44 lines
1.1 KiB
Text
44 lines
1.1 KiB
Text
--
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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 comp (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 [comp.assoc, -{finv f ∘ _}comp.assoc, linv f, comp.left_id, linv g])
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(by rewrite [-comp.assoc, {_ ∘ finv g}comp.assoc, rinv g, comp.right_id, rinv f])
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infixr ` ∘b `:100 := comp
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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 (comp.left_id id) (comp.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, ↑comp, linv])
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end bijection
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