1. FormalConjectures
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  3. IdonealCompleteness
/- Copyright 2026 The Formal Conjectures Authors. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at https://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ import FormalConjecturesUtil

Idoneal numbers completeness conjecture

An integer $D>0$ is idoneal if every integer that can be expressed in exactly one way (up to order and signs) as $x^2 + D y^2$ with gcd(x, Dy)=1 is a prime power or twice a prime power.

The Idoneal Numbers Completeness Conjecture asserts that the following list of 65 numbers is complete: 1,2,3,4,5,6,7,8,9,10,12,13,15,16,18,21,22,24,25,28,30,33,37,40,42,45,48, 57,58,60,70,72,78,85,88,93,102,105,112,120,130,133,165,168,177,190,210,232, 240,253,273,280,312,330,345,357,385,408,462,520,760,840,1320,1365,1848. References:

namespace Idoneal

Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integers $a, b,$ and $c$.

def IsIdoneal (n : ℕ) : Prop := 0 < n ∧ ¬ ∃ a b c : ℕ, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c

The 65 known idoneal numbers that are conjectured to be the only idoneal numbers.

def knownIdonealNumbers : Finset ℕ := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848}

Reduces the unbounded search for a representation n = a*b + b*c + a*c (with 0 < a < b < c) to a bounded, decidable double search over a, b ∈ range (n+1).

The third variable is not searched: for a fixed pair a, b the equation n = a*b + c*(a+b) pins down c = (n - a*b) / (a+b), so the witness c is recovered by exact division. The forward direction uses a, b ≤ n (each pairwise product is at most n) to land the pair in range (n+1).

@[category API, AMS 11] private theorem exists_triple_iff_bounded (n : ℕ) : (∃ a b c : ℕ, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ (∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))) := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) → ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))n:ℕ⊢ (∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))) → ∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) → ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * c⊢ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) have hbn : b ≤ n := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) All goals completed! 🐙 have han : a ≤ n := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) All goals completed! 🐙 have hsum : n - a * b = c * (a + b) := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) have : n = a * b + c * (a + b) := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 4)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 4)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 4)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 3))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 3 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 4)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 3)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4)) (Eq.refl false))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha)))) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl false)))) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + b * c + a * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + b * c) (a * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (b * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul b c)))) (Nat.cast_mul a c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n b) a_1))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 b c) hbc))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ c))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))))han:a ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (id (Eq.mp (Mathlib.Tactic.Zify.natCast_le._simp_1 b n) hbn)))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n a) a_1))))))⊢ n = a * b + a * c + b * c; n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < chbn:b ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 4)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 4)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 4)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 3))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 3 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 4)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 3)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4)) (Eq.refl false))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha)))) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl false)))) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + b * c + a * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + b * c) (a * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (b * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul b c)))) (Nat.cast_mul a c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n b) a_1))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 b c) hbc))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ c))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))))han:a ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (id (Eq.mp (Mathlib.Tactic.Zify.natCast_le._simp_1 b n) hbn)))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n a) a_1))))))heq:n = a * b + a * c + b * c⊢ n = a * b + a * c + b * c; All goals completed! 🐙 All goals completed! 🐙 have hpos : 0 < a + b := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) All goals completed! 🐙 have hc : (n - a * b) / (a + b) = c := n:ℕ⊢ (∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c) ↔ ∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b)) n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 4)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 4)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 4)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 3))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 3 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 4)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 3)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4)) (Eq.refl false))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha)))) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl false)))) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + b * c + a * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + b * c) (a * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (b * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul b c)))) (Nat.cast_mul a c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n b) a_1))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 b c) hbc))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ c))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))))han:a ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (id (Eq.mp (Mathlib.Tactic.Zify.natCast_le._simp_1 b n) hbn)))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n a) a_1))))))hsum:n - a * b = c * (a + b) := have this := Eq.mpr (id (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.zero_mul (a ^ Nat.rawCast 1 * Nat.rawCast 1 + (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c)))))) (Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt n (a * b + a * c + b * c) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq)))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 n (a * b + a * c + b * c)) (congrArg (LT.lt ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1))))))) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 (a * b + a * c + b * c) n) (congrArg (fun x => x < ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1)))))))); Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_1 n a b c heq this a_1hpos:0 < a + b := Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_2 n a b c ha a_1⊢ c * (a + b) / (a + b) = c; All goals completed! 🐙 exact ⟨a, Finset.mem_range.mpr (n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 4)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 4)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 4)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 3))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 3 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 4)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 3)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4)) (Eq.refl false))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha)))) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl false)))) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + b * c + a * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + b * c) (a * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (b * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul b c)))) (Nat.cast_mul a c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n b) a_1))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 b c) hbc))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ c))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))))han:a ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (id (Eq.mp (Mathlib.Tactic.Zify.natCast_le._simp_1 b n) hbn)))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n a) a_1))))))hsum:n - a * b = c * (a + b) := have this := Eq.mpr (id (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.zero_mul (a ^ Nat.rawCast 1 * Nat.rawCast 1 + (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c)))))) (Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt n (a * b + a * c + b * c) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq)))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 n (a * b + a * c + b * c)) (congrArg (LT.lt ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1))))))) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 (a * b + a * c + b * c) n) (congrArg (fun x => x < ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1)))))))); Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_1 n a b c heq this a_1hpos:0 < a + b := Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_2 n a b c ha a_1hc:(n - a * b) / (a + b) = c := Eq.mpr (id (congrArg (fun _a => _a / (a + b) = c) hsum)) (Nat.mul_div_cancel c hpos)⊢ a < n + 1 All goals completed! 🐙), b, Finset.mem_range.mpr (n:ℕa:ℕb:ℕc:ℕha:0 < ahab:a < bhbc:b < cheq:n = a * b + b * c + a * chbn:b ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 4)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 4)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 4)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 4).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 3))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 3 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 4)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 3)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast + 0)))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 3).rawCast) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 3)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑a) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 2 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 2) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_zero_add (Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0)))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 4)) (Eq.refl false))) (Mathlib.Tactic.Linarith.mul_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha)))) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl false)))) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + b * c + a * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + b * c) (a * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (b * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul b c)))) (Nat.cast_mul a c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n b) a_1))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ a)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 b c) hbc))))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ c))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (mul_nonneg_of_nonpos_of_nonpos (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Mathlib.Tactic.Linarith.natCast_nonneg ℤ b)) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 0 a) (congrArg (fun x => x < ↑a) Nat.cast_zero)) ha))))))))han:a ≤ n := le_of_not_gt fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2)))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul ((Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero ((Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑a ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.mul_neg (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 a b) hab))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (id (Eq.mp (Mathlib.Tactic.Zify.natCast_le._simp_1 b n) hbn)))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Mathlib.Tactic.Zify.natCast_lt._simp_1 n a) a_1))))))hsum:n - a * b = c * (a + b) := have this := Eq.mpr (id (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_left c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.zero_mul (a ^ Nat.rawCast 1 * Nat.rawCast 1 + (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c)))))) (Mathlib.Tactic.Linarith.eq_of_not_lt_of_not_gt n (a * b + a * c + b * c) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1)))))))) Mathlib.Tactic.Ring.neg_zero))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (neg_eq_zero.mpr (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq)))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 n (a * b + a * c + b * c)) (congrArg (LT.lt ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1))))))) (Not.intro fun a_1 => Mathlib.Tactic.Linarith.lt_irrefl (Eq.mp (congrArg (fun _a => _a < 0) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_mul (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1))))))) Mathlib.Tactic.Ring.neg_zero))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Int.negOfNat 1).rawCast (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) + 0))))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑a) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf ↑b) (Mathlib.Tactic.Ring.atom_pf ↑c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right (↑c) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (↑c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_lt (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))) (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_gt (↑n ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast) (Mathlib.Tactic.Ring.add_pf_add_zero (↑a ^ Nat.rawCast 1 * (↑b ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑a ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + (↑b ^ Nat.rawCast 1 * (↑c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑a) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑b) (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_overlap_pf_zero (↑c) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.ofNat 0)))))) (Mathlib.Tactic.Ring.add_pf_zero_add 0))))))) (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Mathlib.Tactic.Linarith.add_lt_of_neg_of_le (Mathlib.Tactic.Linarith.lt_of_lt_of_eq (neg_neg_of_pos Mathlib.Tactic.Linarith.zero_lt_one) (sub_eq_zero_of_eq (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_eq._simp_1 n (a * b + a * c + b * c)) (congrArg (Eq ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) (Eq.mp (congrArg (Eq n) (Eq.trans (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right b (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (b ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf b) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left b (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_zero_add (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.atom_pf a) (Mathlib.Tactic.Ring.atom_pf c) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left a (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_pf_right c (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1)))) (Mathlib.Tactic.Ring.mul_zero (a ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))) (Mathlib.Tactic.Ring.zero_mul (c ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (a ^ Nat.rawCast 1 * (b ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_gt (a ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (b ^ Nat.rawCast 1 * (c ^ Nat.rawCast 1 * Nat.rawCast 1) + 0))))) (Eq.trans (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one b)))) (congr (congrArg HAdd.hAdd (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow a) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one a))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (Eq.trans (congrArg (fun x => x + 0) (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow b) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one b))) (Eq.trans (congr (congrArg HMul.hMul (Eq.trans (congrArg (HPow.hPow c) Mathlib.Tactic.RingNF.nat_rawCast_1) (pow_one c))) Mathlib.Tactic.RingNF.nat_rawCast_1) (mul_one c)))) (add_zero (b * c))))) (Mathlib.Tactic.RingNF.add_assoc_rev (a * b) (a * c) (b * c))))) heq))))) (Mathlib.Tactic.Linarith.sub_nonpos_of_le (Int.add_one_le_iff.mpr (id (Eq.mp (Eq.trans (Mathlib.Tactic.Zify.natCast_lt._simp_1 (a * b + a * c + b * c) n) (congrArg (fun x => x < ↑n) (Eq.trans (Nat.cast_add (a * b + a * c) (b * c)) (congr (congrArg HAdd.hAdd (Eq.trans (Nat.cast_add (a * b) (a * c)) (congr (congrArg HAdd.hAdd (Nat.cast_mul a b)) (Nat.cast_mul a c)))) (Nat.cast_mul b c))))) a_1)))))))); Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_1 n a b c heq this a_1hpos:0 < a + b := Decidable.byContradiction fun a_1 => exists_triple_iff_bounded._proof_2 n a b c ha a_1hc:(n - a * b) / (a + b) = c := Eq.mpr (id (congrArg (fun _a => _a / (a + b) = c) hsum)) (Nat.mul_div_cancel c hpos)⊢ b < n + 1 All goals completed! 🐙), ha, hab, hc ▸ hbc, hc ▸ heq⟩ n:ℕ⊢ (∃ a ∈ Finset.range (n + 1), ∃ b ∈ Finset.range (n + 1), 0 < a ∧ a < b ∧ b < (n - a * b) / (a + b) ∧ n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))) → ∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c n:ℕa:ℕleft✝¹:a ∈ Finset.range (n + 1)b:ℕleft✝:b ∈ Finset.range (n + 1)ha:0 < ahab:a < bhbc:b < (n - a * b) / (a + b)heq:n = a * b + b * ((n - a * b) / (a + b)) + a * ((n - a * b) / (a + b))⊢ ∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ n = a * b + b * c + a * c All goals completed! 🐙 set_option maxRecDepth 4096 in

All 65 known idoneal numbers are indeed idoneal.

@[category test, AMS 11] theorem knownIdonealNumbers_are_idoneal : ∀ n ∈ knownIdonealNumbers, IsIdoneal n := ⊢ ∀ n ∈ knownIdonealNumbers, IsIdoneal n intro n n:ℕhn:n ∈ knownIdonealNumbers⊢ IsIdoneal n ⊢ IsIdoneal 1⊢ IsIdoneal 2⊢ IsIdoneal 3⊢ IsIdoneal 4⊢ IsIdoneal 5⊢ IsIdoneal 6⊢ IsIdoneal 7⊢ IsIdoneal 8⊢ IsIdoneal 9⊢ IsIdoneal 10⊢ IsIdoneal 12⊢ IsIdoneal 13⊢ IsIdoneal 15⊢ IsIdoneal 16⊢ IsIdoneal 18⊢ IsIdoneal 21⊢ IsIdoneal 22⊢ IsIdoneal 24⊢ IsIdoneal 25⊢ IsIdoneal 28⊢ IsIdoneal 30⊢ IsIdoneal 33⊢ IsIdoneal 37⊢ IsIdoneal 40⊢ IsIdoneal 42⊢ IsIdoneal 45⊢ IsIdoneal 48⊢ IsIdoneal 57⊢ IsIdoneal 58⊢ IsIdoneal 60⊢ IsIdoneal 70⊢ IsIdoneal 72⊢ IsIdoneal 78⊢ IsIdoneal 85⊢ IsIdoneal 88⊢ IsIdoneal 93⊢ IsIdoneal 102⊢ IsIdoneal 105⊢ IsIdoneal 112⊢ IsIdoneal 120⊢ IsIdoneal 130⊢ IsIdoneal 133⊢ IsIdoneal 165⊢ IsIdoneal 168⊢ IsIdoneal 177⊢ IsIdoneal 190⊢ IsIdoneal 210⊢ IsIdoneal 232⊢ IsIdoneal 240⊢ IsIdoneal 253⊢ IsIdoneal 273⊢ IsIdoneal 280⊢ IsIdoneal 312⊢ IsIdoneal 330⊢ IsIdoneal 345⊢ IsIdoneal 357⊢ IsIdoneal 385⊢ IsIdoneal 408⊢ IsIdoneal 462⊢ IsIdoneal 520⊢ IsIdoneal 760⊢ IsIdoneal 840⊢ IsIdoneal 1320⊢ IsIdoneal 1365⊢ IsIdoneal 1848 ⊢ IsIdoneal 1⊢ IsIdoneal 2⊢ IsIdoneal 3⊢ IsIdoneal 4⊢ IsIdoneal 5⊢ IsIdoneal 6⊢ IsIdoneal 7⊢ IsIdoneal 8⊢ IsIdoneal 9⊢ IsIdoneal 10⊢ IsIdoneal 12⊢ IsIdoneal 13⊢ IsIdoneal 15⊢ IsIdoneal 16⊢ IsIdoneal 18⊢ IsIdoneal 21⊢ IsIdoneal 22⊢ IsIdoneal 24⊢ IsIdoneal 25⊢ IsIdoneal 28⊢ IsIdoneal 30⊢ IsIdoneal 33⊢ IsIdoneal 37⊢ IsIdoneal 40⊢ IsIdoneal 42⊢ IsIdoneal 45⊢ IsIdoneal 48⊢ IsIdoneal 57⊢ IsIdoneal 58⊢ IsIdoneal 60⊢ IsIdoneal 70⊢ IsIdoneal 72⊢ IsIdoneal 78⊢ IsIdoneal 85⊢ IsIdoneal 88⊢ IsIdoneal 93⊢ IsIdoneal 102⊢ IsIdoneal 105⊢ IsIdoneal 112⊢ IsIdoneal 120⊢ IsIdoneal 130⊢ IsIdoneal 133⊢ IsIdoneal 165⊢ IsIdoneal 168⊢ IsIdoneal 177⊢ IsIdoneal 190⊢ IsIdoneal 210⊢ IsIdoneal 232⊢ IsIdoneal 240⊢ IsIdoneal 253⊢ IsIdoneal 273⊢ IsIdoneal 280⊢ IsIdoneal 312⊢ IsIdoneal 330⊢ IsIdoneal 345⊢ IsIdoneal 357⊢ IsIdoneal 385⊢ IsIdoneal 408⊢ IsIdoneal 462⊢ IsIdoneal 520⊢ IsIdoneal 760⊢ IsIdoneal 840⊢ IsIdoneal 1320⊢ IsIdoneal 1365⊢ IsIdoneal 1848 refine ⟨⊢ 0 < 1848 All goals completed! 🐙, ?_⟩ ⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 1 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 2 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 3 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 4 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 5 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 6 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 7 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 8 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 9 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 10 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 12 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 13 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 15 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 16 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 18 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 21 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 22 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 24 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 25 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 28 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 30 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 33 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 37 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 40 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 42 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 45 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 48 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 57 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 58 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 60 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 70 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 72 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 78 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 85 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 88 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 93 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 102 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 105 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 112 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 120 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 130 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 133 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 165 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 168 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 177 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 190 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 210 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 232 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 240 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 253 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 273 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 280 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 312 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 330 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 345 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 357 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 385 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 408 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 462 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 520 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 760 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 840 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 1320 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 1365 = a * b + b * c + a * c⊢ ¬∃ a b c, 0 < a ∧ a < b ∧ b < c ∧ 1848 = a * b + b * c + a * c ⊢ ¬∃ a ∈ Finset.range (1848 + 1), ∃ b ∈ Finset.range (1848 + 1), 0 < a ∧ a < b ∧ b < (1848 - a * b) / (a + b) ∧ 1848 = a * b + b * ((1848 - a * b) / (a + b)) + a * ((1848 - a * b) / (a + b)) ⊢ ¬∃ a ∈ Finset.range (1 + 1), ∃ b ∈ Finset.range (1 + 1), 0 < a ∧ a < b ∧ b < (1 - a * b) / (a + b) ∧ 1 = a * b + b * ((1 - a * b) / (a + b)) + a * ((1 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (2 + 1), ∃ b ∈ Finset.range (2 + 1), 0 < a ∧ a < b ∧ b < (2 - a * b) / (a + b) ∧ 2 = a * b + b * ((2 - a * b) / (a + b)) + a * ((2 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (3 + 1), ∃ b ∈ Finset.range (3 + 1), 0 < a ∧ a < b ∧ b < (3 - a * b) / (a + b) ∧ 3 = a * b + b * ((3 - a * b) / (a + b)) + a * ((3 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (4 + 1), ∃ b ∈ Finset.range (4 + 1), 0 < a ∧ a < b ∧ b < (4 - a * b) / (a + b) ∧ 4 = a * b + b * ((4 - a * b) / (a + b)) + a * ((4 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (5 + 1), ∃ b ∈ Finset.range (5 + 1), 0 < a ∧ a < b ∧ b < (5 - a * b) / (a + b) ∧ 5 = a * b + b * ((5 - a * b) / (a + b)) + a * ((5 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (6 + 1), ∃ b ∈ Finset.range (6 + 1), 0 < a ∧ a < b ∧ b < (6 - a * b) / (a + b) ∧ 6 = a * b + b * ((6 - a * b) / (a + b)) + a * ((6 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (7 + 1), ∃ b ∈ Finset.range (7 + 1), 0 < a ∧ a < b ∧ b < (7 - a * b) / (a + b) ∧ 7 = a * b + b * ((7 - a * b) / (a + b)) + a * ((7 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (8 + 1), ∃ b ∈ Finset.range (8 + 1), 0 < a ∧ a < b ∧ b < (8 - a * b) / (a + b) ∧ 8 = a * b + b * ((8 - a * b) / (a + b)) + a * ((8 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (9 + 1), ∃ b ∈ Finset.range (9 + 1), 0 < a ∧ a < b ∧ b < (9 - a * b) / (a + b) ∧ 9 = a * b + b * ((9 - a * b) / (a + b)) + a * ((9 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (10 + 1), ∃ b ∈ Finset.range (10 + 1), 0 < a ∧ a < b ∧ b < (10 - a * b) / (a + b) ∧ 10 = a * b + b * ((10 - a * b) / (a + b)) + a * ((10 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (12 + 1), ∃ b ∈ Finset.range (12 + 1), 0 < a ∧ a < b ∧ b < (12 - a * b) / (a + b) ∧ 12 = a * b + b * ((12 - a * b) / (a + b)) + a * ((12 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (13 + 1), ∃ b ∈ Finset.range (13 + 1), 0 < a ∧ a < b ∧ b < (13 - a * b) / (a + b) ∧ 13 = a * b + b * ((13 - a * b) / (a + b)) + a * ((13 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (15 + 1), ∃ b ∈ Finset.range (15 + 1), 0 < a ∧ a < b ∧ b < (15 - a * b) / (a + b) ∧ 15 = a * b + b * ((15 - a * b) / (a + b)) + a * ((15 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (16 + 1), ∃ b ∈ Finset.range (16 + 1), 0 < a ∧ a < b ∧ b < (16 - a * b) / (a + b) ∧ 16 = a * b + b * ((16 - a * b) / (a + b)) + a * ((16 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (18 + 1), ∃ b ∈ Finset.range (18 + 1), 0 < a ∧ a < b ∧ b < (18 - a * b) / (a + b) ∧ 18 = a * b + b * ((18 - a * b) / (a + b)) + a * ((18 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (21 + 1), ∃ b ∈ Finset.range (21 + 1), 0 < a ∧ a < b ∧ b < (21 - a * b) / (a + b) ∧ 21 = a * b + b * ((21 - a * b) / (a + b)) + a * ((21 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (22 + 1), ∃ b ∈ Finset.range (22 + 1), 0 < a ∧ a < b ∧ b < (22 - a * b) / (a + b) ∧ 22 = a * b + b * ((22 - a * b) / (a + b)) + a * ((22 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (24 + 1), ∃ b ∈ Finset.range (24 + 1), 0 < a ∧ a < b ∧ b < (24 - a * b) / (a + b) ∧ 24 = a * b + b * ((24 - a * b) / (a + b)) + a * ((24 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (25 + 1), ∃ b ∈ Finset.range (25 + 1), 0 < a ∧ a < b ∧ b < (25 - a * b) / (a + b) ∧ 25 = a * b + b * ((25 - a * b) / (a + b)) + a * ((25 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (28 + 1), ∃ b ∈ Finset.range (28 + 1), 0 < a ∧ a < b ∧ b < (28 - a * b) / (a + b) ∧ 28 = a * b + b * ((28 - a * b) / (a + b)) + a * ((28 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (30 + 1), ∃ b ∈ Finset.range (30 + 1), 0 < a ∧ a < b ∧ b < (30 - a * b) / (a + b) ∧ 30 = a * b + b * ((30 - a * b) / (a + b)) + a * ((30 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (33 + 1), ∃ b ∈ Finset.range (33 + 1), 0 < a ∧ a < b ∧ b < (33 - a * b) / (a + b) ∧ 33 = a * b + b * ((33 - a * b) / (a + b)) + a * ((33 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (37 + 1), ∃ b ∈ Finset.range (37 + 1), 0 < a ∧ a < b ∧ b < (37 - a * b) / (a + b) ∧ 37 = a * b + b * ((37 - a * b) / (a + b)) + a * ((37 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (40 + 1), ∃ b ∈ Finset.range (40 + 1), 0 < a ∧ a < b ∧ b < (40 - a * b) / (a + b) ∧ 40 = a * b + b * ((40 - a * b) / (a + b)) + a * ((40 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (42 + 1), ∃ b ∈ Finset.range (42 + 1), 0 < a ∧ a < b ∧ b < (42 - a * b) / (a + b) ∧ 42 = a * b + b * ((42 - a * b) / (a + b)) + a * ((42 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (45 + 1), ∃ b ∈ Finset.range (45 + 1), 0 < a ∧ a < b ∧ b < (45 - a * b) / (a + b) ∧ 45 = a * b + b * ((45 - a * b) / (a + b)) + a * ((45 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (48 + 1), ∃ b ∈ Finset.range (48 + 1), 0 < a ∧ a < b ∧ b < (48 - a * b) / (a + b) ∧ 48 = a * b + b * ((48 - a * b) / (a + b)) + a * ((48 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (57 + 1), ∃ b ∈ Finset.range (57 + 1), 0 < a ∧ a < b ∧ b < (57 - a * b) / (a + b) ∧ 57 = a * b + b * ((57 - a * b) / (a + b)) + a * ((57 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (58 + 1), ∃ b ∈ Finset.range (58 + 1), 0 < a ∧ a < b ∧ b < (58 - a * b) / (a + b) ∧ 58 = a * b + b * ((58 - a * b) / (a + b)) + a * ((58 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (60 + 1), ∃ b ∈ Finset.range (60 + 1), 0 < a ∧ a < b ∧ b < (60 - a * b) / (a + b) ∧ 60 = a * b + b * ((60 - a * b) / (a + b)) + a * ((60 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (70 + 1), ∃ b ∈ Finset.range (70 + 1), 0 < a ∧ a < b ∧ b < (70 - a * b) / (a + b) ∧ 70 = a * b + b * ((70 - a * b) / (a + b)) + a * ((70 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (72 + 1), ∃ b ∈ Finset.range (72 + 1), 0 < a ∧ a < b ∧ b < (72 - a * b) / (a + b) ∧ 72 = a * b + b * ((72 - a * b) / (a + b)) + a * ((72 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (78 + 1), ∃ b ∈ Finset.range (78 + 1), 0 < a ∧ a < b ∧ b < (78 - a * b) / (a + b) ∧ 78 = a * b + b * ((78 - a * b) / (a + b)) + a * ((78 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (85 + 1), ∃ b ∈ Finset.range (85 + 1), 0 < a ∧ a < b ∧ b < (85 - a * b) / (a + b) ∧ 85 = a * b + b * ((85 - a * b) / (a + b)) + a * ((85 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (88 + 1), ∃ b ∈ Finset.range (88 + 1), 0 < a ∧ a < b ∧ b < (88 - a * b) / (a + b) ∧ 88 = a * b + b * ((88 - a * b) / (a + b)) + a * ((88 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (93 + 1), ∃ b ∈ Finset.range (93 + 1), 0 < a ∧ a < b ∧ b < (93 - a * b) / (a + b) ∧ 93 = a * b + b * ((93 - a * b) / (a + b)) + a * ((93 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (102 + 1), ∃ b ∈ Finset.range (102 + 1), 0 < a ∧ a < b ∧ b < (102 - a * b) / (a + b) ∧ 102 = a * b + b * ((102 - a * b) / (a + b)) + a * ((102 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (105 + 1), ∃ b ∈ Finset.range (105 + 1), 0 < a ∧ a < b ∧ b < (105 - a * b) / (a + b) ∧ 105 = a * b + b * ((105 - a * b) / (a + b)) + a * ((105 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (112 + 1), ∃ b ∈ Finset.range (112 + 1), 0 < a ∧ a < b ∧ b < (112 - a * b) / (a + b) ∧ 112 = a * b + b * ((112 - a * b) / (a + b)) + a * ((112 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (120 + 1), ∃ b ∈ Finset.range (120 + 1), 0 < a ∧ a < b ∧ b < (120 - a * b) / (a + b) ∧ 120 = a * b + b * ((120 - a * b) / (a + b)) + a * ((120 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (130 + 1), ∃ b ∈ Finset.range (130 + 1), 0 < a ∧ a < b ∧ b < (130 - a * b) / (a + b) ∧ 130 = a * b + b * ((130 - a * b) / (a + b)) + a * ((130 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (133 + 1), ∃ b ∈ Finset.range (133 + 1), 0 < a ∧ a < b ∧ b < (133 - a * b) / (a + b) ∧ 133 = a * b + b * ((133 - a * b) / (a + b)) + a * ((133 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (165 + 1), ∃ b ∈ Finset.range (165 + 1), 0 < a ∧ a < b ∧ b < (165 - a * b) / (a + b) ∧ 165 = a * b + b * ((165 - a * b) / (a + b)) + a * ((165 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (168 + 1), ∃ b ∈ Finset.range (168 + 1), 0 < a ∧ a < b ∧ b < (168 - a * b) / (a + b) ∧ 168 = a * b + b * ((168 - a * b) / (a + b)) + a * ((168 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (177 + 1), ∃ b ∈ Finset.range (177 + 1), 0 < a ∧ a < b ∧ b < (177 - a * b) / (a + b) ∧ 177 = a * b + b * ((177 - a * b) / (a + b)) + a * ((177 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (190 + 1), ∃ b ∈ Finset.range (190 + 1), 0 < a ∧ a < b ∧ b < (190 - a * b) / (a + b) ∧ 190 = a * b + b * ((190 - a * b) / (a + b)) + a * ((190 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (210 + 1), ∃ b ∈ Finset.range (210 + 1), 0 < a ∧ a < b ∧ b < (210 - a * b) / (a + b) ∧ 210 = a * b + b * ((210 - a * b) / (a + b)) + a * ((210 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (232 + 1), ∃ b ∈ Finset.range (232 + 1), 0 < a ∧ a < b ∧ b < (232 - a * b) / (a + b) ∧ 232 = a * b + b * ((232 - a * b) / (a + b)) + a * ((232 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (240 + 1), ∃ b ∈ Finset.range (240 + 1), 0 < a ∧ a < b ∧ b < (240 - a * b) / (a + b) ∧ 240 = a * b + b * ((240 - a * b) / (a + b)) + a * ((240 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (253 + 1), ∃ b ∈ Finset.range (253 + 1), 0 < a ∧ a < b ∧ b < (253 - a * b) / (a + b) ∧ 253 = a * b + b * ((253 - a * b) / (a + b)) + a * ((253 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (273 + 1), ∃ b ∈ Finset.range (273 + 1), 0 < a ∧ a < b ∧ b < (273 - a * b) / (a + b) ∧ 273 = a * b + b * ((273 - a * b) / (a + b)) + a * ((273 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (280 + 1), ∃ b ∈ Finset.range (280 + 1), 0 < a ∧ a < b ∧ b < (280 - a * b) / (a + b) ∧ 280 = a * b + b * ((280 - a * b) / (a + b)) + a * ((280 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (312 + 1), ∃ b ∈ Finset.range (312 + 1), 0 < a ∧ a < b ∧ b < (312 - a * b) / (a + b) ∧ 312 = a * b + b * ((312 - a * b) / (a + b)) + a * ((312 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (330 + 1), ∃ b ∈ Finset.range (330 + 1), 0 < a ∧ a < b ∧ b < (330 - a * b) / (a + b) ∧ 330 = a * b + b * ((330 - a * b) / (a + b)) + a * ((330 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (345 + 1), ∃ b ∈ Finset.range (345 + 1), 0 < a ∧ a < b ∧ b < (345 - a * b) / (a + b) ∧ 345 = a * b + b * ((345 - a * b) / (a + b)) + a * ((345 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (357 + 1), ∃ b ∈ Finset.range (357 + 1), 0 < a ∧ a < b ∧ b < (357 - a * b) / (a + b) ∧ 357 = a * b + b * ((357 - a * b) / (a + b)) + a * ((357 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (385 + 1), ∃ b ∈ Finset.range (385 + 1), 0 < a ∧ a < b ∧ b < (385 - a * b) / (a + b) ∧ 385 = a * b + b * ((385 - a * b) / (a + b)) + a * ((385 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (408 + 1), ∃ b ∈ Finset.range (408 + 1), 0 < a ∧ a < b ∧ b < (408 - a * b) / (a + b) ∧ 408 = a * b + b * ((408 - a * b) / (a + b)) + a * ((408 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (462 + 1), ∃ b ∈ Finset.range (462 + 1), 0 < a ∧ a < b ∧ b < (462 - a * b) / (a + b) ∧ 462 = a * b + b * ((462 - a * b) / (a + b)) + a * ((462 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (520 + 1), ∃ b ∈ Finset.range (520 + 1), 0 < a ∧ a < b ∧ b < (520 - a * b) / (a + b) ∧ 520 = a * b + b * ((520 - a * b) / (a + b)) + a * ((520 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (760 + 1), ∃ b ∈ Finset.range (760 + 1), 0 < a ∧ a < b ∧ b < (760 - a * b) / (a + b) ∧ 760 = a * b + b * ((760 - a * b) / (a + b)) + a * ((760 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (840 + 1), ∃ b ∈ Finset.range (840 + 1), 0 < a ∧ a < b ∧ b < (840 - a * b) / (a + b) ∧ 840 = a * b + b * ((840 - a * b) / (a + b)) + a * ((840 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1320 + 1), ∃ b ∈ Finset.range (1320 + 1), 0 < a ∧ a < b ∧ b < (1320 - a * b) / (a + b) ∧ 1320 = a * b + b * ((1320 - a * b) / (a + b)) + a * ((1320 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1365 + 1), ∃ b ∈ Finset.range (1365 + 1), 0 < a ∧ a < b ∧ b < (1365 - a * b) / (a + b) ∧ 1365 = a * b + b * ((1365 - a * b) / (a + b)) + a * ((1365 - a * b) / (a + b))⊢ ¬∃ a ∈ Finset.range (1848 + 1), ∃ b ∈ Finset.range (1848 + 1), 0 < a ∧ a < b ∧ b < (1848 - a * b) / (a + b) ∧ 1848 = a * b + b * ((1848 - a * b) / (a + b)) + a * ((1848 - a * b) / (a + b)) All goals completed! 🐙

Idoneal numbers completeness conjecture.

@[category research open, AMS 11] theorem declaration uses 'sorry'idoneal_numbers_completeness : answer(sorry) ↔ ∀ n : ℕ, IsIdoneal n → n ∈ knownIdonealNumbers := ⊢ True ↔ ∀ (n : ℕ), IsIdoneal n → n ∈ knownIdonealNumbers All goals completed! 🐙 end Idoneal