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