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feat(Logic/Relation): some EqvGen API #40606

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feat(Logic/Relation): some EqvGen API #40606

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18 changes: 18 additions & 0 deletions Mathlib/Logic/Relation.lean
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Expand Up @@ -802,6 +802,24 @@ theorem mono {r p : α → α → Prop} (hrp : ∀ a b, r a b → p a b) (h : Eq
| symm a b _ ih => exact EqvGen.symm _ _ ih
| trans a b c _ _ hab hbc => exact EqvGen.trans _ _ _ hab hbc

lemma map (f : α → β) (r : β → β → Prop) (x y : α)
(h : EqvGen (fun x y ↦ r (f x) (f y)) x y) : EqvGen r (f x) (f y) := by
induction h with
| rel => apply EqvGen.rel; assumption
| refl => exact EqvGen.refl _
| symm => apply EqvGen.symm; assumption
| trans => apply EqvGen.trans <;> assumption
Comment on lines +805 to +811

@chenson2018 chenson2018 Jun 15, 2026

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The convention seems to suggest:

theorem lift {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ a b, r a b → p (f a) (f b))
    (hab : EqvGen r a b) : EqvGen p (f a) (f b) := sorry

theorem lift' {p : β → β → Prop} {a b : α} (f : α → β) (h : ∀ a b, r a b → EqvGen p (f a) (f b))
    (hab : EqvGen r a b) : EqvGen p (f a) (f b) := sorry


@[simp]
lemma idempotent (r : α → α → Prop) : EqvGen (EqvGen r) = EqvGen r := by

@chenson2018 chenson2018 Jun 15, 2026

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Note that if placed at the very end of this module, we can write:

lemma idempotent (r : α → α → Prop) : EqvGen (EqvGen r) = EqvGen r :=
  Equivalence.eqvGen_eq (EqvGen.is_equivalence r)

I'm not 100% sure about the naming and statement. The analogous Relation.reflTransGen_idem uses a different naming, but this is also deprecated for the more general Relation.reflTransGen_eq_self. I think the situation is slightly different here because of Equivalence not being a typeclass that is inferred. If this were using IsEquiv, it'd be straightforward to just keep the generalization, but maybe here we need both.

ext x y
refine ⟨fun h ↦ ?_, fun h ↦ EqvGen.rel _ _ h⟩
induction h with
| rel => assumption
| refl => exact EqvGen.refl _
| symm => apply EqvGen.symm; assumption
| trans => apply EqvGen.trans <;> assumption

end EqvGen

/-- The join of a relation on a single type is a new relation for which
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