bookshelf/Common/Nat/Basic.lean

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2023-08-08 14:56:13 +00:00
import Mathlib.Data.Set.Basic
namespace Nat
/--
The following cancellation law holds for `m`, `n`, and `p` in `ω`:
```
m ⬝ p = n ⬝ p ∧ p ≠ 0 → m = n
```
-/
theorem mul_right_cancel (m n p : ) (hp : 0 < p) : m * p = n * p → m = n := by
intro hmn
match @trichotomous LT.lt _ m n with
| Or.inl h =>
have : m * p < n * p := Nat.mul_lt_mul_of_pos_right h hp
rw [hmn] at this
simp at this
| Or.inr (Or.inl h) => exact h
| Or.inr (Or.inr h) =>
have : n * p < m * p := Nat.mul_lt_mul_of_pos_right h hp
rw [hmn] at this
simp at this
end Nat