diff options
author | Stephane Glondu <steph@glondu.net> | 2012-08-20 18:27:01 +0200 |
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committer | Stephane Glondu <steph@glondu.net> | 2012-08-20 18:27:01 +0200 |
commit | e0d682ec25282a348d35c5b169abafec48555690 (patch) | |
tree | 1a46f0142a85df553388c932110793881f3af52f /theories/Sorting/Heap.v | |
parent | 86535d84cc3cffeee1dcd8545343f234e7285530 (diff) |
Imported Upstream version 8.4dfsgupstream/8.4dfsg
Diffstat (limited to 'theories/Sorting/Heap.v')
-rw-r--r-- | theories/Sorting/Heap.v | 22 |
1 files changed, 11 insertions, 11 deletions
diff --git a/theories/Sorting/Heap.v b/theories/Sorting/Heap.v index 60bb50ce..8b1bdbd4 100644 --- a/theories/Sorting/Heap.v +++ b/theories/Sorting/Heap.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -55,13 +55,13 @@ Section defs. Lemma leA_Tree_Leaf : forall a:A, leA_Tree a Tree_Leaf. Proof. - simpl in |- *; auto with datatypes. + simpl; auto with datatypes. Qed. Lemma leA_Tree_Node : forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D). Proof. - simpl in |- *; auto with datatypes. + simpl; auto with datatypes. Qed. @@ -121,7 +121,7 @@ Section defs. forall (T:Tree) (a b:A), leA a b -> leA_Tree b T -> leA_Tree a T. Proof. simple induction T; auto with datatypes. - intros; simpl in |- *; apply leA_trans with b; auto with datatypes. + intros; simpl; apply leA_trans with b; auto with datatypes. Qed. (** ** Merging two sorted lists *) @@ -213,12 +213,12 @@ Section defs. simple induction 1; intros. apply insert_exist with (Tree_Node a Tree_Leaf Tree_Leaf); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. - simpl in |- *; unfold meq, munion in |- *; auto using node_is_heap with datatypes. + simpl; unfold meq, munion; auto using node_is_heap with datatypes. elim (leA_dec a a0); intros. elim (X a0); intros. apply insert_exist with (Tree_Node a T2 T0); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. - simpl in |- *; apply treesort_twist1; trivial with datatypes. + simpl; apply treesort_twist1; trivial with datatypes. elim (X a); intros T3 HeapT3 ConT3 LeA. apply insert_exist with (Tree_Node a0 T2 T3); auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes. @@ -226,7 +226,7 @@ Section defs. apply low_trans with a; auto with datatypes. apply LeA; auto with datatypes. apply low_trans with a; auto with datatypes. - simpl in |- *; apply treesort_twist2; trivial with datatypes. + simpl; apply treesort_twist2; trivial with datatypes. Qed. @@ -242,10 +242,10 @@ Section defs. Proof. simple induction l. apply (heap_exist nil Tree_Leaf); auto with datatypes. - simpl in |- *; unfold meq in |- *; exact nil_is_heap. + simpl; unfold meq; exact nil_is_heap. simple induction 1. intros T i m; elim (insert T i a). - intros; apply heap_exist with T1; simpl in |- *; auto with datatypes. + intros; apply heap_exist with T1; simpl; auto with datatypes. apply meq_trans with (munion (contents T) (singletonBag a)). apply meq_trans with (munion (singletonBag a) (contents T)). apply meq_right; trivial with datatypes. @@ -269,7 +269,7 @@ Section defs. apply flat_exist with (nil (A:=A)); auto with datatypes. elim X; intros l1 s1 i1 m1; elim X0; intros l2 s2 i2 m2. elim (merge _ s1 _ s2); intros. - apply flat_exist with (a :: l); simpl in |- *; auto with datatypes. + apply flat_exist with (a :: l); simpl; auto with datatypes. apply meq_trans with (munion (list_contents _ eqA_dec l1) (munion (list_contents _ eqA_dec l2) (singletonBag a))). @@ -288,7 +288,7 @@ Section defs. forall l:list A, {m : list A | Sorted leA m & permutation _ eqA_dec l m}. Proof. - intro l; unfold permutation in |- *. + intro l; unfold permutation. elim (list_to_heap l). intros. elim (heap_to_list T); auto with datatypes. |