| import Mathlib | |
| namespace OAI | |
| namespace Problem355.DigitEncoding | |
| theorem matched_indices {T a b c : ℕ} (ha : a < T) (hc : c < T) | |
| (h : a + T * b = (T + 1) * c) : a = c ∧ b = c := by | |
| have h' : a + T * b = c + T * c := by nlinarith | |
| have hmod := congrArg (fun x : ℕ => x % T) h' | |
| have hac : a = c := by | |
| simpa [Nat.add_mod, Nat.mod_eq_of_lt ha, Nat.mod_eq_of_lt hc] using hmod | |
| subst a | |
| constructor | |
| · rfl | |
| · have hT : 0 < T := Nat.zero_lt_of_lt hc | |
| nlinarith | |
| theorem no_small_congruent {h τ s t : ℤ} (hs : τ < |s|) | |
| (hcenter : 2 * |s| < h) (hcong : t ≡ s [ZMOD h]) : τ < |t| := by | |
| by_contra ht | |
| have ht' : |t| ≤ τ := le_of_not_gt ht | |
| obtain ⟨m, hm⟩ := Int.modEq_iff_add_fac.mp hcong | |
| by_cases hm0 : m = 0 | |
| · simp only [hm0, mul_zero, add_zero] at hm | |
| subst s | |
| omega | |
| · have hmabs : 1 ≤ |m| := by | |
| have := abs_pos.mpr hm0 | |
| omega | |
| have hh : 0 < h := lt_of_le_of_lt (by positivity : 0 ≤ 2 * |s|) hcenter | |
| have hprod : h ≤ |h * m| := by | |
| rw [abs_mul, abs_of_pos hh] | |
| nlinarith | |
| have htri : |s - t| ≤ |s| + |t| := abs_sub s t | |
| have heq : s - t = h * m := by omega | |
| rw [heq] at htri | |
| omega | |
| end Problem355.DigitEncoding | |
| end OAI | |