realization of Sets

Circumvent problem with implicits

theories/set/Set.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Add Rec LoadPath "/home/cmarche/recherche/why3/share/theories".
+Add Rec LoadPath "/home/cmarche/recherche/why3/share/modules".
+
+Inductive set_ (X:Type) : Type := mk_set: (X -> Prop) -> set_ X.
+
+Definition set : forall (a:Type), Type.
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact set_.
+(*
+(fun (X: Type) => X -> Prop).
+*)
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Definition mem: forall (a:Type), a -> (set a) -> Prop.
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X x s.
+destruct s.
+exact (P x).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+Hint Unfold mem.
+Notation "x == y" := (infix_eqeq x y) (at level 70, no associativity).
+(*
+Require FunctionalExtensionality.
+Require ProofIrrelevance.
+*)
+(* DO NOT EDIT BELOW *)
+
+Lemma extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+(* YOU MAY EDIT THE PROOF BELOW *)
+admit.
+(*
+intros.
+apply FunctionalExtensionality.functional_extensionality.
+red in H.
+unfold mem in H.
+intro x.
+apply ProofIrrelevance.proof_irrelevance.
+rewrite H.
+; auto.
+*)
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+(*Hint Unfold subset.*)
+(* DO NOT EDIT BELOW *)
+
+Lemma subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+unfold subset; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition empty: forall (a:Type), (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact (fun X => mk_set X (fun _ => False)).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+(*Print Implicit empty.*)
+(* DO NOT EDIT BELOW *)
+
+Lemma empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+unfold is_empty; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition add: forall (a:Type), a -> (set a) -> (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X x s.
+destruct s.
+exact (mk_set _ (fun y => P y \/ y=x)).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments add.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s.
+unfold mem,add; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition remove: forall (a:Type), a -> (set a) -> (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X x s.
+destruct s.
+exact (mk_set _ (fun y => y<>x /\ P y)).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments remove.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s.
+unfold mem,remove; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s.
+unfold subset, remove, mem; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition union: forall (a:Type), (set a) -> (set a) -> (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X s1 s2.
+destruct s1; destruct s2.
+exact (mk_set _ (fun y => P y \/ P0 y)).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments union.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s1; destruct s2.
+unfold union,mem; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition inter: forall (a:Type), (set a) -> (set a) -> (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X s1 s2.
+destruct s1; destruct s2.
+exact (mk_set _ (fun y => P y /\ P0 y)).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments inter.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s1; destruct s2.
+unfold inter, mem; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition diff: forall (a:Type), (set a) -> (set a) -> (set a).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros X s1 s2.
+destruct s1; destruct s2.
+exact (mk_set _ (fun y => P y /\ ~(P0 y))).
+Defined.
+(* DO NOT EDIT BELOW *)
+
+Implicit Arguments diff.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s1; destruct s2.
+unfold diff, mem; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros; destruct s1; destruct s2.
+unfold subset, diff, mem; intuition.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+