diff options
Diffstat (limited to 'theories/Sorting/PermutEq.v')
| -rw-r--r-- | theories/Sorting/PermutEq.v | 40 |
1 files changed, 20 insertions, 20 deletions
diff --git a/theories/Sorting/PermutEq.v b/theories/Sorting/PermutEq.v index f7bd37ee2..9bfe31ed1 100644 --- a/theories/Sorting/PermutEq.v +++ b/theories/Sorting/PermutEq.v @@ -13,22 +13,22 @@ Require Import Omega Relations Setoid List Multiset Permutation. Set Implicit Arguments. (** This file is similar to [PermutSetoid], except that the equality used here - is Coq usual one instead of a setoid equality. In particular, we can then - prove the equivalence between [List.Permutation] and + is Coq usual one instead of a setoid equality. In particular, we can then + prove the equivalence between [List.Permutation] and [Permutation.permutation]. *) Section Perm. - + Variable A : Type. Hypothesis eq_dec : forall x y:A, {x=y} + {~ x=y}. - + Notation permutation := (permutation _ eq_dec). Notation list_contents := (list_contents _ eq_dec). (** we can use [multiplicity] to define [In] and [NoDup]. *) - Lemma multiplicity_In : + Lemma multiplicity_In : forall l a, In a l <-> 0 < multiplicity (list_contents l) a. Proof. induction l. @@ -49,18 +49,18 @@ Section Perm. Lemma multiplicity_In_O : forall l a, ~ In a l -> multiplicity (list_contents l) a = 0. Proof. - intros l a; rewrite multiplicity_In; + intros l a; rewrite multiplicity_In; destruct (multiplicity (list_contents l) a); auto. destruct 1; auto with arith. Qed. - + Lemma multiplicity_In_S : forall l a, In a l -> multiplicity (list_contents l) a >= 1. Proof. intros l a; rewrite multiplicity_In; auto. Qed. - Lemma multiplicity_NoDup : + Lemma multiplicity_NoDup : forall l, NoDup l <-> (forall a, multiplicity (list_contents l) a <= 1). Proof. induction l. @@ -78,7 +78,7 @@ Section Perm. generalize (H a). destruct (eq_dec a a) as [H0|H0]. destruct (multiplicity (list_contents l) a); auto with arith. - simpl; inversion 1. + simpl; inversion 1. inversion H3. destruct H0; auto. rewrite IHl; intros. @@ -86,13 +86,13 @@ Section Perm. destruct (eq_dec a a0); simpl; auto with arith. Qed. - Lemma NoDup_permut : - forall l l', NoDup l -> NoDup l' -> + Lemma NoDup_permut : + forall l l', NoDup l -> NoDup l' -> (forall x, In x l <-> In x l') -> permutation l l'. Proof. intros. red; unfold meq; intros. - rewrite multiplicity_NoDup in H, H0. + rewrite multiplicity_NoDup in H, H0. generalize (H a) (H0 a) (H1 a); clear H H0 H1. do 2 rewrite multiplicity_In. destruct 3; omega. @@ -128,11 +128,11 @@ Section Perm. intro Abs; generalize (permut_In_In _ Abs H). inversion 1. Qed. - - (** When used with [eq], this permutation notion is equivalent to + + (** When used with [eq], this permutation notion is equivalent to the one defined in [List.v]. *) - Lemma permutation_Permutation : + Lemma permutation_Permutation : forall l l', Permutation l l' <-> permutation l l'. Proof. split. @@ -165,7 +165,7 @@ Section Perm. destruct (eq_dec b b) as [H|H]; [ | destruct H; auto]. destruct (eq_dec a b); simpl; auto; intros; discriminate. Qed. - + Lemma permut_length_2 : forall a1 b1 a2 b2, permutation (a1 :: b1 :: nil) (a2 :: b2 :: nil) -> (a1=a2) /\ (b1=b2) \/ (a1=b2) /\ (a2=b1). @@ -177,7 +177,7 @@ Section Perm. apply permut_length_1. red; red; intros. generalize (P a); clear P; simpl. - destruct (eq_dec a1 a) as [H2|H2]; + destruct (eq_dec a1 a) as [H2|H2]; destruct (eq_dec a2 a) as [H3|H3]; auto. destruct H3; transitivity a1; auto. destruct H2; transitivity a2; auto. @@ -187,7 +187,7 @@ Section Perm. apply permut_length_1. red; red; intros. generalize (P a); clear P; simpl. - destruct (eq_dec a1 a) as [H2|H2]; + destruct (eq_dec a1 a) as [H2|H2]; destruct (eq_dec b2 a) as [H3|H3]; auto. simpl; rewrite <- plus_n_Sm; inversion 1; auto. destruct H3; transitivity a1; auto. @@ -210,12 +210,12 @@ Section Perm. Qed. Variable B : Type. - Variable eqB_dec : forall x y:B, { x=y }+{ ~x=y }. + Variable eqB_dec : forall x y:B, { x=y }+{ ~x=y }. (** Permutation is compatible with map. *) Lemma permutation_map : - forall f l1 l2, permutation l1 l2 -> + forall f l1 l2, permutation l1 l2 -> Permutation.permutation _ eqB_dec (map f l1) (map f l2). Proof. intros f; induction l1. |