ad48761032
This PR adds basic lemmas about `Ordering`, describing the interaction of `isLT`/`isLE`/`isGE`/`isGT`, `swap` and the constructors. Additionally, it refactors the instance derivation code such that a `LawfulBEq Ordering` instance is also derived automatically. Some of these lemmas are helpful for the `TreeMap` verification. --------- Co-authored-by: Paul Reichert <6992158+datokrat@users.noreply.github.com>
27 lines
1 KiB
Text
27 lines
1 KiB
Text
/-- `OrientedCmp cmp` asserts that `cmp` is determined by the relation `cmp x y = .lt`. -/
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class OrientedCmp (cmp : α → α → Ordering) : Prop where
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/-- The comparator operation is symmetric, in the sense that if `cmp x y` equals `.lt` then
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`cmp y x = .gt` and vice versa. -/
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symm (x y) : (cmp x y).swap = cmp y x
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namespace OrientedCmp
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theorem cmp_eq_gt [OrientedCmp cmp] : cmp x y = .gt ↔ cmp y x = .lt := by
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rw [← Ordering.swap_inj, symm]; exact .rfl
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end OrientedCmp
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/-- `TransCmp cmp` asserts that `cmp` induces a transitive relation. -/
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class TransCmp (cmp : α → α → Ordering) extends OrientedCmp cmp : Prop where
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/-- The comparator operation is transitive. -/
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le_trans : cmp x y ≠ .gt → cmp y z ≠ .gt → cmp x z ≠ .gt
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namespace TransCmp
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variable [TransCmp cmp]
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open OrientedCmp
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theorem ge_trans (h₁ : cmp x y ≠ .lt) (h₂ : cmp y z ≠ .lt) : cmp x z ≠ .lt := by
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have := @TransCmp.le_trans _ cmp _ z y x
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simp [cmp_eq_gt] at *
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-- `simp` did not fire at `this`
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exact this h₂ h₁
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