(* * Copyright (c) 2009-2010, Andrew Appel, Robert Dockins and Aquinas Hobor. * *)
(** This library reexports portions of the Coq standard libraries used throughtout the proof.  It also defines some convenience tactics. *)
Require Export VST.msl.Extensionality.
Require Export Coq.Lists.List.
Require Export Coq.Bool.Bool.
Require Export Coq.Relations.Relations.
Require Export Omega.
Definition compose (A B C:Type) (g:B -> C) (f:A -> B) := fun x => g (f x).
Arguments compose [A B C] _ _ _.
Infix "oo" := compose (at level 54, right associativity).
Definition id (A:Type) := fun x:A => x.
Record bijection (A B:Type) : Type := Bijection {
bij_f: A -> B;
bij_g: B -> A;
bij_fg: forall x, bij_f (bij_g x) = x;
bij_gf: forall x, bij_g (bij_f x) = x
}.

Lemma bij_fg_id: forall {A B} (f: bijection A B),
(bij_f _ _ f) oo (bij_g _ _ f) = id _.