27 lines
999 B
Text
27 lines
999 B
Text
module
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public section
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class Shelf (α : Type u) where
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act : α → α → α
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self_distrib : ∀ {x y z : α}, act x (act y z) = act (act x y) (act x z)
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class UnitalShelf (α : Type u) extends Shelf α, One α where
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one_act : ∀ a : α, act 1 a = a
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act_one : ∀ a : α, act a 1 = a
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infixr:65 " ◃ " => Shelf.act
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-- Mathlib proof from UnitalShelf.act_act_self_eq
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example {S} [UnitalShelf S] (x y : S) : (x ◃ y) ◃ x = x ◃ y := by
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have h : (x ◃ y) ◃ x = (x ◃ y) ◃ (x ◃ 1) := by rw [UnitalShelf.act_one]
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rw [h, ← Shelf.self_distrib, UnitalShelf.act_one]
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attribute [grind =] UnitalShelf.one_act UnitalShelf.act_one
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-- We actually want the reverse direction of `Shelf.self_distrib`, so don't use the `grind_eq` attribute.
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grind_pattern Shelf.self_distrib => self.act (self.act x y) (self.act x z)
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-- Proof using `grind`:
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example {S} [UnitalShelf S] (x y : S) : (x ◃ y) ◃ x = x ◃ y := by
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have h : (x ◃ y) ◃ x = (x ◃ y) ◃ (x ◃ 1) := by grind
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grind
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