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import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Nat
theorem imo_1981_p6
(f : ℕ → ℕ → ℕ)
(h₀ : ∀ y, f 0 y = y + 1)
(h₁ : ∀ x, f (x + 1) 0 = f x 1)
(h₂ : ∀ x y, f (x + 1) (y + 1) = f x (f (x + 1) y)) :
∀ y, f 4 (y + 1) = 2 ^ (f 4 y + 3) - 3 := by
have h₃: ∀ y, f 1 y = y + 2 := by
intro y
induction' y with n hn
. simp_all only [zero_eq, zero_add]
. nth_rw 1 [← zero_add 1]
rw [h₂ 0 n, h₀ (f (0 + 1) n), hn]
have h₄: ∀ y, f 2 y = 2 * y + 3 := by
intro y
induction' y with n hn
. simp_all only [zero_eq, zero_add, mul_zero]
. rw [h₂, h₃, hn, mul_add]
have h₅: ∀ y, f 3 y = 2 ^ (y + 3) - 3 := by
intro y
induction' y with n hn
. simp_all only [zero_eq, zero_add, mul_zero]
omega
. rw [h₂, h₄, hn]
rw [Nat.mul_sub_left_distrib]
ring_nf
by_cases hn₀: 0 < n
. rw [← Nat.add_sub_assoc, add_comm]
. omega
. have hn₂: 2 ^ 1 ≤ 2 ^ n := by exact Nat.pow_le_pow_of_le (by norm_num) hn₀
linarith
. have hn₁: n = 0 := by linarith
rw [hn₁]
omega
intro y
induction' y with n hn
. simp
rw [h₂, h₁, h₅]
. rw [hn, h₂, h₅, h₂, h₅]
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