46 lines
1.2 KiB
Text
46 lines
1.2 KiB
Text
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import algebra.function
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import logic.funext
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open function
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structure bijection (A : Type) :=
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(func finv : A → A)
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(linv : finv ∘ func = id)
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(rinv : func ∘ finv = id)
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attribute bijection.func [coercion]
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namespace bijection
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variable {A : Type}
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definition compose (f g : bijection A) : bijection A :=
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bijection.mk
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(f ∘ g)
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(finv g ∘ finv f)
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(by rewrite [compose.assoc, -{finv f ∘ _}compose.assoc, linv f, compose.left_id, linv g])
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(by rewrite [-compose.assoc, {_ ∘ finv g}compose.assoc, rinv g, compose.right_id, rinv f])
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infixr `∘b`:100 := compose
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lemma compose.assoc (f g h : bijection A) : (f ∘b g) ∘b h = f ∘b (g ∘b h) := rfl
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definition id : bijection A :=
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bijection.mk id id (compose.left_id id) (compose.left_id id)
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lemma id.left_id (f : bijection A) : id ∘b f = f :=
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bijection.rec_on f (λx x x x, rfl)
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lemma id.right_id (f : bijection A) : f ∘b id = f :=
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bijection.rec_on f (λx x x x, rfl)
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definition inv (f : bijection A) : bijection A :=
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bijection.mk
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(finv f)
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(func f)
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(rinv f)
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(linv f)
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lemma inv.linv (f : bijection A) : inv f ∘b f = id :=
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bijection.rec_on f (λfunc finv linv rinv, by rewrite [↑inv, ↑compose, linv])
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end bijection
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