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Diffstat (limited to 'theories/Sorting/Permutation.v')
| -rw-r--r-- | theories/Sorting/Permutation.v | 148 |
1 files changed, 117 insertions, 31 deletions
diff --git a/theories/Sorting/Permutation.v b/theories/Sorting/Permutation.v index b3287cd1..0f2e02b5 100644 --- a/theories/Sorting/Permutation.v +++ b/theories/Sorting/Permutation.v @@ -6,30 +6,39 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Permutation.v 5920 2004-07-16 20:01:26Z herbelin $ i*) +(*i $Id: Permutation.v 8823 2006-05-16 16:17:43Z letouzey $ i*) Require Import Relations. Require Import List. Require Import Multiset. +Require Import Arith. + +(** This file define a notion of permutation for lists, based on multisets: + there exists a permutation between two lists iff every elements have + the same multiplicities in the two lists. + + Unlike [List.Permutation], the present notion of permutation requires + a decidable equality. At the same time, this definition can be used + with a non-standard equality, whereas [List.Permutation] cannot. + + The present file contains basic results, obtained without any particular + assumption on the decidable equality used. + + File [PermutSetoid] contains additional results about permutations + with respect to an setoid equality (i.e. an equivalence relation). + + Finally, file [PermutEq] concerns Coq equality : this file is similar + to the previous one, but proves in addition that [List.Permutation] + and [permutation] are equivalent in this context. +*) Set Implicit Arguments. Section defs. Variable A : Set. -Variable leA : relation A. Variable eqA : relation A. - -Let gtA (x y:A) := ~ leA x y. - -Hypothesis leA_dec : forall x y:A, {leA x y} + {~ leA x y}. Hypothesis eqA_dec : forall x y:A, {eqA x y} + {~ eqA x y}. -Hypothesis leA_refl : forall x y:A, eqA x y -> leA x y. -Hypothesis leA_trans : forall x y z:A, leA x y -> leA y z -> leA x z. -Hypothesis leA_antisym : forall x y:A, leA x y -> leA y x -> eqA x y. - -Hint Resolve leA_refl: default. -Hint Immediate eqA_dec leA_dec leA_antisym: default. Let emptyBag := EmptyBag A. Let singletonBag := SingletonBag _ eqA_dec. @@ -63,6 +72,12 @@ unfold permutation in |- *; auto with datatypes. Qed. Hint Resolve permut_refl. +Lemma permut_sym : + forall l1 l2 : list A, permutation l1 l2 -> permutation l2 l1. +Proof. +unfold permutation, meq; intros; apply sym_eq; trivial. +Qed. + Lemma permut_tran : forall l m n:list A, permutation l m -> permutation m n -> permutation l n. Proof. @@ -70,51 +85,122 @@ unfold permutation in |- *; intros. apply meq_trans with (list_contents m); auto with datatypes. Qed. -Lemma permut_right : +Lemma permut_cons : forall l m:list A, permutation l m -> forall a:A, permutation (a :: l) (a :: m). Proof. unfold permutation in |- *; simpl in |- *; auto with datatypes. Qed. -Hint Resolve permut_right. +Hint Resolve permut_cons. Lemma permut_app : forall l l' m m':list A, permutation l l' -> permutation m m' -> permutation (l ++ m) (l' ++ m'). Proof. unfold permutation in |- *; intros. -apply meq_trans with (munion (list_contents l) (list_contents m)); +apply meq_trans with (munion (list_contents l) (list_contents m)); auto with datatypes. -apply meq_trans with (munion (list_contents l') (list_contents m')); +apply meq_trans with (munion (list_contents l') (list_contents m')); auto with datatypes. apply meq_trans with (munion (list_contents l') (list_contents m)); auto with datatypes. Qed. Hint Resolve permut_app. -Lemma permut_cons : - forall l m:list A, - permutation l m -> forall a:A, permutation (a :: l) (a :: m). +Lemma permut_add_inside : + forall a l1 l2 l3 l4, + permutation (l1 ++ l2) (l3 ++ l4) -> + permutation (l1 ++ a :: l2) (l3 ++ a :: l4). Proof. -intros l m H a. -change (permutation ((a :: nil) ++ l) ((a :: nil) ++ m)) in |- *. -apply permut_app; auto with datatypes. +unfold permutation, meq in *; intros. +generalize (H a0); clear H. +do 4 rewrite list_contents_app. +simpl. +destruct (eqA_dec a a0); simpl; auto with arith. +do 2 rewrite <- plus_n_Sm; f_equal; auto. +Qed. + +Lemma permut_add_cons_inside : + forall a l l1 l2, + permutation l (l1 ++ l2) -> + permutation (a :: l) (l1 ++ a :: l2). +Proof. +intros; +replace (a :: l) with (nil ++ a :: l); trivial; +apply permut_add_inside; trivial. Qed. -Hint Resolve permut_cons. Lemma permut_middle : forall (l m:list A) (a:A), permutation (a :: l ++ m) (l ++ a :: m). Proof. -unfold permutation in |- *. -simple induction l; simpl in |- *; auto with datatypes. -intros. -apply meq_trans with - (munion (singletonBag a) - (munion (singletonBag a0) (list_contents (l0 ++ m)))); - auto with datatypes. -apply munion_perm_left; auto with datatypes. +intros; apply permut_add_cons_inside; auto. Qed. Hint Resolve permut_middle. +Lemma permut_sym_app : + forall l1 l2, permutation (l1 ++ l2) (l2 ++ l1). +Proof. +intros l1 l2; +unfold permutation, meq; +intro a; do 2 rewrite list_contents_app; simpl; +auto with arith. +Qed. + +Lemma permut_rev : + forall l, permutation l (rev l). +Proof. +induction l. +simpl; auto. +simpl. +apply permut_add_cons_inside. +rewrite <- app_nil_end; auto. +Qed. + +(** Some inversion results. *) +Lemma permut_conv_inv : + forall e l1 l2, permutation (e :: l1) (e :: l2) -> permutation l1 l2. +Proof. +intros e l1 l2; unfold permutation, meq; simpl; intros H a; +generalize (H a); apply plus_reg_l. +Qed. + +Lemma permut_app_inv1 : + forall l l1 l2, permutation (l1 ++ l) (l2 ++ l) -> permutation l1 l2. +Proof. +intros l l1 l2; unfold permutation, meq; simpl; +intros H a; generalize (H a); clear H. +do 2 rewrite list_contents_app. +simpl. +intros; apply plus_reg_l with (multiplicity (list_contents l) a). +rewrite plus_comm; rewrite H; rewrite plus_comm. +trivial. +Qed. + +Lemma permut_app_inv2 : + forall l l1 l2, permutation (l ++ l1) (l ++ l2) -> permutation l1 l2. +Proof. +intros l l1 l2; unfold permutation, meq; simpl; +intros H a; generalize (H a); clear H. +do 2 rewrite list_contents_app. +simpl. +intros; apply plus_reg_l with (multiplicity (list_contents l) a). +trivial. +Qed. + +Lemma permut_remove_hd : + forall l l1 l2 a, + permutation (a :: l) (l1 ++ a :: l2) -> permutation l (l1 ++ l2). +Proof. +intros l l1 l2 a; unfold permutation, meq; simpl; intros H a0; generalize (H a0); clear H. +do 2 rewrite list_contents_app; simpl; intro H. +apply plus_reg_l with (if eqA_dec a a0 then 1 else 0). +rewrite H; clear H. +symmetry; rewrite plus_comm. +repeat rewrite <- plus_assoc; f_equal. +apply plus_comm. +Qed. + End defs. +(* For compatibilty *) +Notation permut_right := permut_cons. Unset Implicit Arguments. |