diff options
Diffstat (limited to 'theories7/Sorting/Permutation.v')
-rw-r--r-- | theories7/Sorting/Permutation.v | 111 |
1 files changed, 111 insertions, 0 deletions
diff --git a/theories7/Sorting/Permutation.v b/theories7/Sorting/Permutation.v new file mode 100644 index 00000000..46b8da00 --- /dev/null +++ b/theories7/Sorting/Permutation.v @@ -0,0 +1,111 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(*i $Id: Permutation.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*) + +Require Relations. +Require PolyList. +Require Multiset. + +Set Implicit Arguments. + +Section defs. + +Variable A : Set. +Variable leA : (relation A). +Variable eqA : (relation A). + +Local gtA := [x,y:A]~(leA x y). + +Hypothesis leA_dec : (x,y:A){(leA x y)}+{~(leA x y)}. +Hypothesis eqA_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}. +Hypothesis leA_refl : (x,y:A) (eqA x y) -> (leA x y). +Hypothesis leA_trans : (x,y,z:A) (leA x y) -> (leA y z) -> (leA x z). +Hypothesis leA_antisym : (x,y:A)(leA x y) -> (leA y x) -> (eqA x y). + +Hints Resolve leA_refl : default. +Hints Immediate eqA_dec leA_dec leA_antisym : default. + +Local emptyBag := (EmptyBag A). +Local singletonBag := (SingletonBag eqA_dec). + +(** contents of a list *) + +Fixpoint list_contents [l:(list A)] : (multiset A) := + Cases l of + nil => emptyBag + | (cons a l) => (munion (singletonBag a) (list_contents l)) + end. + +Lemma list_contents_app : (l,m:(list A)) + (meq (list_contents (app l m)) (munion (list_contents l) (list_contents m))). +Proof. +Induction l; Simpl; Auto with datatypes. +Intros. +Apply meq_trans with + (munion (singletonBag a) (munion (list_contents l0) (list_contents m))); Auto with datatypes. +Qed. +Hints Resolve list_contents_app. + +Definition permutation := [l,m:(list A)](meq (list_contents l) (list_contents m)). + +Lemma permut_refl : (l:(list A))(permutation l l). +Proof. +Unfold permutation; Auto with datatypes. +Qed. +Hints Resolve permut_refl. + +Lemma permut_tran : (l,m,n:(list A)) + (permutation l m) -> (permutation m n) -> (permutation l n). +Proof. +Unfold permutation; Intros. +Apply meq_trans with (list_contents m); Auto with datatypes. +Qed. + +Lemma permut_right : (l,m:(list A)) + (permutation l m) -> (a:A)(permutation (cons a l) (cons a m)). +Proof. +Unfold permutation; Simpl; Auto with datatypes. +Qed. +Hints Resolve permut_right. + +Lemma permut_app : (l,l',m,m':(list A)) + (permutation l l') -> (permutation m m') -> + (permutation (app l m) (app l' m')). +Proof. +Unfold permutation; Intros. +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. +Apply meq_trans with (munion (list_contents l') (list_contents m)); Auto with datatypes. +Qed. +Hints Resolve permut_app. + +Lemma permut_cons : (l,m:(list A))(permutation l m) -> + (a:A)(permutation (cons a l) (cons a m)). +Proof. +Intros l m H a. +Change (permutation (app (cons a (nil A)) l) (app (cons a (nil A)) m)). +Apply permut_app; Auto with datatypes. +Qed. +Hints Resolve permut_cons. + +Lemma permut_middle : (l,m:(list A)) + (a:A)(permutation (cons a (app l m)) (app l (cons a m))). +Proof. +Unfold permutation. +Induction l; Simpl; Auto with datatypes. +Intros. +Apply meq_trans with (munion (singletonBag a) + (munion (singletonBag a0) (list_contents (app l0 m)))); Auto with datatypes. +Apply munion_perm_left; Auto with datatypes. +Qed. +Hints Resolve permut_middle. + +End defs. +Unset Implicit Arguments. + |