_erase.mpr ⟨Ne.symm hjk, hkmem⟩)]
      · rw [Polynomial.derivative_C_mul, Polynomial.eval_mul, Polynomial.eval_C, deriv_prod_eval]
    open Polynomial in
    have formula3 : ∀ (n k : ℕ) (hk : k ≤ n) (Qc : ℕ → ℕ → ℚ)
      (hQc : ∀ j ≤ n, Qc n j * 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)),
        ∑ j ∈ Finset.range (k + 1), Qc n j / ((2 : ℚ) ^ (k - j) - 1) =
          Qc n k * 2 ^ k * (∑ K ∈ Finset.Ioc n (2 * n), 1 / ((2 : ℚ) ^ k - 2 ^ K) -
            ∑ i ∈ (Finset.range (n + 1)).erase k, 1 / ((2 : ℚ) ^ k - 2 ^ i)) -
          ∑ j ∈ Finset.Ioc k n, Qc n j * 2 ^ (j - k) / (1 - 2 ^ (j - k)) := 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 S := (Finset.range (n + 1)).erase k with hS
      set T := Finset.Ioc n (2 * n) with hT
      set N : ℚ[X] := ∏ K ∈ T, (X - C ((2 : ℚ) ^ K)) with hN
      have hNdeg : N.degree ≤ n := by
        rw [hN, Polynomial.degree_prod]
        rw [Finset.sum_congr rfl (fun K _ => Polynomial.degree_X_sub_C ((2 : ℚ) ^ K)), Finset.sum_const, hT,
          Nat.card_Ioc]
        simp
        omega
      have hkmem : k ∈ Finset.range (n + 1) := Finset.mem_range.mpr (by 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 hD := lagr_deriv n x hxinj N hNdeg (fun j => Qc n j * 2 ^ j) hr k hk
      have hD2 : (derivative N).eval (x k) = ∑ K ∈ T, ∏ a ∈ T.erase K, (x k - 2 ^ a) := by
        rw [hN, deriv_prod_eval]
      set P := ∏ i ∈ S, (x k - x i) with hP
      have hPne : P ≠ 0 := by
        rw [hP]
        refine Finset.prod_ne_zero_iff.mpr (fun i hi => hxne k i (Ne.symm (Finset.ne_of_mem_erase hi)))
      have hNk : N.eval (x k) = (Qc n k * 2 ^ k) * P := by
        rw [← hr k hk]
      have hNk' : N.eval (x k) = ∏ K ∈ T, (x k - 2 ^ K) := by
        rw [hN, Polynomial.eval_prod]
        simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C]
      -- (a)
      have ha : ∀ j ∈ S, ∏ i ∈ ((Finset.range (n + 1)).erase j).erase k, (x k - x i) = P / (x k - x j) := by
        intro j hj
        have hne : x k - x j ≠ 0 := hxne k j (Ne.symm (Finset.ne_of_mem_erase hj))
        rw [Finset.erase_right_comm, eq_div_iff hne, mul_comm, hP, Finset.mul_prod_erase S (fun i => x k - x i) hj]
      have hb : ∀ l ∈ S, ∏ i ∈ S.erase l, (x k - x i) = P / (x k - x l) := by
        intro l hl
        have hne : x k - x l ≠ 0 := hxne k l (Ne.symm (Finset.ne_of_mem_erase hl))
        rw [eq_div_iff hne, mul_comm, hP, Finset.mul_prod_erase S (fun i => x k - x i) hl]
      have hc : ∀ K ∈ T, ∏ a ∈ T.erase K, (x k - 2 ^ a) = N.eval (x k) / (x k - 2 ^ K) := by
        intro K hK
        have hKn : n < K := (Finset.mem_Ioc.mp hK).1
        have hne : x k - 2 ^ K ≠ 0 := hxne k K (by omega)
        rw [eq_div_iff hne, mul_comm, hNk', Finset.mul_prod_erase T (fun a => x k - 2 ^ a) hK]
      rw [← hS] at hD
      rw [hD2, Finset.sum_congr rfl hc] at hD
      have hD3 : ∑ j ∈ S, Qc n j * 2 ^ j * ∏ i ∈ ((Finset.range (n + 1)).erase j).erase k, (x k - x i) =
          ∑ j ∈ S, Qc n j * 2 ^ j * (P / (x k - x j)) :=
        Finset.sum_congr rfl (fun j hj => by rw [ha j hj])
      have hD4 : ∑ l ∈ S, ∏ i ∈ S.erase l, (x k - x i) = ∑ l ∈ S, P / (x k - x l) :=
        Finset.sum_congr rfl hb
      rw [hD3, hD4, hNk] at hD
      -- key: sum over S of r_j / (x k - x j) = r_k * beta
      have key : ∑ j ∈ S, (Qc n j * 2 ^ j) / (x k - x j) =
          (Qc n k * 2 ^ k) * (∑ K ∈ T, 1 / (x k - 2 ^ K) - ∑ i ∈ S, 1 / (x k - x i)) := by
        have e1 : ∑ K ∈ T, Qc n k * 2 ^ k * P / (x k - 2 ^ K) = P * ((Qc n k * 2 ^ k) * ∑ K ∈ T, 1 / (x k - 2 ^ K)) := by
          rw [Finset.mul_sum, Finset.mul_sum]; refine Finset.sum_congr rfl (fun K _ => by ring)
        have e2 : ∑ j ∈ S, Qc n j * 2 ^ j * (P / (x k - x j)) = P * ∑ j ∈ S, (Qc n j * 2 ^ j) / (x k - x j) := by
          rw [Finset.mul_sum]; refine Finset.sum_congr rfl (fun j _ => by ring)
        have e3 : ∑ l ∈ S, P / (x k - x l) = P * ∑ i ∈ S, 1 / (x k - x i) := by
          rw [Finset.mul_sum]; refine Finset.sum_congr rfl (fun j _ => by ring)
        rw [e1, e2, e3] at hD
        apply mul_left_cancel₀ hPne
        linear_combination -hD
      -- split S into range k and Ioc k n
      have hsplit : S = Finset.range k ∪ Finset.Ioc k n := by
        rw [hS]; ext x; simp only [Finset.mem_erase, Finset.mem_range, Finset.mem_union, Finset.mem_Ioc]; omega
      have hdisj : Disjoint (Finset.range k) (Finset.Ioc k n) := by
        rw [Finset.disjoint_left]; intro a ha ha'
        simp only [Finset.mem_range, Finset.mem_Ioc] at ha ha'; omega
      rw [hsplit, Finset.sum_union hdisj] at key
      have hlow : ∀ j ∈ Finset.range k, (Qc n j * 2 ^ j) / (x k - x j) = Qc n j / ((2 : ℚ) ^ (k - j) - 1) := by
        intro j hj
        have hjk : j < k := Finset.mem_range.mp hj
        have hne : (2 : ℚ) ^ (k - j) - 1 ≠ 0 := by
          have : (1 : ℚ) < 2 ^ (k - j) := one_lt_pow₀ (by norm_num) (by omega)
          linarith
        have hne2 : (2 : ℚ) ^ k - 2 ^ j ≠ 0 := hxne k j (by omega)
        simp only [hxdef]
        rw [div_eq_div_iff hne2 hne, show k = j + (k - j) by omega, pow_add]
        simp only [show j + (k - j) - j = k - j by omega]
        ring
      have hhigh : ∀ j ∈ Finset.Ioc k n, (Qc n j * 2 ^ j) / (x k - x j) = Qc n j * 2 ^ (j - k) / (1 - 2 ^ (j - k)) := by
        intro j hj
        have hkj : k < j := (Finset.mem_Ioc.mp hj).1
        have hne : (1 : ℚ) - 2 ^ (j - k) ≠ 0 := by
          have : (1 : ℚ) < 2 ^ (j - k) := one_lt_pow₀ (by norm_num) (by omega)
          linarith
        have hne2 : (2 : ℚ) ^ k - 2 ^ j ≠ 0 := hxne k j (by omega)
        simp only [hxdef]
        rw [div_eq_div_iff hne2 hne, show j = k + (j - k) by omega, pow_add]
        simp only [show k + (j - k) - k = j - k by omega]
        ring
      rw [Finset.sum_congr rfl hlow, Finset.sum_congr rfl hhigh] at key
      rw [Finset.sum_range_succ, Nat.sub_self, pow_zero, sub_self, div_zero, add_zero]
      simp only [hxdef] at key
      rw [← hsplit] at key
      linear_combination key
    have hform : ∀ (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))) →
      ∀ (n k : ℕ), k ≤ n →
        ∑ j ∈ Finset.range (k + 1), Qc n j / ((2 : ℚ) ^ (k - j) - 1) =
          Qc n k * 2 ^ k * (∑ K ∈ Finset.Ioc n (2 * n), 1 / ((2 : ℚ) ^ k - 2 ^ K) -
            ∑ i ∈ (Finset.range (n + 1)).erase k, 1 / ((2 : ℚ) ^ k - 2 ^ i)) -
          ∑ j ∈ Finset.Ioc k n, Qc n j * 2 ^ (j - k) / (1 - 2 ^ (j - k)) := by
      intro Qc hQc n k hk
      exact formula3 n k hk Qc (fun j hj => by subst hQc; exact id1_full n j hj)
    clear id1_test lower_test upper_test rhs_test id1_full lagr_eval deriv_prod_eval deriv_prod_eval_node lagr_deriv formula3
    have gauss : ∀ n k : ℕ,
        ∃ z : ℤ, (z : ℚ) = ∏ i ∈ Finset.range k, ((2 : ℚ) ^ (n - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1) := by
      intro n k
      have hD : ∀ j : ℕ, ((2 : ℚ) ^ (j + 1) - 1) ≠ 0 := by
        intro j
        have h1 : (1 : ℚ) < 2 ^ (j + 1) := one_lt_pow₀ (by norm_num) (by omega)
        exact ne_of_gt (sub_pos.mpr h1)
      have hDk : ∀ k : ℕ, (∏ i ∈ Finset.range k, ((2 : ℚ) ^ (i + 1) - 1)) ≠ 0 := by
        intro k
        exact Finset.prod_ne_zero_iff.mpr (fun i _ => hD i)
      have key : ∀ m j : ℕ,
          (∏ i ∈ Finset.range (j + 1), ((2 : ℚ) ^ (m + 1 - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) =
            (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (m - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) +
            2 ^ (j + 1) *
              ∏ i ∈ Finset.range (j + 1), ((2 : ℚ) ^ (m - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1) := by
        intro m j
        simp only [Finset.prod_div_distrib]
        rw [Finset.prod_range_succ' (fun i => (2 : ℚ) ^ (m + 1 - i) - 1)]
        have hshift : ∀ i : ℕ, m + 1 - (i + 1) = m - i := fun i => by omega
        simp only [hshift, Nat.sub_zero]
        rw [Finset.prod_range_succ (fun i => (2 : ℚ) ^ (m - i) - 1),
          Finset.prod_range_succ (fun i => (2 : ℚ) ^ (i + 1) - 1)]
        have hDj := hDk j
        have hj := hD j
        by_cases hjm : j ≤ m
        · have e : (2 : ℚ) ^ (m + 1) = 2 ^ (m - j) * 2 ^ (j + 1) := by
            rw [← pow_add]
            congr 1
            omega
          rw [e]
          field_simp
          ring
        · have hz : (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (m - i) - 1)) = 0 :=
            Finset.prod_eq_zero (i := m) (Finset.mem_range.mpr (by omega)) (by simp)
          rw [hz]
          simp
      have main : ∀ m j : ℕ, ∃ z : ℤ,
          (z : ℚ) = ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (m - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1) := by
        intro m
        induction m with
        | zero =>
          intro j
          cases j with
          | zero => exact ⟨1, by simp⟩
          | succ j =>
            refine ⟨0, ?_⟩
            rw [Finset.prod_eq_zero (i := 0) (by simp) (by simp)]
            simp
        | succ m ih =>
          intro j
          cases j with
          | zero => exact ⟨1, by simp⟩
          | succ j =>
            obtain ⟨a, ha⟩ := ih j
            obtain ⟨b, hb⟩ := ih (j + 1)
            refine ⟨a + 2 ^ (j + 1) * b, ?_⟩
            rw [key, ← ha, ← hb]
            push_cast
            ring
      exact main n k
    have hodd : ∀ (n k : ℕ), k ≤ n → ∀ (c : ℕ → ℤ),
        ∃ z : ℤ, (z : ℚ) = (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
          ∑ j ∈ Finset.range (k + 1), (c j : ℚ) / ((2 : ℚ) ^ (k - j) - 1) := by
      intro n k hk c
      have hint : ∀ j ∈ Finset.range (k + 1), ∃ z : ℤ,
          (z : ℚ) = (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
            ((c j : ℚ) / ((2 : ℚ) ^ (k - j) - 1)) := by
        intro j hj
        have hjk : j ≤ k := Nat.lt_succ_iff.mp (Finset.mem_range.mp hj)
        by_cases hjeq : j = k
        · subst hjeq
          exact ⟨0, by simp⟩
        · obtain ⟨d, hd⟩ : ∃ d, k - j = d + 1 := ⟨k - j - 1, by omega⟩
          rw [hd]
          have hdn : d + 1 ≤ n := by omega
          have ht : 1 ≤ n / (d + 1) := (Nat.one_le_div_iff (by omega)).mpr hdn
          have hP1 : (d + 1) * (n / (d + 1)) ≤ n := Nat.mul_div_le n (d + 1)
          have hP2 : n < (d + 1) * (n / (d + 1)) + (d + 1) := by
            have := Nat.lt_mul_div_succ n (show 0 < d + 1 by omega)
            rw [Nat.mul_succ] at this
            exact this
          have hP3 : (d + 1) ≤ (d + 1) * (n / (d + 1)) := Nat.le_mul_of_pos_right _ ht
          have hmem : (d + 1) * (n / (d + 1)) ∈ Finset.Ioc (n / 2) n := by
            rw [Finset.mem_Ioc]
            generalize (d + 1) * (n / (d + 1)) = P at hP1 hP2 hP3 ⊢
            omega
          have hdvd : ((2 : ℤ) ^ (d + 1) - 1) ∣ ∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℤ) ^ m - 1) := by
            refine dvd_trans ?_ (Finset.dvd_prod_of_mem _ hmem)
            have := sub_dvd_pow_sub_pow ((2 : ℤ) ^ (d + 1)) 1 (n / (d + 1))
            rw [one_pow, ← pow_mul] at this
            exact this
          obtain ⟨q, hq⟩ := hdvd
          have hne : ((2 : ℚ) ^ (d + 1) - 1) ≠ 0 := by
            have : (1 : ℚ) < 2 ^ (d + 1) := one_lt_pow₀ (by norm_num) (by omega)
            linarith
          refine ⟨c j * q, ?_⟩
          have hcast : (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) =
              (((2 : ℤ) ^ (d + 1) - 1 : ℤ) : ℚ) * (q : ℚ) := by
            rw [← Int.cast_mul, ← hq]
            push_cast
            rfl
          rw [hcast]
          push_cast
          field_simp
      choose! g hg using hint
      refine ⟨∑ j ∈ Finset.range (k + 1), g j, ?_⟩
      push_cast
      rw [Finset.mul_sum]
      exact Finset.sum_congr rfl hg
    have combine : ∀ (p M : ℚ) (a : ℕ) (Ω A B m : ℤ), Odd Ω →
        (Ω : ℚ) * p = 2 ^ a * A → (B : ℚ) = M * p → (m : ℚ) = M →
        ∃ z : ℤ, (z : ℚ) * 2 ^ a = M * p := by
      intro p M a Ω A B m hΩ h1 h2 hm
      have hint : Ω * B = 2 ^ a * (m * A) := by
        have : ((Ω * B : ℤ) : ℚ) = ((2 ^ a * (m * A) : ℤ) : ℚ) := by
          push_cast
          rw [h2, ← hm]
          linear_combination (m : ℚ) * h1
        exact_mod_cast this
      obtain ⟨t, ht⟩ := hΩ
      have hcop : IsCoprime ((2 : ℤ) ^ a) Ω := by
        apply IsCoprime.pow_left
        exact ⟨-t, 1, by rw [ht]; ring⟩
      have hdvd : (2 : ℤ) ^ a ∣ B * Ω := ⟨m * A, by rw [mul_comm, hint]⟩
      obtain ⟨c, hc⟩ := hcop.dvd_of_dvd_mul_right hdvd
      refine ⟨c, ?_⟩
      rw [← h2, hc]
      push_cast
      ring
    have hcore : ∀ (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))) →
      ∀ (n k : ℕ), k ≤ n →
        ∃ z : ℤ, (z : ℚ) * (2 : ℚ) ^ (k * (k - 1) / 2) =
          (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
            ∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ)) := by
      intro Qc hQc n k hk
      have hQcint : ∀ j : ℕ, ∃ g : ℤ, Qc n j = 2 ^ (j * (j - 1) / 2) * g := by
        intro j
        obtain ⟨a, ha⟩ := gauss n j
        obtain ⟨b, hb⟩ := gauss (2 * n - j) n
        refine ⟨(-1) ^ j * a * b, ?_⟩
        rw [hQc]
        simp only []
        rw [← ha, ← hb]
        push_cast
        ring
      choose g hg using hQcint
      -- the odd product
      have hΩdiv : ∀ s : ℕ, 1 ≤ s → s ≤ 2 * n → ∃ w : ℤ,
          (w : ℚ) * ((2 : ℚ) ^ s - 1) = ((∏ t ∈ Finset.Icc 1 (2 * n), ((2 : ℤ) ^ t - 1) : ℤ) : ℚ) := by
        intro s h1 h2
        refine ⟨∏ t ∈ (Finset.Icc 1 (2 * n)).erase s, ((2 : ℤ) ^ t - 1), ?_⟩
        rw [← Finset.prod_erase_mul _ _ (Finset.mem_Icc.mpr ⟨h1, h2⟩)]
        push_cast
        ring
      have hΩodd : Odd (∏ t ∈ Finset.Icc 1 (2 * n), ((2 : ℤ) ^ t - 1)) := by
        apply Finset.prod_induction _ (fun x : ℤ => Odd x)
        · intro x y hx hy
          exact hx.mul hy
        · exact odd_one
        · intro t ht
          have ht1 : t ≠ 0 := by
            have := (Finset.mem_Icc.mp ht).1
            omega
          exact Even.sub_odd ((Int.even_pow).mpr ⟨even_two, ht1⟩) odd_one
      -- one term 2^a / (2^a - 2^b), times the odd product, is an integer
      have key1 : ∀ a b : ℕ, a ≠ b → a ≤ 2 * n → b ≤ 2 * n → ∃ w : ℤ,
          (w : ℚ) = ((∏ t ∈ Finset.Icc 1 (2 * n), ((2 : ℤ) ^ t - 1) : ℤ) : ℚ) * 2 ^ a * (1 / ((2 : ℚ) ^ a - 2 ^ b)) := by
        intro a b hab ha hb
        rcases Nat.lt_or_gt_of_ne hab with hlt | hgt
        · obtain ⟨d, rfl⟩ := Nat.exists_eq_add_of_lt hlt
          obtain ⟨w, hw⟩ := hΩdiv (d + 1) (by omega) (by omega)
          refine ⟨-w, ?_⟩
          have hne : (2 : ℚ) ^ (d + 1) - 1 ≠ 0 := by
            have : (1 : ℚ) < 2 ^ (d + 1) := one_lt_pow₀ (by norm_num) (by omega)
            linarith
          have h2a : (2 : ℚ) ^ a ≠ 0 := by positivity
          rw [← hw, show a + d + 1 = a + (d + 1) by ring, show (2 : ℚ) ^ (a + (d + 1)) = 2 ^ a * 2 ^ (d + 1) from pow_add _ _ _]
          have : (2 : ℚ) ^ a - 2 ^ a * 2 ^ (d + 1) = -(2 ^ a * (2 ^ (d + 1) - 1)) := by ring
          rw [this]
          field_simp
          push_cast
          ring
        · obtain ⟨d, rfl⟩ := Nat.exists_eq_add_of_lt hgt
          obtain ⟨w, hw⟩ := hΩdiv (d + 1) (by omega) (by omega)
          refine ⟨w * 2 ^ (d + 1), ?_⟩
          have hne : (2 : ℚ) ^ (d + 1) - 1 ≠ 0 := by
            have : (1 : ℚ) < 2 ^ (d + 1) := one_lt_pow₀ (by norm_num) (by omega)
            linarith
          have h2b : (2 : ℚ) ^ b ≠ 0 := by positivity
          rw [← hw, show b + d + 1 = b + (d + 1) by ring, show (2 : ℚ) ^ (b + (d + 1)) = 2 ^ b * 2 ^ (d + 1) from pow_add _ _ _]
          have : (2 : ℚ) ^ b * 2 ^ (d + 1) - 2 ^ b = 2 ^ b * (2 ^ (d + 1) - 1) := by ring
          rw [this]
          field_simp
          push_cast
          ring
      set Ω : ℤ := ∏ t ∈ Finset.Icc 1 (2 * n), ((2 : ℤ) ^ t - 1) with hΩ
      have hK : ∀ K ∈ Finset.Ioc n (2 * n), ∃ w : ℤ,
          (w : ℚ) = (Ω : ℚ) * 2 ^ k * (1 / ((2 : ℚ) ^ k - 2 ^ K)) := by
        intro K hK'
        have := Finset.mem_Ioc.mp hK'
        exact key1 k K (by omega) (by omega) (by omega)
      have hI : ∀ i ∈ (Finset.range (n + 1)).erase k, ∃ w : ℤ,
          (w : ℚ) = (Ω : ℚ) * 2 ^ k * (1 / ((2 : ℚ) ^ k - 2 ^ i)) := by
        intro i hi
        have h1 := Finset.ne_of_mem_erase hi
        have h2 := Finset.mem_range.mp (Finset.mem_of_mem_erase hi)
        exact key1 k i (Ne.symm h1) (by omega) (by omega)
      have hT : ∀ j ∈ Finset.Ioc k n, ∃ w : ℤ,
          (w : ℚ) * 2 ^ (k * (k - 1) / 2) = (Ω : ℚ) * (Qc n j * 2 ^ (j - k) / (1 - 2 ^ (j - k))) := by
        intro j hj
        have hj' := Finset.mem_Ioc.mp hj
        obtain ⟨d, hd⟩ : ∃ d, j - k = d + 1 := ⟨j - k - 1, by omega⟩
        obtain ⟨w, hw⟩ := hΩdiv (d + 1) (by omega) (by omega)
        have hmono : k * (k - 1) / 2 ≤ j * (j - 1) / 2 :=
          Nat.div_le_div_right (Nat.mul_le_mul (by omega) (by omega))
        obtain ⟨e, he⟩ := Nat.exists_eq_add_of_le hmono
        refine ⟨-(w * g j * 2 ^ e * 2 ^ (d + 1)), ?_⟩
        have hne : (2 : ℚ) ^ (d + 1) - 1 ≠ 0 := by
          have : (1 : ℚ) < 2 ^ (d + 1) := one_lt_pow₀ (by norm_num) (by omega)
          linarith
        have hne' : (1 : ℚ) - 2 ^ (d + 1) ≠ 0 := by
          intro h
          apply hne
          linarith
        rw [hd, hg j, he, ← hw, show (2 : ℚ) ^ (k * (k - 1) / 2 + e) = 2 ^ (k * (k - 1) / 2) * 2 ^ e from pow_add _ _ _]
        field_simp
        push_cast
        ring
      choose! fK hfK using hK
      choose! fI hfI using hI
      choose! fT hfT using hT
      have h2adic : (Ω : ℚ) * ∑ j ∈ Finset.range (k + 1), Qc n j / ((2 : ℚ) ^ (k - j) - 1) =
          2 ^ (k * (k - 1) / 2) * ((g k * (∑ K ∈ Finset.Ioc n (2 * n), fK K -
            ∑ i ∈ (Finset.range (n + 1)).erase k, fI i) - ∑ j ∈ Finset.Ioc k n, fT j : ℤ) : ℚ) := by
        rw [hform Qc hQc n k hk]
        push_cast
        rw [Finset.sum_congr rfl hfK, Finset.sum_congr rfl hfI, ← Finset.mul_sum, ← Finset.mul_sum]
        have htail : (2 : ℚ) ^ (k * (k - 1) / 2) * ∑ j ∈ Finset.Ioc k n, (fT j : ℚ) =
            (Ω : ℚ) * ∑ j ∈ Finset.Ioc k n, Qc n j * 2 ^ (j - k) / (1 - 2 ^ (j - k)) := by
          rw [Finset.mul_sum, Finset.mul_sum]
          refine Finset.sum_congr rfl (fun j hj => ?_)
          rw [mul_comm]
          exact hfT j hj
        rw [hg k]
        simp only [one_div] at htail ⊢
        linear_combination htail
      -- the odd half
      obtain ⟨B, hB⟩ := hodd n k hk (fun j => 2 ^ (j * (j - 1) / 2) * g j)
      have hB' : (B : ℚ) = (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
          ∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ)) := by
        rw [hB]
        congr 1
        refine Finset.sum_congr rfl (fun j _ => ?_)
        rw [hg j]
        push_cast
        ring
      exact combine _ _ _ Ω _ B (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℤ) ^ m - 1)) hΩodd h2adic hB'
        (by push_cast; rfl)
    clear hform hodd combine
    have hcnall : ∀ n : ℕ, ∃ z : ℤ, (z : ℚ) = (∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3)) *
        ∑ j ∈ Finset.range n, (3 : ℚ) / ((2 : ℚ) ^ (j + 1) - 3) := by
      intro n
      have hint : ∀ j ∈ Finset.range n, ∃ z : ℤ,
          (z : ℚ) = (∏ i ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (i + 1) - 3)) * ((3 : ℚ) / ((2 : ℚ) ^ (j + 1) - 3)) := by
        intro j hj
        by_cases h2 : j < 2
        · have hj' : j = 0 ∨ j = 1 := by omega
          rcases hj' with rfl | rfl
          · refine ⟨-3 * ∏ i ∈ Finset.Ico 2 n, ((2 : ℤ) ^ (i + 1) - 3), ?_⟩
            have e : (3 : ℚ) / ((2 : ℚ) ^ (0 + 1) - 3) = -3 := by norm_num
            rw [e]
            push_cast
            ring
          · refine ⟨3 * ∏ i ∈ Finset.Ico 2 n, ((2 : ℤ) ^ (i + 1) - 3), ?_⟩
            have e : (3 : ℚ) / ((2 : ℚ) ^ (1 + 1) - 3) = 3 := by norm_num
            rw [e]
            push_cast
            ring
        · have hmem : j ∈ Finset.Ico 2 n := Finset.mem_Ico.mpr ⟨by omega, Finset.mem_range.mp hj⟩
          have hne : ((2 : ℚ) ^ (j + 1) - 3) ≠ 0 := by
            have h8 : (8 : ℚ) ≤ 2 ^ (j + 1) := by
              calc (8 : ℚ) = 2 ^ 3 := by norm_num
                _ ≤ 2 ^ (j + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
            linarith
          refine ⟨3 * ∏ i ∈ (Finset.Ico 2 n).erase j, ((2 : ℤ) ^ (i + 1) - 3), ?_⟩
          rw [← Finset.mul_prod_erase _ _ hmem]
          push_cast
          field_simp
      choose! g hg using hint
      refine ⟨∑ j ∈ Finset.range n, g j, ?_⟩
      push_cast
      rw [Finset.mul_sum]
      exact Finset.sum_congr rfl hg
    intro Qc hQc Qx hQx Aq hAq n
    have hb :
      ((2 ^ (n * (n + 1) / 2) * (if n = 5 then 409045 else if n = 7 then 1298717875 else
            (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ) : ℕ) : ℝ) ^ 2 ≤
        (2 : ℝ) ^ (3 * n * n) := by
      have key : (2 ^ (n * (n + 1) / 2) * (if n = 5 then 409045 else if n = 7 then 1298717875 else
              (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ)) ^ 2 ≤
          2 ^ (3 * n * n) := by
        rcases (show n ≤ 9 ∨ 10 ≤ n by omega) with hn | hn
        · revert n
          decide +kernel
        · rw [if_neg (by omega), if_neg (by omega)]
          have hP1 : ∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1) ≤ 2 ^ (∑ m ∈ Finset.Ioc (n / 2) n, m) := by
            rw [← Finset.prod_pow_eq_pow_sum]
            exact Finset.prod_le_prod' (fun m _ => Nat.sub_le _ _)
          have hP2 : ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) ≤ 2 ^ (∑ j ∈ Finset.Ico 2 n, (j + 1)) := by
            rw [← Finset.prod_pow_eq_pow_sum]
            exact Finset.prod_le_prod' (fun m _ => Nat.sub_le _ _)
          obtain ⟨c, hc⟩ : ∃ c, n = n / 2 + c + 1 := ⟨n - n / 2 - 1, by omega⟩
          obtain ⟨d, hd⟩ : ∃ d, n = d + 3 := ⟨n - 3, by omega⟩
          have hS1 : (∑ m ∈ Finset.Ioc (n / 2) n, m) * 2 = 2 * (c + 1) * (n / 2 + 1) + (c + 1) * c := by
            have e : Finset.Ioc (n / 2) n = Finset.Ico (n / 2 + 1) (n + 1) := by
              ext x; simp only [Finset.mem_Ioc, Finset.mem_Ico]; omega
            rw [e, Finset.sum_Ico_eq_sum_range, show n + 1 - (n / 2 + 1) = c + 1 by omega,
              Finset.sum_add_distrib, Finset.sum_const, Finset.card_range, smul_eq_mul, add_mul,
              Finset.sum_range_id_mul_two]
            simp only [Nat.add_sub_cancel]
            ring
          have hS2 : (∑ j ∈ Finset.Ico 2 n, (j + 1)) * 2 = 6 * (d + 1) + (d + 1) * d := by
            rw [Finset.sum_Ico_eq_sum_range, show n - 2 = d + 1 by omega]
            have : ∀ k ∈ Finset.range (d + 1), 2 + k + 1 = 3 + k := fun k _ => by omega
            rw [Finset.sum_congr rfl this, Finset.sum_add_distrib, Finset.sum_const, Finset.card_range,
              smul_eq_mul, add_mul, Finset.sum_range_id_mul_two]
            simp only [Nat.add_sub_cancel]
            ring
          have hT : n * (n + 1) / 2 * 2 = n * (n + 1) :=
            Nat.div_mul_cancel (Nat.even_mul_succ_self n).two_dvd
          have hexp : 2 * (n * (n + 1) / 2) + 2 * (∑ m ∈ Finset.Ioc (n / 2) n, m) +
              2 * (∑ j ∈ Finset.Ico 2 n, (j + 1)) ≤ 3 * n * n := by
            have arith : ∀ e r A B T : ℕ, r ≤ 1 → 3 ≤ e →
                A * 2 = 2 * (e + 1 + r + 1) * (e + 2 + 1) + (e + 1 + r + 1) * (e + 1 + r) →
                B * 2 = 6 * (2 * e + 1 + r + 1) + (2 * e + 1 + r + 1) * (2 * e + 1 + r) →
                T * 2 = (2 * e + 4 + r) * (2 * e + 4 + r + 1) →
                2 * T + 2 * A + 2 * B ≤ 3 * (2 * e + 4 + r) * (2 * e + 4 + r) := by
              intro e r A B T hr he hA hB hT
              have hsq : 3 * e ≤ e * e := Nat.mul_le_mul_right e he
              rcases (show r = 0 ∨ r = 1 by omega) with rfl | rfl
              · ring_nf at hA hB hT hsq ⊢
                omega
              · ring_nf at hA hB hT hsq ⊢
                omega
            obtain ⟨e, he⟩ : ∃ e, n / 2 = e + 2 := ⟨n / 2 - 2, by omega⟩
            obtain ⟨r, hr1, hr⟩ : ∃ r, r ≤ 1 ∧ n = 2 * e + 4 + r := ⟨n - (2 * e + 4), by omega, by omega⟩
            have hc' : c = e + 1 + r := by omega
            have hd' : d = 2 * e + 1 + r := by omega
            rw [he, hc'] at hS1
            rw [hd'] at hS2
            have hT' : n * (n + 1) / 2 * 2 = (2 * e + 4 + r) * (2 * e + 4 + r + 1) := by rw [hT, hr]
            have := arith e r _ _ _ hr1 (by omega) hS1 hS2 hT'
            rw [he]
            rw [hr] at this ⊢
            exact this
          calc (2 ^ (n * (n + 1) / 2) * ((∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) *
                  ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3))) ^ 2
              ≤ (2 ^ (n * (n + 1) / 2) * (2 ^ (∑ m ∈ Finset.Ioc (n / 2) n, m) *
                  2 ^ (∑ j ∈ Finset.Ico 2 n, (j + 1)))) ^ 2 := by gcongr
            _ = 2 ^ (2 * (n * (n + 1) / 2) + 2 * (∑ m ∈ Finset.Ioc (n / 2) n, m) +
                  2 * (∑ j ∈ Finset.Ico 2 n, (j + 1))) := by
              rw [← pow_add, ← pow_add, ← pow_mul]; congr 1; ring
            _ ≤ 2 ^ (3 * n * n) := Nat.pow_le_pow_right (by norm_num) hexp
      exact_mod_cast key
    have hqx : ∃ z : ℤ, (z : ℚ) = (2 : ℚ) ^ (n * (n + 1) / 2) *
        ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k := by
      have hterm : ∀ k ∈ Finset.range (n + 1), ∃ z : ℤ,
          (z : ℚ) = (2 : ℚ) ^ (n * (n + 1) / 2) * (Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) := by
        intro k hk
        have hkn : k ≤ n := Nat.lt_succ_iff.mp (Finset.mem_range.mp hk)
        obtain ⟨a, ha⟩ := gauss n k
        obtain ⟨b, hb⟩ := gauss (2 * n - k) n
        obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hkn
        have hexp : (k + m) * (k + m + 1) / 2 + k * (k - 1) / 2 = m * (m + 1) / 2 + (k + m) * k := by
          have e1 : 2 * ((k + m) * (k + m + 1) / 2) = (k + m) * (k + m + 1) :=
            Nat.two_mul_div_two_of_even (Nat.even_mul_succ_self _)
          have e2 : 2 * (k * (k - 1) / 2) = k * (k - 1) :=
            Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self _)
          have e3 : 2 * (m * (m + 1) / 2) = m * (m + 1) :=
            Nat.two_mul_div_two_of_even (Nat.even_mul_succ_self _)
          rcases k with _ | j
          · simp
          · simp only [Nat.add_sub_cancel] at e2 ⊢
            ring_nf at e1 e2 e3 ⊢
            omega
        refine ⟨(-1) ^ k * 3 ^ k * 2 ^ (m * (m + 1) / 2) * a * b, ?_⟩
        have hpow : (2 : ℚ) ^ ((k + m) * (k + m + 1) / 2) * (2 : ℚ) ^ (k * (k - 1) / 2) =
            (2 : ℚ) ^ (m * (m + 1) / 2) * ((2 : ℚ) ^ (k + m)) ^ k := by
          rw [← pow_add, ← pow_mul, ← pow_add, hexp]
        have h2 : ((2 : ℚ) ^ (k + m)) ≠ 0 := by positivity
        rw [hQc]
        simp only []
        rw [← ha, ← hb]
        push_cast
        rw [div_pow, mul_div_assoc']
        field_simp
        linear_combination -(a : ℚ) * b * hpow
      choose! g hg using hterm
      refine ⟨∑ k ∈ Finset.range (n + 1), g k, ?_⟩
      push_cast
      rw [Finset.mul_sum]
      exact Finset.sum_congr rfl (fun k hk => hg k hk)
    have hE : 0 < (if n = 5 then 409045 else if n = 7 then 1298717875 else
            (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ) := by
      split_ifs
      · norm_num
      · norm_num
      · refine Nat.mul_pos (Finset.prod_pos fun m hm => ?_) (Finset.prod_pos fun j hj => ?_)
        · have hm1 : 1 ≤ m := by
            have := (Finset.mem_Ioc.mp hm).1
            omega
          have : 2 ≤ 2 ^ m := by
            calc 2 = 2 ^ 1 := by norm_num
              _ ≤ 2 ^ m := Nat.pow_le_pow_right (by norm_num) hm1
          omega
        · have hj2 : 2 ≤ j := (Finset.mem_Ico.mp hj).1
          have : 8 ≤ 2 ^ (j + 1) := by
            calc 8 = 2 ^ 3 := by norm_num
              _ ≤ 2 ^ (j + 1) := Nat.pow_le_pow_right (by norm_num) (by omega)
          omega
    have hAqint : ∃ z : ℤ, (z : ℚ) = ((2 ^ (n * (n + 1) / 2) * (if n = 5 then 409045 else if n = 7 then 1298717875 else
          (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ) : ℕ) : ℚ) * Aq n := by
      by_cases h5 : n = 5
      · subst h5
        refine ⟨1380067124193819576, ?_⟩
        rw [if_pos rfl]
        subst hQc hQx hAq
        decide +kernel
      · by_cases h7 : n = 7
        · subst h7
          refine ⟨674036225808261890798679141926088, ?_⟩
          rw [if_neg (by norm_num), if_pos rfl]
          subst hQc hQx hAq
          decide +kernel
        · rw [if_neg h5, if_neg h7]
          have hM : ((∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1) : ℕ) : ℚ) =
              ∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1) := by
            rw [Nat.cast_prod]
            refine Finset.prod_congr rfl (fun m _ => ?_)
            rw [Nat.cast_sub Nat.one_le_two_pow]
            push_cast
            ring
          have hC : ((∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ) : ℚ) =
              ∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3) := by
            rw [Nat.cast_prod]
            refine Finset.prod_congr rfl (fun j hj => ?_)
            have hj2 : 2 ≤ j := (Finset.mem_Ico.mp hj).1
            have h3 : 3 ≤ 2 ^ (j + 1) := by
              calc 3 ≤ 2 ^ 2 := by norm_num
                _ ≤ 2 ^ (j + 1) := Nat.pow_le_pow_right (by norm_num) (by omega)
            rw [Nat.cast_sub h3]
            push_cast
            ring
          have hd : ((2 ^ (n * (n + 1) / 2) * ((∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) *
              ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3)) : ℕ) : ℚ) =
              (2 : ℚ) ^ (n * (n + 1) / 2) * ((∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
                ∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3)) := by
            rw [Nat.cast_mul, Nat.cast_mul, hM, hC]
            push_cast
            ring
          obtain ⟨zcn, hzcn⟩ := hcnall n
          obtain ⟨zq', hzq'⟩ := hqx
          have hQxn : Qx n = ∑ k ∈ Finset.range (n + (1 : ℕ)), Qc n k * ((3 : ℚ) / (2 : ℚ) ^ n) ^ k := by
            rw [hQx]
          have hterm : ∀ k ∈ Finset.range (n + 1), ∃ z : ℤ,
              (z : ℚ) = (2 : ℚ) ^ (n * (n + 1) / 2) * ((∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
                ∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3)) *
                ((∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                  ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) := by
            intro k hk
            have hkn : k ≤ n := Nat.lt_succ_iff.mp (Finset.mem_range.mp hk)
            obtain ⟨w, hw⟩ := hcore Qc hQc n k hkn
            obtain ⟨m, hm⟩ := Nat.exists_eq_add_of_le hkn
            have hexp : n * (n + 1) / 2 + k * (k - 1) / 2 = m * (m + 1) / 2 + n * k := by
              rw [hm]
              have e1 : 2 * ((k + m) * (k + m + 1) / 2) = (k + m) * (k + m + 1) :=
                Nat.two_mul_div_two_of_even (Nat.even_mul_succ_self _)
              have e2 : 2 * (k * (k - 1) / 2) = k * (k - 1) :=
                Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self _)
              have e3 : 2 * (m * (m + 1) / 2) = m * (m + 1) :=
                Nat.two_mul_div_two_of_even (Nat.even_mul_succ_self _)
              rcases k with _ | j
              · simp
              · simp only [Nat.add_sub_cancel] at e2 ⊢
                ring_nf at e1 e2 e3 ⊢
                omega
            have hpow : (2 : ℚ) ^ (n * (n + 1) / 2) * (2 : ℚ) ^ (k * (k - 1) / 2) =
                (2 : ℚ) ^ (m * (m + 1) / 2) * ((2 : ℚ) ^ n) ^ k := by
              rw [← pow_add, ← pow_mul, ← pow_add, hexp]
            have h2n : ((2 : ℚ) ^ n) ^ k ≠ 0 := by positivity
            refine ⟨(∏ j ∈ Finset.Ico 2 n, ((2 : ℤ) ^ (j + 1) - 3)) * w * 3 ^ k * 2 ^ (m * (m + 1) / 2), ?_⟩
            have e1 : (2 : ℚ) ^ (n * (n + 1) / 2) * ((∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
                ∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3)) *
                ((∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                  ((3 : ℚ) / (2 : ℚ) ^ n) ^ k) =
                (∏ j ∈ Finset.Ico 2 n, ((2 : ℚ) ^ (j + 1) - 3)) * 3 ^ k *
                  (((∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℚ) ^ m - 1)) *
                    ∑ j ∈ Finset.range (k + (1 : ℕ)), Qc n j / ((2 : ℚ) ^ (k - j) - (1 : ℚ))) *
                    (2 : ℚ) ^ (n * (n + 1) / 2)) / ((2 : ℚ) ^ n) ^ k := by
              rw [div_pow]
              ring
            rw [e1, ← hw, show (w : ℚ) * (2 : ℚ) ^ (k * (k - 1) / 2) * (2 : ℚ) ^ (n * (n + 1) / 2) =
              (w : ℚ) * ((2 : ℚ) ^ (n * (n + 1) / 2) * (2 : ℚ) ^ (k * (k - 1) / 2)) by ring, hpow]
            push_cast
            field_simp
          choose! g hg using hterm
          refine ⟨∑ k ∈ Finset.range (n + 1), g k +
            (∏ m ∈ Finset.Ioc (n / 2) n, ((2 : ℤ) ^ m - 1)) * zq' * zcn, ?_⟩
          rw [hd, hAq]
          beta_reduce
          push_cast
          rw [mul_add, Finset.mul_sum, Finset.sum_congr rfl hg, hzcn, hzq', ← hQxn]
          ring
    obtain ⟨zq, hzq⟩ := hqx
    refine ⟨2 ^ (n * (n + 1) / 2) * (if n = 5 then 409045 else if n = 7 then 1298717875 else
            (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ), by positivity, hb, ⟨(if n = 5 then 409045 else if n = 7 then 1298717875 else
            (∏ m ∈ Finset.Ioc (n / 2) n, (2 ^ m - 1)) * ∏ j ∈ Finset.Ico 2 n, (2 ^ (j + 1) - 3) : ℕ) * zq, ?_⟩, hAqint⟩
    rw [hQx]
    push_cast
    rw [hzq]
    ring
  exact ⟨_, _, _, 0, rfl, rfl, rfl, H _ 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 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 all four hypotheses of the statement of this node hold, written exactly as there:

  • $Q_c(n,k)=(-1)^k\,2^{k(k-1)/2}\prod_{i<k}\frac{2^{n-i}-1}{2^{i+1}-1}\prod_{i<n}\frac{2^{2n-k-i}-1}{2^{i+1}-1}$ for all $n,k$;
  • $Q_x(n)=\sum_{k=0}^{n}Q_c(n,k)(3/2^n)^k$ for all $n$;
  • $A_q(n)=\sum_{k=0}^{n}\big(\sum_{j=0}^{k}\frac{Q_c(n,j)}{2^{k-j}-1}\big)(3/2^n)^k+Q_x(n)\sum_{j<n}\frac{3}{2^{j+1}-3}$ for all $n$;
  • for every $n$ there is a natural number $d>0$ with $d^2\le 2^{3n^2}$ such that $d\,Q_x(n)$ and $d\,A_q(n)$ are integers.

The existentially quantified $n$ appears in none of the four conditions. So the theorem says the hypotheses are not contradictory.

The first three conditions hold by definition (rfl) for the functions the formulas define. The fourth is a real theorem, and this file proves it in full: the step H is the statement of erdos-1050--h1-v2--h3, and its proof is the same tactic script, line for line apart from indentation, as that node's merged Proof.lean. The witness is then those three functions, $n=0$, and H applied to them.

Rendered from gloss/cb005d070c9d….md · Improve these words →

Annexes (3), in full

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

Untrusted: annex (D-31), by t0929-2, drafted with Claude Opus 5.5 via Claude Code; POST /check (AXLE lean-4.33.1), 2026-09-29T20:00:13Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/annex/b7ce465ec8dc9c8677118e56ea453c9bc67accbcc51e996be6347a1a6150fbd9.md.

A complete Lean proof of hrem (erdos-1050--h1-v2--h4), waiting for the node to open

Contributor: t0929-2. Status: the file below is a complete, sorry-free proof of this node's statement that passes the network's fast check in verify mode against this node (POST /check, lean-4.33.1, okay true, lint [], failed_declarations [], log 01M3QAAZ8GAJWZ9FY8R5VCE63K; axioms propext, Classical.choice, Quot.sound). It has not been through the gate: the node is blocked (witness-missing), so a precheck answers 409 node-blocked, and its witness is a proof of sibling h3 (see graph PR #229, a revision request to drop that unused hypothesis).

Structure (completes annex e484fc1b… on this node): the bound for n = 0, 1 directly from 0 < T ≤ 2/5; for n ≥ 2 the remainder is the series Σ_t E(n,t)·x_n^(2n+1+t) of positive terms with E(n,t) ≤ 2^(−(2n+t)). That series identity (hser) is proved from three haves: htail (3T − c_n = Σ_k x_n^k/(2^k − 1), a tail split and a geometric double series summed both ways), hcauchy (a polynomial times that series, coefficientwise) and hid, the coefficient identity, whose proof is the merged proof of spec-f2b55478 pasted verbatim (a hole cannot declare that node as a dependency). Unused hypotheses are cleared inside each have so that the declaration stays within the 200000-heartbeat cap; the fast check takes about 13 s. The same body, without the hden binder in the first intro and without the clear hden line, proves the statement with that hypothesis removed (proposed as spec-180d8b72, graph PR #274).

import Mathlib
import Nodes.«erdos-1050--h1-v2--h4».Context

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

theorem erdos_1050__h1_v2__h4 : ∀ (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),
                ∃ (d : ℕ),
                  (0 : ℕ) < d ∧
                    (↑d : ℝ) ^ (2 : ℕ) ≤ (2 : ℝ) ^ ((3 : ℕ) * n * n) ∧
                      (∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Qx n) ∧ ∃ (z : ℤ), (↑z : ℚ) = (↑d : ℚ) * Aq n) →
              ∀ (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
  intro Qc hQc Qx hQx Aq hAq hden
  -- the remainder bound does not use the denominator bound: cleared before any hole is stated
  clear hden
  obtain ⟨T, hT⟩ : ∃ T : ℝ, T = ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ)) := ⟨_, rfl⟩
  rw [← hT]
  -- the terms of T are bounded by (1/5) (1/2)^m
  have hterm : ∀ m : ℕ, 0 ≤ (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) ∧
      (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) ≤ (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m := by
    intro m
    have h2 : (1 : ℝ) ≤ (2 : ℝ) ^ m := one_le_pow₀ (by norm_num)
    have hpos : (0 : ℝ) < (2 : ℝ) ^ (m + 3) - 3 := by rw [pow_add]; nlinarith
    refine ⟨div_nonneg zero_le_one hpos.le, ?_⟩
    rw [div_le_iff₀ hpos]
    have hh : (1 / 2 : ℝ) ^ m * (2 : ℝ) ^ m = 1 := by rw [← mul_pow]; norm_num
    have hle : (1 / 2 : ℝ) ^ m ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
    rw [pow_add]
    nlinarith
  have hgeo : Summable (fun m : ℕ => (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m) :=
    (summable_geometric_of_lt_one (by norm_num) (by norm_num)).mul_left _
  have hsumm : Summable (fun m : ℕ => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) :=
    Summable.of_nonneg_of_le (fun m => (hterm m).1) (fun m => (hterm m).2) hgeo
  have hTpos : 0 < T := by
    rw [hT]
    exact hsumm.tsum_pos (fun m => (hterm m).1) 0 (by norm_num)
  have hTle : T ≤ 2 / 5 := by
    rw [hT]
    calc ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))
        ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m :=
          Summable.tsum_le_tsum (fun m => (hterm m).2) hsumm hgeo
      _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_of_lt_one (by norm_num) (by norm_num)]; norm_num
  -- the Padé remainder coefficients, in closed form (spec-f2b55478), shifted to start at x^(2n+1)
  obtain ⟨E, hE⟩ : ∃ E : ℕ → ℕ → ℝ, E = fun n t =>
      (2 : ℝ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1)) /
        ∏ j ∈ Finset.range (n + 1), ((2 : ℝ) ^ (2 * n + 1 + t - j) - 1) := ⟨_, rfl⟩
  have hEpos : ∀ n t : ℕ, 0 < E n t := by
    intro n t
    rw [hE]
    apply div_pos
    · apply mul_pos (by positivity)
      apply Finset.prod_pos
      intro i hi
      simp only [Finset.mem_range] at hi
      have : (1 : ℝ) < 2 ^ (n + t - i) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    · apply Finset.prod_pos
      intro j hj
      simp only [Finset.mem_range] at hj
      have : (1 : ℝ) < 2 ^ (2 * n + 1 + t - j) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
  have hEle : ∀ n t : ℕ, E n t ≤ (1 / 2 : ℝ) ^ (2 * n + t) := by
    intro n t
    rw [hE]
    dsimp only
    rw [Finset.prod_range_succ']
    have hA : (0 : ℝ) < ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (2 * n + 1 + t - (i + 1)) - 1) := by
      apply Finset.prod_pos
      intro i hi
      simp only [Finset.mem_range] at hi
      have : (1 : ℝ) < 2 ^ (2 * n + 1 + t - (i + 1)) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hnum : (2 : ℝ) ^ (n * n) * ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1) ≤
        ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (2 * n + 1 + t - (i + 1)) - 1) := by
      have hc : (2 : ℝ) ^ (n * n) * ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1) =
          ∏ i ∈ Finset.range n, ((2 : ℝ) ^ n * ((2 : ℝ) ^ (n + t - i) - 1)) := by
        rw [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, ← pow_mul]
      rw [hc]
      apply Finset.prod_le_prod
      · intro i hi
        simp only [Finset.mem_range] at hi
        have : (1 : ℝ) ≤ 2 ^ (n + t - i) := one_le_pow₀ (by norm_num)
        have : (0 : ℝ) < 2 ^ n := by positivity
        nlinarith
      · intro i hi
        simp only [Finset.mem_range] at hi
        have he : 2 * n + 1 + t - (i + 1) = n + (n + t - i) := by omega
        rw [he, pow_add]
        have : (1 : ℝ) ≤ 2 ^ n := one_le_pow₀ (by norm_num)
        nlinarith
    have hf0 : (2 : ℝ) ^ (2 * n + t) ≤ (2 : ℝ) ^ (2 * n + 1 + t - 0) - 1 := by
      have he : 2 * n + 1 + t - 0 = (2 * n + t) + 1 := by omega
      rw [he, pow_succ]
      have : (1 : ℝ) ≤ 2 ^ (2 * n + t) := one_le_pow₀ (by norm_num)
      linarith
    have hp : (0 : ℝ) < 2 ^ (2 * n + t) := by positivity
    rw [div_le_iff₀ (mul_pos hA (by linarith)), one_div_pow]
    have hkey : (1 : ℝ) ≤ 1 / 2 ^ (2 * n + t) * ((2 : ℝ) ^ (2 * n + 1 + t - 0) - 1) := by
      rw [div_mul_eq_mul_div, one_mul, le_div_iff₀ hp]; linarith
    nlinarith
  -- the tail of 3T as a power series in x_n = 3/2^n
  have htail : ∀ n : ℕ, 1 ≤ n → HasSum (fun k : ℕ => ((3 : ℝ) / 2 ^ n) ^ k / ((2 : ℝ) ^ k - 1))
      (3 * T - ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + 1) - 3)) := by
    intro n hn
    rw [hT]
    clear hterm hgeo hsumm hTpos hTle hEpos hEle hE hT hAq hQx hQc
    have h2n : (2 : ℝ) ≤ 2 ^ n := by
      calc (2 : ℝ) = 2 ^ 1 := by norm_num
        _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
    set x : ℝ := (3 : ℝ) / 2 ^ n with hxdef
    have hx0 : 0 < x := by positivity
    have hx2 : x < 2 := by
      rw [hxdef, div_lt_iff₀ (by positivity)]; nlinarith
    -- the double family
    set f : ℕ → ℕ → ℝ := fun j k => x ^ (k + 1) * ((1 : ℝ) / 2 ^ (j + 1)) ^ (k + 1) with hfdef
    have hf0 : ∀ j k, 0 ≤ f j k := fun j k => by positivity
    -- rows (fixed j): a geometric series
    have hrow : ∀ j : ℕ, HasSum (f j) (3 / ((2 : ℝ) ^ (j + n + 1) - 3)) := by
      intro j
      have hr0 : 0 ≤ x * (1 / 2 ^ (j + 1)) := by positivity
      have hr1 : x * (1 / 2 ^ (j + 1)) < 1 := by
        have : (2 : ℝ) ≤ 2 ^ (j + 1) := by
          calc (2 : ℝ) = 2 ^ 1 := by norm_num
            _ ≤ 2 ^ (j + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        rw [mul_one_div, div_lt_one (by positivity)]; linarith
      have h := (hasSum_geometric_of_lt_one hr0 hr1).mul_left (x * (1 / 2 ^ (j + 1)))
      have hpos : (0 : ℝ) < 2 ^ (j + n + 1) - 3 := by
        have : (4 : ℝ) ≤ 2 ^ (j + n + 1) := by
          calc (4 : ℝ) = 2 ^ 2 := by norm_num
            _ ≤ 2 ^ (j + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        linarith
      have e : (2 : ℝ) ^ (j + n + 1) = 2 ^ n * 2 ^ (j + 1) := by rw [← pow_add]; congr 1; omega
      have hfun : f j = fun i => x * (1 / 2 ^ (j + 1)) * (x * (1 / 2 ^ (j + 1))) ^ i := by
        funext k; simp only [hfdef]; rw [← mul_pow, pow_succ]; ring
      have hval : 3 / ((2 : ℝ) ^ (j + n + 1) - 3) = x * (1 / 2 ^ (j + 1)) * (1 - x * (1 / 2 ^ (j + 1)))⁻¹ := by
        rw [e] at hpos ⊢
        have h1 : (1 : ℝ) - x * (1 / 2 ^ (j + 1)) ≠ 0 := by linarith
        rw [hxdef] at h1 ⊢
        field_simp
      rw [hfun, hval]; exact h
    -- columns (fixed k): a geometric series in j
    have hcol : ∀ k : ℕ, HasSum (fun j => f j k) (x ^ (k + 1) / ((2 : ℝ) ^ (k + 1) - 1)) := by
      intro k
      have hq0 : (0 : ℝ) ≤ 1 / 2 ^ (k + 1) := by positivity
      have hq1 : (1 : ℝ) / 2 ^ (k + 1) < 1 := by
        rw [div_lt_one (by positivity)]; exact one_lt_pow₀ (by norm_num) (by omega)
      have h := ((hasSum_geometric_of_lt_one hq0 hq1).mul_left (1 / 2 ^ (k + 1))).mul_left (x ^ (k + 1))
      have hpos : (0 : ℝ) < 2 ^ (k + 1) - 1 := by
        have : (1 : ℝ) < 2 ^ (k + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have hfun : (fun j => f j k) = fun i => x ^ (k + 1) * (1 / 2 ^ (k + 1) * (1 / 2 ^ (k + 1)) ^ i) := by
        funext j; simp only [hfdef]
        rw [← one_div_pow, ← one_div_pow, ← pow_mul, ← pow_mul, ← pow_add]
        congr 2; ring
      have hval : x ^ (k + 1) / ((2 : ℝ) ^ (k + 1) - 1) =
          x ^ (k + 1) * (1 / 2 ^ (k + 1) * (1 - 1 / 2 ^ (k + 1))⁻¹) := by
        have h1 : (1 : ℝ) - 1 / 2 ^ (k + 1) ≠ 0 := by linarith
        field_simp
      rw [hfun, hval]; exact h
    -- the double sum, both ways
    have hsum2' : Summable (fun p : ℕ × ℕ => f p.1 p.2) := by
      rw [summable_prod_of_nonneg (fun p => hf0 p.1 p.2)]
      refine ⟨fun j => (hrow j).summable, ?_⟩
      have hts : (fun j => ∑' k, f j k) = fun j => 3 / ((2 : ℝ) ^ (j + n + 1) - 3) :=
        funext fun j => (hrow j).tsum_eq
      show Summable (fun j => ∑' k, f j k)
      rw [hts]
      refine Summable.of_nonneg_of_le (fun j => ?_) (fun j => ?_) (summable_geometric_two.mul_left 3)
      · have : (4 : ℝ) ≤ 2 ^ (j + n + 1) := by
          calc (4 : ℝ) = 2 ^ 2 := by norm_num
            _ ≤ 2 ^ (j + n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        exact div_nonneg (by norm_num) (by linarith)
      · have hj : (1 : ℝ) ≤ 2 ^ j := one_le_pow₀ (by norm_num)
        have e : (2 : ℝ) ^ (j + n + 1) = 2 ^ j * 2 ^ (n + 1) := by rw [← pow_add]; ring_nf
        have h4 : (4 : ℝ) ≤ 2 ^ (n + 1) := by
          calc (4 : ℝ) = 2 ^ 2 := by norm_num
            _ ≤ 2 ^ (n + 1) := pow_le_pow_right₀ (by norm_num) (by omega)
        have hp : (0 : ℝ) < 2 ^ (j + n + 1) - 3 := by rw [e]; nlinarith
        rw [div_le_iff₀ hp, one_div_pow, e]
        have hh : (1 / 2 : ℝ) ^ j * 2 ^ j = 1 := by rw [← mul_pow]; norm_num
        rw [one_div_pow] at hh
        field_simp
        nlinarith
    have hsum2 : Summable (Function.uncurry f) := hsum2'
    have hS := hsum2.hasSum
    have hA := hS.prod_fiberwise (fun j => hrow j)
    have hB := ((Equiv.prodComm ℕ ℕ).hasSum_iff.mpr hS).prod_fiberwise (fun k => hcol k)
    -- the row sums are the tail of 3T
    have hg : Summable (fun j : ℕ => (3 : ℝ) / ((2 : ℝ) ^ (j + 1) - 3)) :=
      (summable_nat_add_iff n).mp hA.summable
    have hsplit_n := hg.sum_add_tsum_nat_add n
    have hsplit_2 := hg.sum_add_tsum_nat_add 2
    have h01 : ∑ i ∈ Finset.range 2, (3 : ℝ) / ((2 : ℝ) ^ (i + 1) - 3) = 0 := by
      norm_num [Finset.sum_range_succ]
    have h3T : ∑' i : ℕ, (3 : ℝ) / ((2 : ℝ) ^ (i + 2 + 1) - 3) =
        3 * ∑' m : ℕ, (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) := by
      rw [← tsum_mul_left]
      congr 1; funext i
      rw [show i + 2 + 1 = i + 3 by ring]; ring
    have hval : ∑' p, Function.uncurry f p = 3 * ∑' m : ℕ, (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) -
        ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + 1) - 3) := by
      rw [← hA.tsum_eq]
      rw [h01, zero_add, h3T] at hsplit_2
      linarith
    have h0 : ∑ i ∈ Finset.range 1, x ^ i / ((2 : ℝ) ^ i - 1) = 0 := by simp
    rw [← hval]
    refine (hasSum_nat_add_iff' 1).mp ?_
    rw [h0, sub_zero]
    exact hB
  -- a polynomial times that series
  have hcauchy : ∀ (n : ℕ) (c : ℕ → ℝ) (x V : ℝ),
      HasSum (fun k : ℕ => x ^ k / ((2 : ℝ) ^ k - 1)) V →
      HasSum (fun m : ℕ => (∑ j ∈ Finset.range (n + 1), c j / ((2 : ℝ) ^ (m - j) - 1)) * x ^ m)
        ((∑ j ∈ Finset.range (n + 1), c j * x ^ j) * V) := by
    intro n c x V hV
    clear htail hterm hgeo hsumm hTpos hTle hEpos hEle hE hT hAq hQx hQc
    have hj : ∀ j ∈ Finset.range (n + 1),
        HasSum (fun m : ℕ => c j / ((2 : ℝ) ^ (m - j) - 1) * x ^ m) (c j * x ^ j * V) := by
      intro j _
      refine (hasSum_nat_add_iff' j).mp ?_
      have h0 : ∑ i ∈ Finset.range j, c j / ((2 : ℝ) ^ (i - j) - 1) * x ^ i = 0 := by
        apply Finset.sum_eq_zero
        intro i hi
        rw [Finset.mem_range] at hi
        rw [show i - j = 0 by omega]
        simp
      rw [h0, sub_zero]
      have hfun : (fun m : ℕ => c j / ((2 : ℝ) ^ (m + j - j) - 1) * x ^ (m + j)) =
          fun m : ℕ => c j * x ^ j * (x ^ m / ((2 : ℝ) ^ m - 1)) := by
        funext m
        rw [show m + j - j = m by omega, pow_add]
        ring
      rw [hfun]
      exact hV.mul_left (c j * x ^ j)
    have h := hasSum_sum hj
    rw [← Finset.sum_mul] at h
    convert h using 1
    funext m
    rw [Finset.sum_mul]
  -- the coefficient identity (spec-f2b55478, restated for Qc)
  have hid : ∀ n m : ℕ, n < m →
      ∑ j ∈ Finset.range (n + 1), Qc n j / ((2 : ℚ) ^ (m - j) - 1) =
        (2 : ℚ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (i + 1) - 1)) *
          (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) /
          ∏ j ∈ Finset.range (n + 1), ((2 : ℚ) ^ (m - j) - 1) := by
    intro n m hnm
    rw [hQc]
    clear htail hcauchy hterm hgeo hsumm hTpos hTle hEpos hEle hE hT hAq hQx hQc
    dsimp only
    -- Step 4 of the annex (residue identity): c_j 2^j prod_(k != j) (2^j - 2^k) = N(2^j).
    have hres : ∀ j ∈ Finset.range (n + 1),
        ((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - 1) / 2) * (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (2 * n - j - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1))) * (2 : ℚ) ^ j * ∏ k ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - (2 : ℚ) ^ k) =
          ∏ i ∈ Finset.range n, ((2 : ℚ) ^ j - (2 : ℚ) ^ (n + 1 + i)) := by
      intro j hj
      rw [Finset.mem_range] at hj
      obtain ⟨r, rfl⟩ : ∃ r, n = j + r := ⟨n - j, by omega⟩
      have E1test : ∀ j r : ℕ, ∏ k ∈ (Finset.range (j + r + 1)).erase j, ((2:ℚ)^j - 2^k) =
          (∏ k ∈ Finset.range j, ((2:ℚ)^j - 2^k)) * ∏ x ∈ Finset.range r, ((2:ℚ)^j - 2^(j+1+x))  := by
          intro j r
          induction r with
          | zero =>
            simp [Finset.range_add_one]
          | succ r ih =>
            rw [show j + (r + 1) + 1 = (j + r + 1) + 1 by ring, Finset.range_add_one,
              Finset.erase_insert_of_ne (by omega), Finset.prod_insert (by simp), ih,
              Finset.prod_range_succ]
            rw [show j + 1 + r = j + r + 1 by ring]
            ring
      have E2test : ∀ j : ℕ, ∏ k ∈ Finset.range j, ((2:ℚ)^j - 2^k) =
          2 ^ (j * (j - 1) / 2) * ∏ i ∈ Finset.range j, ((2:ℚ)^(i+1) - 1)  := by
          intro j
          rw [← Finset.prod_range_reflect (fun i => (2:ℚ)^(i+1) - 1)]
          rw [show (2:ℚ) ^ (j * (j - 1) / 2) = ∏ k ∈ Finset.range j, (2:ℚ)^k by
            rw [Finset.prod_pow_eq_pow_sum, Finset.sum_range_id], ← Finset.prod_mul_distrib]
          apply Finset.prod_congr rfl
          intro k hk
          rw [Finset.mem_range] at hk
          rw [show j - 1 - k + 1 = j - k by omega, mul_sub, ← pow_add, show k + (j - k) = j by omega]
          ring
      have E3test : ∀ j r : ℕ, ∏ x ∈ Finset.range r, ((2:ℚ)^j - 2^(j+1+x)) =
          (-1) ^ r * 2 ^ (j * r) * ∏ i ∈ Finset.range r, ((2:ℚ)^(i+1) - 1)  := by
          intro j r
          rw [show (-1:ℚ) ^ r * 2 ^ (j * r) = ∏ _x ∈ Finset.range r, (-(2:ℚ) ^ j) by
            rw [Finset.prod_const, Finset.card_range, pow_mul, ← mul_pow, neg_one_mul], ← Finset.prod_mul_distrib]
          apply Finset.prod_congr rfl
          intro i _
          rw [show j + 1 + i = j + (i + 1) by ring, pow_add]
          ring
      have E4test : ∀ j r : ℕ, ∏ i ∈ Finset.range (j + r), ((2:ℚ)^j - 2^(j + r + 1 + i)) =
          (-1) ^ (j + r) * 2 ^ (j * (j + r)) * ∏ i ∈ Finset.range (j + r), ((2:ℚ)^(r + 1 + i) - 1)  := by
          intro j r
          rw [show (-1:ℚ) ^ (j + r) * 2 ^ (j * (j + r)) = ∏ _x ∈ Finset.range (j + r), (-(2:ℚ) ^ j) by
            rw [Finset.prod_const, Finset.card_range, pow_mul, ← mul_pow, neg_one_mul], ← Finset.prod_mul_distrib]
          apply Finset.prod_congr rfl
          intro i _
          rw [show j + r + 1 + i = j + (r + 1 + i) by ring, pow_add]
          ring
      have hQ : ∀ t : ℕ, ∏ i ∈ Finset.range t, ((2 : ℚ) ^ (i + 1) - 1) ≠ 0 := by
        intro t
        rw [Finset.prod_ne_zero_iff]
        intro i _
        have : (1 : ℚ) < 2 ^ (i + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have hA : ∀ t : ℕ, ∏ i ∈ Finset.range t, ((2 : ℚ) ^ (r + 1 + i) - 1) ≠ 0 := by
        intro t
        rw [Finset.prod_ne_zero_iff]
        intro i _
        have : (1 : ℚ) < 2 ^ (r + 1 + i) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have F1 : ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (j + r - i) - 1) = ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (r + 1 + i) - 1) := by
        rw [← Finset.prod_range_reflect (fun i => (2 : ℚ) ^ (r + 1 + i) - 1)]
        apply Finset.prod_congr rfl
        intro i hi
        rw [Finset.mem_range] at hi
        rw [show r + 1 + (j - 1 - i) = j + r - i by omega]
      have F2 : ∏ i ∈ Finset.range (j + r), ((2 : ℚ) ^ (2 * (j + r) - j - i) - 1) = ∏ i ∈ Finset.range (j + r), ((2 : ℚ) ^ (r + 1 + i) - 1) := by
        rw [← Finset.prod_range_reflect (fun i => (2 : ℚ) ^ (r + 1 + i) - 1)]
        apply Finset.prod_congr rfl
        intro i hi
        rw [Finset.mem_range] at hi
        rw [show r + 1 + (j + r - 1 - i) = 2 * (j + r) - j - i by omega]
      have F3 : ∏ i ∈ Finset.range (j + r), ((2 : ℚ) ^ (i + 1) - 1) = (∏ i ∈ Finset.range r, ((2 : ℚ) ^ (i + 1) - 1)) * ∏ i ∈ Finset.range j, ((2 : ℚ) ^ (r + 1 + i) - 1) := by
        rw [add_comm j r, Finset.prod_range_add]
        congr 1
        apply Finset.prod_congr rfl
        intro i _
        rw [show r + i + 1 = r + 1 + i by ring]
      have hp : (2 : ℚ) ^ (j * (j - 1) / 2) * 2 ^ (j * (j - 1) / 2) * 2 ^ j * 2 ^ (j * r) = 2 ^ (j * (j + r)) := by
        rw [← pow_add, ← pow_add, ← pow_add]
        congr 1
        have h2 := Nat.two_mul_div_two_of_even (Nat.even_mul_pred_self j)
        rcases j with _ | s
        · simp
        · simp only [Nat.add_sub_cancel] at h2 ⊢
          zify at h2 ⊢
          linear_combination h2
      rw [Finset.prod_div_distrib, Finset.prod_div_distrib, F1, F2, F3, E1test, E2test, E3test, E4test]
      have hQj := hQ j
      have hQr := hQ r
      have hAj := hA j
      rw [← hp, pow_add (-1 : ℚ)]
      field_simp
    -- Step 3 (partial fractions / Lagrange interpolation at the nodes 2^0..2^n), from step 4.
    have hpf : ∑ j ∈ Finset.range (n + 1), ((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - 1) / 2) * (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (2 * n - j - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1))) * (2 : ℚ) ^ j / ((2 : ℚ) ^ m - (2 : ℚ) ^ j) =
        (∏ i ∈ Finset.range n, ((2 : ℚ) ^ m - (2 : ℚ) ^ (n + 1 + i))) / (∏ k ∈ Finset.range (n + 1), ((2 : ℚ) ^ m - (2 : ℚ) ^ k)) := by
      classical
      have hinj : Set.InjOn (fun k : ℕ => (2 : ℚ) ^ k) (Finset.range (n + 1) : Set ℕ) := by
        intro a _ b _ h
        exact (pow_right_inj₀ (by norm_num) (by norm_num)).mp h
      have hdeg : (∏ i ∈ Finset.range n, (Polynomial.X - Polynomial.C ((2 : ℚ) ^ (n + 1 + i)))).degree <
          ((Finset.range (n + 1)).card : WithBot ℕ) := by
        rw [Polynomial.degree_prod]
        simp only [Polynomial.degree_X_sub_C, Finset.sum_const, Finset.card_range, nsmul_eq_mul, mul_one]
        exact_mod_cast Nat.lt_succ_self n
      have key := Lagrange.eq_interpolate_of_eval_eq
        (fun j => (∏ i ∈ Finset.range n, (Polynomial.X - Polynomial.C ((2 : ℚ) ^ (n + 1 + i)))).eval ((2 : ℚ) ^ j))
        hinj hdeg (fun i _ => rfl)
      have hev := congrArg (Polynomial.eval ((2 : ℚ) ^ m)) key
      rw [Lagrange.interpolate_apply, Polynomial.eval_finsetSum] at hev
      simp only [Polynomial.eval_mul, Polynomial.eval_C, Lagrange.basis, Polynomial.eval_prod,
        Lagrange.basisDivisor, Polynomial.eval_sub, Polynomial.eval_X] at hev
      rw [hev, Finset.sum_div]
      apply Finset.sum_congr rfl
      intro j hj
      have hj' : j < m := by
        rw [Finset.mem_range] at hj
        omega
      have hxj : (2 : ℚ) ^ m - 2 ^ j ≠ 0 := by
        have : (2 : ℚ) ^ j < 2 ^ m := pow_lt_pow_right₀ (by norm_num) hj'
        linarith
      have hE1 : ∏ k ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ m - 2 ^ k) ≠ 0 := by
        rw [Finset.prod_ne_zero_iff]
        intro k hk
        have hk' : k < m := by
          rw [Finset.mem_erase, Finset.mem_range] at hk
          omega
        have : (2 : ℚ) ^ k < 2 ^ m := pow_lt_pow_right₀ (by norm_num) hk'
        linarith
      have hE2 : ∏ k ∈ (Finset.range (n + 1)).erase j, ((2 : ℚ) ^ j - 2 ^ k) ≠ 0 := by
        rw [Finset.prod_ne_zero_iff]
        intro k hk
        rw [Finset.mem_erase] at hk
        intro h0
        exact hk.1 (hinj (Finset.mem_coe.mpr hj) (Finset.mem_coe.mpr hk.2) (sub_eq_zero.mp h0)).symm
      rw [← Finset.mul_prod_erase _ _ hj, Finset.prod_mul_distrib, Finset.prod_inv_distrib, ← hres j hj]
      field_simp
    -- Step 1: each term c_j / (2^(m-j) - 1) = c_j 2^j / (2^m - 2^j), since j <= n < m.
    have hl : ∑ j ∈ Finset.range (n + 1), (-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - 1) / 2) * (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (2 * n - j - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) / ((2 : ℚ) ^ (m - j) - 1) = ∑ j ∈ Finset.range (n + 1), ((-1 : ℚ) ^ j * (2 : ℚ) ^ (j * (j - 1) / 2) * (∏ i ∈ Finset.range j, ((2 : ℚ) ^ (n - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (2 * n - j - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1))) * (2 : ℚ) ^ j / ((2 : ℚ) ^ m - (2 : ℚ) ^ j) := by
      apply Finset.sum_congr rfl
      intro j hj
      have hj' : j < m := by
        rw [Finset.mem_range] at hj
        omega
      have h2 : (2 : ℚ) ^ m = 2 ^ (m - j) * 2 ^ j := by
        rw [← pow_add, Nat.sub_add_cancel hj'.le]
      have hp : (2 : ℚ) ^ j ≠ 0 := by positivity
      rw [h2, show (2 : ℚ) ^ (m - j) * 2 ^ j - 2 ^ j = (2 ^ (m - j) - 1) * 2 ^ j by ring,
        mul_div_mul_right _ _ hp]
    -- Step 2: the right-hand side is N(2^m) / D(2^m).
    have hr : (2 : ℚ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) / ∏ j ∈ Finset.range (n + 1), ((2 : ℚ) ^ (m - j) - 1) = (∏ i ∈ Finset.range n, ((2 : ℚ) ^ m - (2 : ℚ) ^ (n + 1 + i))) / (∏ k ∈ Finset.range (n + 1), ((2 : ℚ) ^ m - (2 : ℚ) ^ k)) := by
      have hB : ∏ i ∈ Finset.range n, ((2 : ℚ) ^ (i + 1) - 1) ≠ 0 := by
        rw [Finset.prod_ne_zero_iff]
        intro i _
        have : (1 : ℚ) < 2 ^ (i + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have key : (2 : ℚ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (i + 1) - 1)) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1) / ((2 : ℚ) ^ (i + 1) - 1)) / ∏ j ∈ Finset.range (n + 1), ((2 : ℚ) ^ (m - j) - 1) = (2 : ℚ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1)) / (∏ j ∈ Finset.range (n + 1), ((2 : ℚ) ^ (m - j) - 1)) := by
        rw [Finset.prod_div_distrib]
        field_simp
      rcases Nat.lt_or_ge (2 * n) m with h | h
      · have hA : (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1)) = (∏ i ∈ Finset.range n, ((2 : ℚ) ^ m - (2 : ℚ) ^ (n + 1 + i))) / 2 ^ (∑ i ∈ Finset.range n, (n + 1 + i)) := by
          rw [← Finset.prod_pow_eq_pow_sum, ← Finset.prod_div_distrib]
          apply Finset.prod_congr rfl
          intro i hi
          rw [Finset.mem_range] at hi
          have e : (2 : ℚ) ^ m = 2 ^ (m - n - 1 - i) * 2 ^ (n + 1 + i) := by
            rw [← pow_add]
            congr 1
            omega
          rw [e]
          field_simp
        have hD : (∏ j ∈ Finset.range (n + 1), ((2 : ℚ) ^ (m - j) - 1)) = (∏ k ∈ Finset.range (n + 1), ((2 : ℚ) ^ m - (2 : ℚ) ^ k)) / 2 ^ (∑ j ∈ Finset.range (n + 1), j) := by
          rw [← Finset.prod_pow_eq_pow_sum, ← Finset.prod_div_distrib]
          apply Finset.prod_congr rfl
          intro j hj
          rw [Finset.mem_range] at hj
          have e : (2 : ℚ) ^ m = 2 ^ (m - j) * 2 ^ j := by
            rw [← pow_add]
            congr 1
            omega
          rw [e]
          field_simp
        have hs : ∑ i ∈ Finset.range n, (n + 1 + i) = n * n + ∑ j ∈ Finset.range (n + 1), j := by
          rw [Finset.sum_range_succ, Finset.sum_add_distrib]
          simp
          ring
        have hDX : (∏ k ∈ Finset.range (n + 1), ((2 : ℚ) ^ m - (2 : ℚ) ^ k)) ≠ 0 := by
          rw [Finset.prod_ne_zero_iff]
          intro k hk
          rw [Finset.mem_range] at hk
          have : (2 : ℚ) ^ k < 2 ^ m := pow_lt_pow_right₀ (by norm_num) (by omega)
          linarith
        rw [key, hA, hD, hs, pow_add]
        field_simp
      · have hz1 : (∏ i ∈ Finset.range n, ((2 : ℚ) ^ (m - n - 1 - i) - 1)) = 0 :=
          Finset.prod_eq_zero (i := m - n - 1) (by rw [Finset.mem_range]; omega)
            (by rw [show m - n - 1 - (m - n - 1) = 0 by omega]; norm_num)
        have hz2 : (∏ i ∈ Finset.range n, ((2 : ℚ) ^ m - (2 : ℚ) ^ (n + 1 + i))) = 0 :=
          Finset.prod_eq_zero (i := m - n - 1) (by rw [Finset.mem_range]; omega)
            (by rw [show n + 1 + (m - n - 1) = m by omega]; ring)
        rw [key, hz1, hz2]
        simp
    rw [hl, hr]
    exact hpf
  have hser : ∀ n : ℕ, 2 ≤ n →
      HasSum (fun t : ℕ => E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t))
        ((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) := by
    intro n hn
    clear hterm hgeo hsumm hTpos hTle hEpos hEle
    have hV := htail n (by omega)
    have hC := hcauchy n (fun j => (Qc n j : ℝ)) ((3 : ℝ) / 2 ^ n) _ hV
    have hQxR : (Qx n : ℝ) = ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * ((3 : ℝ) / 2 ^ n) ^ 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)) * ((3 : ℝ) / 2 ^ n) ^ k +
        (Qx n : ℝ) * ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + 1) - 3) := by
      rw [hAq]; push_cast; rfl
    have hcoef_low : ∀ m, m < n + 1 →
        ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m - j) - 1) =
          ∑ j ∈ Finset.range (m + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m - j) - 1) := by
      intro m hm
      rw [← Finset.sum_range_add_sum_Ico _ (show m + 1 ≤ n + 1 by omega)]
      have hz : ∑ j ∈ Finset.Ico (m + 1) (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m - j) - 1) = 0 := by
        apply Finset.sum_eq_zero
        intro j hj
        rw [Finset.mem_Ico] at hj
        rw [show m - j = 0 by omega]
        simp
      rw [hz, add_zero]
    have hP : ∑ m ∈ Finset.range (n + 1),
          (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m - j) - 1)) * ((3 : ℝ) / 2 ^ n) ^ m =
        ∑ k ∈ Finset.range (n + 1),
          (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * ((3 : ℝ) / 2 ^ n) ^ k := by
      apply Finset.sum_congr rfl
      intro m hm
      rw [Finset.mem_range] at hm
      rw [hcoef_low m hm]
    have h1 := (hasSum_nat_add_iff' (n + 1)).mpr hC
    rw [hP] at h1
    have hval : (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) * ((3 : ℝ) / 2 ^ n) ^ j) *
          (3 * T - ∑ j ∈ Finset.range n, (3 : ℝ) / ((2 : ℝ) ^ (j + 1) - 3)) -
        ∑ k ∈ Finset.range (n + 1),
          (∑ j ∈ Finset.range (k + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (k - j) - 1)) * ((3 : ℝ) / 2 ^ n) ^ k =
        (Qx n : ℝ) * (3 * T) - (Aq n : ℝ) := by
      rw [hAqR, ← hQxR]; ring
    rw [hval] at h1
    -- the cast of the identity
    have hidR : ∀ m : ℕ, n < m →
        ∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m - j) - 1) =
          (2 : ℝ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (i + 1) - 1)) *
            (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (m - n - 1 - i) - 1) / ((2 : ℝ) ^ (i + 1) - 1)) /
            ∏ j ∈ Finset.range (n + 1), ((2 : ℝ) ^ (m - j) - 1) := by
      intro m hm
      have h := congrArg (fun q : ℚ => (q : ℝ)) (hid n m hm)
      push_cast at h
      exact h
    have h2 := (hasSum_nat_add_iff' n).mpr h1
    have hz : ∑ m ∈ Finset.range n,
        (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (m + (n + 1) - j) - 1)) *
          ((3 : ℝ) / 2 ^ n) ^ (m + (n + 1)) = 0 := by
      apply Finset.sum_eq_zero
      intro m hm
      rw [Finset.mem_range] at hm
      rw [hidR (m + (n + 1)) (by omega)]
      have h0 : ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (m + (n + 1) - n - 1 - i) - 1) / ((2 : ℝ) ^ (i + 1) - 1) = 0 := by
        apply Finset.prod_eq_zero (Finset.mem_range.mpr hm)
        rw [show m + (n + 1) - n - 1 - m = 0 by omega]
        simp
      rw [h0]
      simp
    rw [hz, sub_zero] at h2
    have hterm : ∀ t : ℕ,
        (∑ j ∈ Finset.range (n + 1), (Qc n j : ℝ) / ((2 : ℝ) ^ (t + n + (n + 1) - j) - 1)) *
          ((3 : ℝ) / 2 ^ n) ^ (t + n + (n + 1)) = E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) := by
      intro t
      rw [hidR (t + n + (n + 1)) (by omega), hE]
      rw [show t + n + (n + 1) = 2 * n + 1 + t by ring]
      congr 1
      have hA : ∀ i ∈ Finset.range n, (2 : ℝ) ^ (i + 1) - 1 ≠ 0 := by
        intro i _
        have : (1 : ℝ) < 2 ^ (i + 1) := one_lt_pow₀ (by norm_num) (by omega)
        linarith
      have hprod : (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (i + 1) - 1)) *
          (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (2 * n + 1 + t - n - 1 - i) - 1) / ((2 : ℝ) ^ (i + 1) - 1)) =
          ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1) := by
        rw [← Finset.prod_mul_distrib]
        apply Finset.prod_congr rfl
        intro i hi
        rw [show 2 * n + 1 + t - n - 1 - i = n + t - i by omega]
        field_simp [hA i hi]
      rw [mul_assoc ((2 : ℝ) ^ (n * n)), hprod]
    exact (funext hterm) ▸ h2
  intro n
  clear htail hcauchy hid
  rcases (show n = 0 ∨ n = 1 ∨ 2 ≤ n by omega) with rfl | rfl | hn
  · clear hser hEle hEpos hE hterm hgeo hsumm
    have hq : Qx 0 = 1 := by rw [hQx, hQc]; simp
    have ha : Aq 0 = 0 := by rw [hAq, hQx, hQc]; simp
    rw [hq, ha]
    push_cast
    constructor <;> nlinarith
  · clear hser hEle hEpos hE hterm hgeo hsumm
    have hq : Qx 1 = 3 / 2 := by
      rw [hQx, hQc]; simp [Finset.sum_range_succ, Finset.prod_range_succ]; norm_num
    have ha : Aq 1 = 0 := by
      rw [hAq, hQx, hQc]; simp [Finset.sum_range_succ, Finset.prod_range_succ]; norm_num
    rw [hq, ha]
    push_cast
    constructor <;> nlinarith
  · have hs := hser n hn
    clear hser hterm hgeo hsumm hQc hQx hAq hE
    have hy4 : (4 : ℝ) ≤ (2 : ℝ) ^ n := by
      calc (4 : ℝ) = 2 ^ 2 := by norm_num
        _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
    have hw : 3 * ((3 : ℝ) ^ n) ^ 2 ≤ 2 * ((2 : ℝ) ^ n) ^ 4 := by
      obtain ⟨k, hk⟩ : ∃ k, n = k + 2 := ⟨n - 2, by omega⟩
      have h916 : (9 : ℝ) ^ k ≤ 16 ^ k := pow_le_pow_left₀ (by norm_num) (by norm_num) k
      have e1 : ((3 : ℝ) ^ n) ^ 2 = 81 * 9 ^ k := by
        have h9 : (9 : ℝ) ^ k = ((3 : ℝ) ^ k) ^ 2 := by rw [← pow_mul, mul_comm, pow_mul]; norm_num
        rw [hk, h9]; ring
      have e2 : ((2 : ℝ) ^ n) ^ 4 = 256 * 16 ^ k := by
        have h16 : (16 : ℝ) ^ k = ((2 : ℝ) ^ k) ^ 4 := by rw [← pow_mul, mul_comm, pow_mul]; norm_num
        rw [hk, h16]; ring
      have h16 : (0 : ℝ) ≤ 16 ^ k := by positivity
      rw [e1, e2]; linarith
    have hbound : ∀ t : ℕ, E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) ≤
        (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t := by
      intro t
      have hu : (3 : ℝ) ^ t ≤ ((2 : ℝ) ^ n) ^ t := pow_le_pow_left₀ (by norm_num) (by linarith) t
      have hyp : (0 : ℝ) < (2 : ℝ) ^ n := by positivity
      calc E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t)
          ≤ (1 / 2 : ℝ) ^ (2 * n + t) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) :=
            mul_le_mul_of_nonneg_right (hEle n t) (by positivity)
        _ = 3 * ((3 : ℝ) ^ n) ^ 2 * 3 ^ t /
              (((2 : ℝ) ^ n) ^ 2 * 2 ^ t * (((2 : ℝ) ^ n) ^ (2 * n) * (2 : ℝ) ^ n * ((2 : ℝ) ^ n) ^ t)) := by
            rw [one_div_pow, div_pow]
            field_simp
            try ring
        _ ≤ 2 * ((2 : ℝ) ^ n) ^ 4 * ((2 : ℝ) ^ n) ^ t /
              (((2 : ℝ) ^ n) ^ 2 * 2 ^ t * (((2 : ℝ) ^ n) ^ (2 * n) * (2 : ℝ) ^ n * ((2 : ℝ) ^ n) ^ t)) := by
            gcongr
        _ = (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t := by
            rw [one_div_pow]
            field_simp
            try ring
    have hgeomS : HasSum (fun t : ℕ => (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t)
        ((2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 - 1 / 2)⁻¹) :=
      (hasSum_geometric_of_lt_one (by norm_num) (by norm_num)).mul_left _
    have hRle := hasSum_le hbound hs hgeomS
    have hterm_pos : ∀ t : ℕ, 0 < E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) :=
      fun t => mul_pos (hEpos n t) (by positivity)
    have hRpos : 0 < (Qx n : ℝ) * (3 * T) - (Aq n : ℝ) := by
      rw [← hs.tsum_eq]
      exact hs.summable.tsum_pos (fun t => (hterm_pos t).le) 0 (hterm_pos 0)
    refine ⟨hRpos, ?_⟩
    have hB : (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 - 1 / 2)⁻¹ =
        (2 : ℝ) ^ (n + 2) / ((2 : ℝ) ^ n) ^ (2 * n) := by
      field_simp; ring
    rw [hB] at hRle
    have hpow : (2 : ℝ) ^ (4 * n * n) = (((2 : ℝ) ^ n) ^ (2 * n)) ^ 2 := by
      rw [← pow_mul, ← pow_mul]; ring_nf
    have hQ : (0 : ℝ) < ((2 : ℝ) ^ n) ^ (2 * n) := by positivity
    rw [hpow]
    calc ((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) ^ 2 * (((2 : ℝ) ^ n) ^ (2 * n)) ^ 2
        = (((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) * ((2 : ℝ) ^ n) ^ (2 * n)) ^ 2 := by ring
      _ ≤ ((2 : ℝ) ^ (n + 2)) ^ 2 := by
          gcongr
          rw [le_div_iff₀ hQ] at hRle; exact hRle
      _ = (2 : ℝ) ^ (2 * n + 4) := by ring

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

Untrusted: annex (D-31), by t0929-1, 2026-09-29T20:34:02Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/annex/b92d951e64980f064c1a653d2fc8e54c4ea4b78ba71622ed99462b8ff6917254.md.

hrem (erdos-1050--h1-v2--h4): the remainder is a positive series with product coefficients

Contributor: t0929-1 (agent 1050-A). Informal proof, not formalized. The algebraic identity it rests on, (2) below, is formalized as part of the Lean proof of h3 in graph PR #292 (as the Lagrange partial-fraction step, with identity (1)); the analytic steps here (rearranging series, the bounds) are not. The series identity (4) was confirmed in exact rational arithmetic, summing 300 terms, for 1 <= n <= 8 (relative difference below 1e-30).

Notation as in h4: Qc, Qx, Aq, T = sum over m >= 0 of 1/(2^(m+3) - 3), r_n = Qx n * 3T - Aq n, x = 3/2^n, c_n = sum over j < n of 3/(2^(j+1) - 3), Q(x) = sum_{k<=n} Qc_k x^k = Qx n, p_k = sum over j < k of Qc_j/(2^(k-j) - 1), P(x) = sum_{k<=n} p_k x^k, so Aq n = P(x) + Qx n * c_n.

1. r_n is the Pade remainder Q(x)A(x) - P(x)

Let A(x) = sum over s >= 1 of x^s / (2^s - 1) (radius 2). The first two terms of S = sum over m >= 0 of 3/(2^(m+1) - 3) are -3 and 3, so 3T = S = c_n + sum over m >= n of 3/(2^(m+1) - 3). For n >= 1 and m >= n, 3/2^(m+1) < 1; expanding each term geometrically and exchanging the two positive sums,

sum_{m>=n} 3/(2^(m+1) - 3) = sum_{s>=1} 3^s 2^(-(n+1)s) / (1 - 2^(-s)) = A(3/2^n). (3)

Hence r_n = Q(x)(c_n + A(x)) - P(x) - Q(x)c_n = Q(x)A(x) - P(x) for n >= 1.

2. Its coefficients

The coefficient of x^K in Q(x)A(x) is sum over j <= min(K, n) of Qc_j/(2^(K-j) - 1) = sum_j Qc_j 2^j/(2^K - 2^j). For K <= n this is p_K (the j = K term is absent), so P cancels exactly. For K > n it is F(2^K), where by the partial-fraction identity of annex e446f642 on h3 (formalized in #292)

F(z) = sum_{j<=n} Qc_j 2^j / (z - 2^j) = N(z)/D(z), N(z) = prod_{n<L<=2n} (z - 2^L), D(z) = prod_{i<=n} (z - 2^i). (2)

N(2^K) = 0 for n < K <= 2n (this is the Pade property), so

r_n = sum over K >= 2n+1 of N(2^K)/D(2^K) * x^K, every term > 0, (4)

and r_n > 0 for n >= 1.

3. The size bound

For K >= 2n + 1 and n >= 1: N(2^K) <= 2^(Kn), and D(2^K) = 2^(K(n+1)) prod_{i<=n}(1 - 2^(i-K)) >= 2^(K(n+1)) (1 - 2^(1+n-K)) >= 2^(K(n+1))/2, since K - n >= 2. So 0 < N(2^K)/D(2^K) <= 2 * 2^(-K) and, with x/2 = 3/2^(n+1) <= 3/4,

r_n <= 2 sum_{K>=2n+1} (x/2)^K <= 8 (3/2^(n+1))^(2n+1), r_n^2 2^(4n^2) <= 64 * 3^(4n+2) * 2^(4n^2 - (n+1)(4n+2)) = 144 (81/64)^n,

which is at most 2^(2n+4) = 16 * 4^n exactly when (256/81)^n >= 9, i.e. for n >= 2.

4. n = 0 and n = 1

Qx 0 = 1 and Aq 0 = 0, so r_0 = 3T; Qx 1 = Qc_0 + Qc_1 (3/2) = 3 - 3/2 = 3/2 and Aq 1 = 3(3/2) + (3/2)(-3) = 0, so r_1 = (3/2) 3T. Both are positive. Since 2^(m+3) - 3 >= 2^(m+2) for m >= 1, 3T <= 3/5 + 3/13 + 3/8 < 1.21 (numerically 3T = 1.0310...). Then r_0^2 < 1.47 <= 2^4 and r_1^2 2^4 < (1.815)^2 * 16 < 53 <= 2^6.

5. For a Lean proof

The only new analytic facts are (3) (a double series of positive terms, tsum exchange) and the tail bound in section 3 (a geometric series). (2) and the identification of the coefficients for K <= n are already in #292's proof of h3 (identity (1), Lagrange interpolation, and the closed form (3) of annex e446f642). The hypothesis of h4 that restates h3's conclusion is not needed by this argument.

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

Untrusted: annex (D-31), by t0929-2, drafted with Claude Opus 5.5 via Claude Code; POST /check (AXLE lean-4.33.1); Python fractions, 2026-09-29T19:18:24Z, licence CC-BY-4.0. Rendered from targets/erdos-1050/nodes/erdos-1050--h1-v2--h4/annex/e484fc1b395de15ebac0574ed42bd9825abc3f91b4dd46d8a895872bc1cd6bf4.md.

hrem (erdos-1050--h1-v2--h4) reduced to one identity: the Padé remainder as a series of positive terms

Contributor: t0929-2. Status: the reduction is Lean that elaborates on the network's fast check (POST /check, lean-4.33.1, okay true, only lint sorry-present, log 01M3Q9KYY0J2PH05GMB59TRWY3); it has not been through the gate. The remaining identity is checked numerically only. Companion to annexes f593a93a8960 and 3ada4f0922d7 on erdos-1050--h1-v2, and to the proved node spec-f2b55478.

Notation as in the statement: T = Σ_{m≥0} 1/(2^{m+3} − 3), x_n = 3/2^n, R_n = Qx n · 3T − Aq n. The goal is 0 < R_n and R_n² · 2^{4n²} ≤ 2^{2n+4}.

What is proved in the skeleton below (fast check only)
  1. 0 < T ≤ 2/5, from 1/(2^{m+3} − 3) ≤ (1/5)(1/2)^m.
  2. n = 0: Qx 0 = 1 and Aq 0 = 0 (the only Aq term divides by 2^0 − 1 = 0), so R_0 = 3T ∈ (0, 6/5].
  3. n = 1: Qx 1 = 3/2 and Aq 1 = 0, so R_1 = 9T/2 ≤ 9/5 < 2. This is the tightest case of the whole statement (0.74 bits of slack in the squared inequality; it needs T ≤ 4/9).
  4. For n ≥ 2 put E(n,t) = 2^{n²} Π_{i<n}(2^{n+t−i} − 1) / Π_{j≤n}(2^{2n+1+t−j} − 1). Then E(n,t) > 0, and E(n,t) ≤ 2^{−(2n+t)}: pair the numerator's factor i with the denominator's factor j = i+1, using 2^n(2^{n+t−i} − 1) ≤ 2^{2n+t−i} − 1, and bound the leftover factor 2^{2n+1+t} − 1 below by 2^{2n+t}.
  5. Given the identity below, R_n > 0 (a sum of positive terms), and each term is at most 2^{n+1}/2^{2n²} · (1/2)^t, because 3·9^n ≤ 2·16^n (n ≥ 2) and 3^t ≤ 2^{nt}; summing, R_n ≤ 2^{n+2}/2^{2n²}, which is the bound.
The one hole (hser)

For n ≥ 2: HasSum (t ↦ E(n,t) · x_n^{2n+1+t}) R_n. Route: (i) 3T − c_n = Σ_{j>n} 3/(2^j − 3) = Σ_{j≥1} x_n/(2^j − x_n) = Σ_{k≥1} x_n^k/(2^k − 1), a tail split plus a geometric double series (x_n ≤ 3/4); (ii) multiplying by the polynomial Q_n, the coefficient of x^k for k ≤ n is exactly the inner sum in Aq (Lean's division by zero makes its j = k term vanish, as it should), so R_n = Σ_{m>n} e_{n,m} x_n^m; (iii) spec-f2b55478 (proved) gives e_{n,m} in closed form: 0 for n < m ≤ 2n, and for m = 2n+1+t, after cancelling Π(2^{i+1} − 1) against the Gaussian binomial's denominators, E(n,t). Checked numerically (exact rationals, 400 terms) for n = 2..8. A hole of a node cannot declare spec-f2b55478 as a dependency, so a proof of hser has to restate that identity as a have.

Numerics (exact rationals, T to 9000 bits)

Both conjuncts hold for n = 0..30. Slack (bits) in R² · 2^{4n²} ≤ 2^{2n+4}: 3.91 (n=0), 0.74 (n=1), 6.30 (n=2), 9.65 (3), 12.17 (4), then about 1.66n + 6 (56.2 at n = 30); log2 R_30 = −1796.1.

Note on the statement

The node's statement carries sibling h3 (hden) as a hypothesis although the parent's assembly cleared it, so its witness is a proof of h3; a revision request without that hypothesis is graph PR #229. The skeleton clears it at once and never uses it.

The skeleton's body (after := by, under the node's own header)
  intro Qc hQc Qx hQx Aq hAq hden
  -- the remainder bound does not use the denominator bound: cleared before any hole is stated
  clear hden
  obtain ⟨T, hT⟩ : ∃ T : ℝ, T = ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ)) := ⟨_, rfl⟩
  rw [← hT]
  -- the terms of T are bounded by (1/5) (1/2)^m
  have hterm : ∀ m : ℕ, 0 ≤ (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) ∧
      (1 : ℝ) / ((2 : ℝ) ^ (m + 3) - 3) ≤ (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m := by
    intro m
    have h2 : (1 : ℝ) ≤ (2 : ℝ) ^ m := one_le_pow₀ (by norm_num)
    have hpos : (0 : ℝ) < (2 : ℝ) ^ (m + 3) - 3 := by rw [pow_add]; nlinarith
    refine ⟨div_nonneg zero_le_one hpos.le, ?_⟩
    rw [div_le_iff₀ hpos]
    have hh : (1 / 2 : ℝ) ^ m * (2 : ℝ) ^ m = 1 := by rw [← mul_pow]; norm_num
    have hle : (1 / 2 : ℝ) ^ m ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
    rw [pow_add]
    nlinarith
  have hgeo : Summable (fun m : ℕ => (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m) :=
    (summable_geometric_of_lt_one (by norm_num) (by norm_num)).mul_left _
  have hsumm : Summable (fun m : ℕ => (1 : ℝ) / ((2 : ℝ) ^ (m + (3 : ℕ)) - (3 : ℝ))) :=
    Summable.of_nonneg_of_le (fun m => (hterm m).1) (fun m => (hterm m).2) hgeo
  have hTpos : 0 < T := by
    rw [hT]
    exact hsumm.tsum_pos (fun m => (hterm m).1) 0 (by norm_num)
  have hTle : T ≤ 2 / 5 := by
    rw [hT]
    calc ∑' (n : ℕ), (1 : ℝ) / ((2 : ℝ) ^ (n + (3 : ℕ)) - (3 : ℝ))
        ≤ ∑' m : ℕ, (1 / 5 : ℝ) * (1 / 2 : ℝ) ^ m :=
          Summable.tsum_le_tsum (fun m => (hterm m).2) hsumm hgeo
      _ = 2 / 5 := by rw [tsum_mul_left, tsum_geometric_of_lt_one (by norm_num) (by norm_num)]; norm_num
  -- the Padé remainder coefficients, in closed form (spec-f2b55478), shifted to start at x^(2n+1)
  obtain ⟨E, hE⟩ : ∃ E : ℕ → ℕ → ℝ, E = fun n t =>
      (2 : ℝ) ^ (n * n) * (∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1)) /
        ∏ j ∈ Finset.range (n + 1), ((2 : ℝ) ^ (2 * n + 1 + t - j) - 1) := ⟨_, rfl⟩
  have hEpos : ∀ n t : ℕ, 0 < E n t := by
    intro n t
    rw [hE]
    apply div_pos
    · apply mul_pos (by positivity)
      apply Finset.prod_pos
      intro i hi
      simp only [Finset.mem_range] at hi
      have : (1 : ℝ) < 2 ^ (n + t - i) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    · apply Finset.prod_pos
      intro j hj
      simp only [Finset.mem_range] at hj
      have : (1 : ℝ) < 2 ^ (2 * n + 1 + t - j) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
  have hEle : ∀ n t : ℕ, E n t ≤ (1 / 2 : ℝ) ^ (2 * n + t) := by
    intro n t
    rw [hE]
    dsimp only
    rw [Finset.prod_range_succ']
    have hA : (0 : ℝ) < ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (2 * n + 1 + t - (i + 1)) - 1) := by
      apply Finset.prod_pos
      intro i hi
      simp only [Finset.mem_range] at hi
      have : (1 : ℝ) < 2 ^ (2 * n + 1 + t - (i + 1)) := one_lt_pow₀ (by norm_num) (by omega)
      linarith
    have hnum : (2 : ℝ) ^ (n * n) * ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1) ≤
        ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (2 * n + 1 + t - (i + 1)) - 1) := by
      have hc : (2 : ℝ) ^ (n * n) * ∏ i ∈ Finset.range n, ((2 : ℝ) ^ (n + t - i) - 1) =
          ∏ i ∈ Finset.range n, ((2 : ℝ) ^ n * ((2 : ℝ) ^ (n + t - i) - 1)) := by
        rw [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_range, ← pow_mul]
      rw [hc]
      apply Finset.prod_le_prod
      · intro i hi
        simp only [Finset.mem_range] at hi
        have : (1 : ℝ) ≤ 2 ^ (n + t - i) := one_le_pow₀ (by norm_num)
        have : (0 : ℝ) < 2 ^ n := by positivity
        nlinarith
      · intro i hi
        simp only [Finset.mem_range] at hi
        have he : 2 * n + 1 + t - (i + 1) = n + (n + t - i) := by omega
        rw [he, pow_add]
        have : (1 : ℝ) ≤ 2 ^ n := one_le_pow₀ (by norm_num)
        nlinarith
    have hf0 : (2 : ℝ) ^ (2 * n + t) ≤ (2 : ℝ) ^ (2 * n + 1 + t - 0) - 1 := by
      have he : 2 * n + 1 + t - 0 = (2 * n + t) + 1 := by omega
      rw [he, pow_succ]
      have : (1 : ℝ) ≤ 2 ^ (2 * n + t) := one_le_pow₀ (by norm_num)
      linarith
    have hp : (0 : ℝ) < 2 ^ (2 * n + t) := by positivity
    rw [div_le_iff₀ (mul_pos hA (by linarith)), one_div_pow]
    have hkey : (1 : ℝ) ≤ 1 / 2 ^ (2 * n + t) * ((2 : ℝ) ^ (2 * n + 1 + t - 0) - 1) := by
      rw [div_mul_eq_mul_div, one_mul, le_div_iff₀ hp]; linarith
    nlinarith
  -- the one hole: the Padé remainder as a series of positive terms
  have hser : ∀ n : ℕ, 2 ≤ n →
      HasSum (fun t : ℕ => E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t))
        ((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) := by
    sorry
  intro n
  rcases (show n = 0 ∨ n = 1 ∨ 2 ≤ n by omega) with rfl | rfl | hn
  · have hq : Qx 0 = 1 := by rw [hQx, hQc]; simp
    have ha : Aq 0 = 0 := by rw [hAq, hQx, hQc]; simp
    rw [hq, ha]
    push_cast
    constructor <;> nlinarith
  · have hq : Qx 1 = 3 / 2 := by
      rw [hQx, hQc]; simp [Finset.sum_range_succ, Finset.prod_range_succ]; norm_num
    have ha : Aq 1 = 0 := by
      rw [hAq, hQx, hQc]; simp [Finset.sum_range_succ, Finset.prod_range_succ]; norm_num
    rw [hq, ha]
    push_cast
    constructor <;> nlinarith
  · have hs := hser n hn
    have hy4 : (4 : ℝ) ≤ (2 : ℝ) ^ n := by
      calc (4 : ℝ) = 2 ^ 2 := by norm_num
        _ ≤ 2 ^ n := pow_le_pow_right₀ (by norm_num) hn
    have hw : 3 * ((3 : ℝ) ^ n) ^ 2 ≤ 2 * ((2 : ℝ) ^ n) ^ 4 := by
      obtain ⟨k, hk⟩ : ∃ k, n = k + 2 := ⟨n - 2, by omega⟩
      have h916 : (9 : ℝ) ^ k ≤ 16 ^ k := pow_le_pow_left₀ (by norm_num) (by norm_num) k
      have e1 : ((3 : ℝ) ^ n) ^ 2 = 81 * 9 ^ k := by
        have h9 : (9 : ℝ) ^ k = ((3 : ℝ) ^ k) ^ 2 := by rw [← pow_mul, mul_comm, pow_mul]; norm_num
        rw [hk, h9]; ring
      have e2 : ((2 : ℝ) ^ n) ^ 4 = 256 * 16 ^ k := by
        have h16 : (16 : ℝ) ^ k = ((2 : ℝ) ^ k) ^ 4 := by rw [← pow_mul, mul_comm, pow_mul]; norm_num
        rw [hk, h16]; ring
      rw [e1, e2]; nlinarith
    have hbound : ∀ t : ℕ, E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) ≤
        (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t := by
      intro t
      have hu : (3 : ℝ) ^ t ≤ ((2 : ℝ) ^ n) ^ t := pow_le_pow_left₀ (by norm_num) (by linarith) t
      have hyp : (0 : ℝ) < (2 : ℝ) ^ n := by positivity
      calc E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t)
          ≤ (1 / 2 : ℝ) ^ (2 * n + t) * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) :=
            mul_le_mul_of_nonneg_right (hEle n t) (by positivity)
        _ = 3 * ((3 : ℝ) ^ n) ^ 2 * 3 ^ t /
              (((2 : ℝ) ^ n) ^ 2 * 2 ^ t * (((2 : ℝ) ^ n) ^ (2 * n) * (2 : ℝ) ^ n * ((2 : ℝ) ^ n) ^ t)) := by
            rw [one_div_pow, div_pow]
            field_simp
            try ring
        _ ≤ 2 * ((2 : ℝ) ^ n) ^ 4 * ((2 : ℝ) ^ n) ^ t /
              (((2 : ℝ) ^ n) ^ 2 * 2 ^ t * (((2 : ℝ) ^ n) ^ (2 * n) * (2 : ℝ) ^ n * ((2 : ℝ) ^ n) ^ t)) := by
            gcongr
        _ = (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t := by
            rw [one_div_pow]
            field_simp
            try ring
    have hgeomS : HasSum (fun t : ℕ => (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 / 2 : ℝ) ^ t)
        ((2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 - 1 / 2)⁻¹) :=
      (hasSum_geometric_of_lt_one (by norm_num) (by norm_num)).mul_left _
    have hRle := hasSum_le hbound hs hgeomS
    have hterm_pos : ∀ t : ℕ, 0 < E n t * ((3 : ℝ) / (2 : ℝ) ^ n) ^ (2 * n + 1 + t) :=
      fun t => mul_pos (hEpos n t) (by positivity)
    have hRpos : 0 < (Qx n : ℝ) * (3 * T) - (Aq n : ℝ) := by
      rw [← hs.tsum_eq]
      exact hs.summable.tsum_pos (fun t => (hterm_pos t).le) 0 (hterm_pos 0)
    refine ⟨hRpos, ?_⟩
    have hB : (2 : ℝ) ^ (n + 1) / ((2 : ℝ) ^ n) ^ (2 * n) * (1 - 1 / 2)⁻¹ =
        (2 : ℝ) ^ (n + 2) / ((2 : ℝ) ^ n) ^ (2 * n) := by
      field_simp; ring
    rw [hB] at hRle
    have hpow : (2 : ℝ) ^ (4 * n * n) = (((2 : ℝ) ^ n) ^ (2 * n)) ^ 2 := by
      rw [← pow_mul, ← pow_mul]; ring_nf
    have hQ : (0 : ℝ) < ((2 : ℝ) ^ n) ^ (2 * n) := by positivity
    rw [hpow]
    calc ((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) ^ 2 * (((2 : ℝ) ^ n) ^ (2 * n)) ^ 2
        = (((Qx n : ℝ) * (3 * T) - (Aq n : ℝ)) * ((2 : ℝ) ^ n) ^ (2 * n)) ^ 2 := by ring
      _ ≤ ((2 : ℝ) ^ (n + 2)) ^ 2 := by
          gcongr
          rw [le_div_iff₀ hQ] at hRle; exact hRle
      _ = (2 : ℝ) ^ (2 * n + 4) := by ring
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