c06af84d9f
This PR resolves a defeq diamond, which caused a problem in Mathlib:
```
import Mathlib
example (R : Type) [I : Ring R] :
@AddCommGroup.toGrindIntModule R (@Ring.toAddCommGroup R I) =
@Lean.Grind.Ring.instIntModule R (@Ring.toGrindRing R I) := rfl -- fails
```
20 lines
765 B
Text
20 lines
765 B
Text
-- I'll start collecting particularly horrific examples
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-- involving `Nat` modulo and div with variable denominators.
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example (a b n : Nat) (ha : a < n) : (n - a) * b % n = (n - a * b % n) % n := by
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rw [Nat.sub_mul]
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rw [Nat.mod_eq_mod_iff]
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match b with
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| 0 => refine ⟨1, 0, by simp⟩
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| b+1 =>
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refine ⟨a*(b+1)/n, b, ?_⟩
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rw [Nat.mod_def, Nat.mul_add_one, Nat.mul_comm _ n, Nat.mul_comm b n]
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have : n * (a * (b + 1) / n) ≤ a * (b + 1) := Nat.mul_div_le (a * (b + 1)) n
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have := Nat.lt_mul_div_succ (a * (b + 1)) (show 0 < n by omega)
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rw [Nat.mul_add_one n] at this
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have : a * (b + 1) ≤ n * b + n := by
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rw [Nat.mul_add_one]
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have := Nat.mul_le_mul_right b ha
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rw [Nat.succ_mul] at this
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omega
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omega
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