mod

Stream: mathlib4

[vxctyxeha]

how to launch m = 5 ∨ m = 7 ∨ m = 35 from  m ∣ 35

import Mathlib

open Equiv Subgroup
open Int Finset
open Int

theorem consistency_of_simultaneous_congruences_specific_values (m : ℤ) (hm_ge_2 : m ≥ 2) :
    (∃ (x y z : ℤ),
        6*x + 4*y + 13*z ≡ 5 [ZMOD m] ∧
        9*x + 6*y      ≡ 7 [ZMOD m] ∧
        12*x + 8*y - z  ≡ 12 [ZMOD m])
    →m = 5 ∨ m = 7 ∨ m = 35 := by

    rintro ⟨x, y, z, h1, h2, h3⟩ -- 假设存在解 x, y, z 满足 h1, h2, h3
    have h3m13 : 13 * (12*x + 8*y - z) ≡ 13 * 12 [ZMOD m] := ModEq.mul_left 13 h3
    have hC_lhs_calc : (6*x + 4*y + 13*z) + 13 * (12*x + 8*y - z) = 162*x + 108*y := by ring
    have hC_rhs_calc : 5 + 13 * 12 = 161 := by norm_num
    have hC : 162*x + 108*y ≡ 161 [ZMOD m] := by
      rw [← hC_lhs_calc]
      exact ModEq.add h1 h3m13
    have h2m18 : 18 * (9*x + 6*y) ≡ 18 * 7 [ZMOD m] := ModEq.mul_left 18 h2
    have hC_alt_lhs : 18 * (9*x + 6*y) = 162*x + 108*y := by ring
    have hC_alt_rhs : 18 * 7 = 126 := by norm_num
    have hC_alt : 162*x + 108*y ≡ 126 [ZMOD m] := by
      rw [← hC_alt_lhs]
      exact h2m18
    have h_compat : 161 ≡ 126 [ZMOD m] := ModEq.trans hC.symm hC_alt -- 161 ≡ 162x+108y ≡ 126
    have hm35 : m ∣ 161 - 126 := by
    -- 应用模同余的定义 (a ≡ b ↔ m | a - b)
      exact ModEq.dvd (id (ModEq.symm h_compat))
    ring_nf at hm35

[Edison Xie]

hC_alt_rhs didn't do anything and is this another piece of code generated by LLM?

[Ilmārs Cīrulis]

Did it, was a bit fun.

[vxctyxeha]

Edison Xie said:

hC_alt_rhs didn't do anything and is this another piece of code generated by LLM?

part of