Add new files theories/Program/Basics.v and theories/Classes/Relations.v

for basic functional programming and relation
definitions respectively. 
Classes.Relations also includes the definition of Morphism
and instances for the standard morphisms and relations (eq, iff, impl,
inverse and complement). The class_setoid.ml4 [setoid_rewrite] tactic
has been reimplemented on top of these definitions, hence it doesn't
require a setoid implementation anymore. It also generates obligations
for missing reflexivity proofs, like the current setoid_rewrite. It has
not been tested on large examples but it should handle directions and
occurences. Works with in if no obligations are generated at this time.
What's missing is being able to rewrite by a lemma instead of a simple
hypothesis with no premises.


git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10502 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 143 insertions, 0 deletions

diff --git a/theories/Program/Basics.v b/theories/Program/Basics.v
new file mode 100644
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+(* -*- coq-prog-args: ("-emacs-U" "-nois") -*- *)
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(* Standard functions and proofs about them.
+ * Author: Matthieu Sozeau
+ * Institution: LRI, CNRS UMR 8623 - UniversitĂcopyright Paris Sud
+ *              91405 Orsay, France *)
+
+(* $Id: FSetAVL_prog.v 616 2007-08-08 12:28:10Z msozeau $ *)
+
+Set Implicit Arguments.
+Unset Strict Implicit.
+
+Require Export Coq.Program.FunctionalExtensionality.
+
+Definition compose `A B C` (g : B -> C) (f : A -> B) := fun x : A => g (f x).
+
+Definition arrow (A B : Type) := A -> B.
+
+Definition impl (A B : Prop) : Prop := A -> B.
+
+Definition id `A` := fun x : A => x.
+
+Hint Unfold compose.
+
+Notation " g 'o' f " := (compose g f)  (at level 50, left associativity) : program_scope.
+
+Open Scope program_scope.
+
+Lemma compose_id_left : forall A B (f : A -> B), id o f = f.
+Proof.
+  intros.
+  unfold id, compose.
+  symmetry ; apply eta_expansion.
+Qed.
+
+Lemma compose_id_right : forall A B (f : A -> B), f o id = f.
+Proof.
+  intros.
+  unfold id, compose.
+  symmetry ; apply eta_expansion.
+Qed.
+
+Lemma compose_assoc : forall A B C D (f : A -> B) (g : B -> C) (h : C -> D), 
+  h o (g o f) = h o g o f.
+Proof.
+  reflexivity.
+Qed.
+
+Hint Rewrite @compose_id_left @compose_id_right @compose_assoc : core.
+
+Notation " f '#' x " := (f x) (at level 100, x at level 200, only parsing).
+
+Definition const `A B` (a : A) := fun x : B => a.
+
+Definition flip `A B C` (f : A -> B -> C) x y := f y x.
+
+Lemma flip_flip : forall A B C (f : A -> B -> C), flip (flip f) = f.
+Proof.
+  unfold flip.
+  intros.
+  extensionality x ; extensionality y.
+  reflexivity.
+Qed.
+
+Definition apply `A B` (f : A -> B) (x : A) := f x.
+
+(** Notations for the unit type and value. *)
+
+Notation " () " := Datatypes.unit : type_scope.
+Notation " () " := tt.
+
+(** Set maximally inserted implicit arguments for standard definitions. *)
+
+Implicit Arguments eq [[A]].
+
+Implicit Arguments Some [[A]].
+Implicit Arguments None [[A]].
+
+Implicit Arguments inl [[A] [B]].
+Implicit Arguments inr [[A] [B]].
+
+Implicit Arguments left [[A] [B]].
+Implicit Arguments right [[A] [B]].
+
+(** Curryfication. *)
+
+Definition curry `a b c` (f : a -> b -> c) (p : prod a b) : c :=
+  let (x, y) := p in f x y.
+
+Definition uncurry `a b c` (f : prod a b -> c) (x : a) (y : b) : c :=
+  f (x, y).
+
+Lemma uncurry_curry : forall a b c (f : a -> b -> c), uncurry (curry f) = f.
+Proof.
+  simpl ; intros.
+  unfold uncurry, curry.
+  extensionality x ; extensionality y.
+  reflexivity.
+Qed.
+
+Lemma curry_uncurry : forall a b c (f : prod a b -> c), curry (uncurry f) = f.
+Proof.
+  simpl ; intros.
+  unfold uncurry, curry.
+  extensionality x. 
+  destruct x ; simpl ; reflexivity.
+Qed.
+
+(** Useful implicits and notations for lists. *)
+
+Require Export Coq.Lists.List.
+
+Implicit Arguments nil [[A]].
+Implicit Arguments cons [[A]].
+
+Notation " [] " := nil.
+Notation " [ x ] " := (cons x nil).
+Notation " [ x ; .. ; y ] " := (cons x .. (cons y nil) ..).
+
+(** n-ary exists ! *)
+
+Notation "'exists' x y , p" := (ex (fun x => (ex (fun y => p))))
+  (at level 200, x ident, y ident, right associativity) : type_scope.
+
+Notation "'exists' x y z , p" := (ex (fun x => (ex (fun y => (ex (fun z => p))))))
+  (at level 200, x ident, y ident, z ident, right associativity) : type_scope.
+
+Notation "'exists' x y z w , p" := (ex (fun x => (ex (fun y => (ex (fun z => (ex (fun w => p))))))))
+  (at level 200, x ident, y ident, z ident, w ident, right associativity) : type_scope.
+
+Tactic Notation "exist" constr(x) := exists x.
+Tactic Notation "exist" constr(x) constr(y) := exists x ; exists y.
+Tactic Notation "exist" constr(x) constr(y) constr(z) := exists x ; exists y ; exists z.
+Tactic Notation "exist" constr(x) constr(y) constr(z) constr(w) := exists x ; exists y ; exists z ; exists w.
+
+Notation " ! A " := (notT A) (at level 200, A at level 100) : type_scope.