2023-08-08 14:56:13 +00:00
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import Mathlib.Data.Set.Basic
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2023-08-15 02:37:09 +00:00
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import Mathlib.Tactic.NormNum
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2023-08-08 14:56:13 +00:00
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namespace Nat
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2023-08-15 02:37:09 +00:00
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/--
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If `n < m⁺`, then `n < m` or `n = m`.
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-/
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theorem lt_or_eq_of_lt {n m : Nat} (h : n < m.succ) : n < m ∨ n = m :=
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lt_or_eq_of_le (lt_succ.mp h)
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2023-08-08 14:56:13 +00:00
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/--
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The following cancellation law holds for `m`, `n`, and `p` in `ω`:
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```
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m ⬝ p = n ⬝ p ∧ p ≠ 0 → m = n
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```
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-/
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theorem mul_right_cancel (m n p : ℕ) (hp : 0 < p) : m * p = n * p → m = n := by
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intro hmn
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match @trichotomous ℕ LT.lt _ m n with
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| Or.inl h =>
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have : m * p < n * p := Nat.mul_lt_mul_of_pos_right h hp
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rw [hmn] at this
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simp at this
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| Or.inr (Or.inl h) => exact h
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| Or.inr (Or.inr h) =>
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have : n * p < m * p := Nat.mul_lt_mul_of_pos_right h hp
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rw [hmn] at this
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simp at this
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end Nat
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