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1c3ffd8 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 | import Mathlib
set_option linter.unusedVariables.analyzeTactics true
open Nat
theorem imo_1982_p1
(f : ℕ → ℤ)
(h₀ : ∀ m n, (0 < m ∧ 0 < n) → f (m + n) - f m - f n = 0 ∨ f (m + n) - f m - f n = 1)
(h₁ : f 2 = 0)
(h₂ : 0 < f 3)
(h₃ : f 9999 = 3333) :
f 1982 = 660 := by
have h₀₀: ∀ m n, (0 < m ∧ 0 < n) → f m + f n ≤ f (m + n) := by
intros m n hmn
have g₀: f (m + n) - f m - f n = 0 ∨ f (m + n) - f m - f n = 1 := by
exact h₀ m n hmn
omega
have h₀₁: ∀ m k, (0 < m ∧ 0 < k) → k * f m ≤ f (k * m) := by
intros m k hmk
have g₁: 1 ≤ k := by linarith
refine Nat.le_induction ?_ ?_ k g₁
. simp
. intros n hmn g₂
rw [cast_add]
rw [add_mul, add_mul, one_mul]
simp
have g₃: f (n * m) + f (m) ≤ f (n * m + m) := by
refine h₀₀ (n * m) m ?_
constructor
. refine mul_pos ?_ hmk.1
exact hmn
. exact hmk.1
refine le_trans ?_ g₃
exact (Int.add_le_add_iff_right (f m)).mpr g₂
have h₄: f 3 = 1 := by
have g₀ : 3333 * f 3 ≤ f (9999) := by
refine h₀₁ 3 3333 ?_
omega
linarith
have h₅: f 1980 = 660 := by
have h₅₀: f 1980 ≤ 660 := by
have g₀ : f (5 * 1980) + f 99 ≤ f (9999) := by
refine h₀₀ (5 * 1980) 99 (by omega)
have g₁: 5 * f (1980) ≤ f (5 * 1980) := by
exact h₀₁ 1980 5 (by omega)
have g₂: 33 * f 3 ≤ f 99 := by
exact h₀₁ 3 33 (by omega)
rw [h₃] at g₀
linarith
have h₅₁: 660 ≤ f 1980 := by
have g₀ : 660 * f 3 ≤ f (1980) := by
refine h₀₁ 3 660 ?_
omega
rw [h₄] at g₀
exact g₀
exact le_antisymm h₅₀ h₅₁
have h₆: f 1982 - f 1980 - f 2 = 0 ∨ f 1982 - f 1980 - f 2 = 1 := by
refine h₀ 1980 2 ?_
omega
cases' h₆ with h₆₀ h₆₁
. linarith
. exfalso
rw [h₅, h₁] at h₆₁
have h₆₂: f 1982 = 661 := by
linarith
have h₆₃: 5 * f 1982 + 29 ≤ 3333 := by
have g₀ : f (5 * 1982) + f 89 ≤ f 9999 := by
refine h₀₀ (5 * 1982) 89 (by omega)
have g₁: f (29 * 3) + f 2 ≤ f 89 := by
refine h₀₀ (29 * 3) 2 (by omega)
have g₂: 5 * f (1982) ≤ f (5 * 1982) := by
exact h₀₁ 1982 5 (by omega)
have g₃: 29 * f 3 ≤ f (87) := by
exact h₀₁ 3 29 (by omega)
linarith
rw [h₆₂] at h₆₃
linarith
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