--- a/src/HOL/Library/Permutation.thy Thu Jun 24 17:52:55 2004 +0200
+++ b/src/HOL/Library/Permutation.thy Thu Jun 24 17:54:53 2004 +0200
@@ -1,15 +1,10 @@
(* Title: HOL/Library/Permutation.thy
- ID: $Id$
- Author: Lawrence C Paulson and Thomas M Rasmussen
- Copyright 1995 University of Cambridge
-
-TODO: it would be nice to prove (for "multiset", defined on
-HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)
+ Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
*)
header {* Permutations *}
-theory Permutation = Main:
+theory Permutation = Multiset:
consts
perm :: "('a list * 'a list) set"
@@ -27,9 +22,7 @@
trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
lemma perm_refl [iff]: "l <~~> l"
- apply (induct l)
- apply auto
- done
+by (induct l, auto)
subsection {* Some examples of rule induction on permutations *}
@@ -41,9 +34,7 @@
done
lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
- apply (insert xperm_empty_imp_aux)
- apply blast
- done
+by (insert xperm_empty_imp_aux, blast)
text {*
@@ -51,24 +42,16 @@
*}
lemma perm_length: "xs <~~> ys ==> length xs = length ys"
- apply (erule perm.induct)
- apply simp_all
- done
+by (erule perm.induct, simp_all)
lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
- apply (drule perm_length)
- apply auto
- done
+by (drule perm_length, auto)
lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
- apply (erule perm.induct)
- apply auto
- done
+by (erule perm.induct, auto)
lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
- apply (erule perm.induct)
- apply auto
- done
+by (erule perm.induct, auto)
subsection {* Ways of making new permutations *}
@@ -78,44 +61,35 @@
*}
lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
- apply (induct xs)
- apply auto
- done
+by (induct xs, auto)
lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
- apply (induct xs)
- apply simp_all
+ apply (induct xs, simp_all)
apply (blast intro: perm_append_Cons)
done
lemma perm_append_single: "a # xs <~~> xs @ [a]"
apply (rule perm.trans)
prefer 2
- apply (rule perm_append_swap)
- apply simp
+ apply (rule perm_append_swap, simp)
done
lemma perm_rev: "rev xs <~~> xs"
- apply (induct xs)
- apply simp_all
+ apply (induct xs, simp_all)
apply (blast intro!: perm_append_single intro: perm_sym)
done
lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
- apply (induct l)
- apply auto
- done
+by (induct l, auto)
lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
- apply (blast intro!: perm_append_swap perm_append1)
- done
+by (blast intro!: perm_append_swap perm_append1)
subsection {* Further results *}
lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
- apply (blast intro: perm_empty_imp)
- done
+by (blast intro: perm_empty_imp)
lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
apply auto
@@ -123,17 +97,13 @@
done
lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
- apply (erule perm.induct)
- apply auto
- done
+by (erule perm.induct, auto)
lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
- apply (blast intro: perm_sing_imp)
- done
+by (blast intro: perm_sing_imp)
lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
- apply (blast dest: perm_sym)
- done
+by (blast dest: perm_sym)
subsection {* Removing elements *}
@@ -145,35 +115,26 @@
"remove x (y # ys) = (if x = y then ys else y # remove x ys)"
lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
- apply (induct ys)
- apply auto
- done
+by (induct ys, auto)
lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
- apply (induct l)
- apply auto
- done
+by (induct l, auto)
text {* \medskip Congruence rule *}
lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
- apply (erule perm.induct)
- apply auto
- done
+by (erule perm.induct, auto)
lemma remove_hd [simp]: "remove z (z # xs) = xs"
apply auto
done
lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
- apply (drule_tac z = z in perm_remove_perm)
- apply auto
- done
+by (drule_tac z = z in perm_remove_perm, auto)
lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
- apply (blast intro: cons_perm_imp_perm)
- done
+by (blast intro: cons_perm_imp_perm)
lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
apply (induct zs rule: rev_induct)
@@ -182,8 +143,7 @@
done
lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
- apply (blast intro: append_perm_imp_perm perm_append1)
- done
+by (blast intro: append_perm_imp_perm perm_append1)
lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
apply (safe intro!: perm_append2)
@@ -193,4 +153,47 @@
apply (blast intro: perm_append_swap)
done
+(****************** Norbert Voelker 17 June 2004 **************)
+
+consts
+ multiset_of :: "'a list \<Rightarrow> 'a multiset"
+primrec
+ "multiset_of [] = {#}"
+ "multiset_of (a # x) = multiset_of x + {# a #}"
+
+lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
+ by (induct_tac x, auto)
+
+lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
+ by (induct_tac x, auto)
+
+lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
+ by (induct_tac x, auto)
+
+lemma multiset_of_remove[simp]:
+ "multiset_of (remove a x) = multiset_of x - {#a#}"
+ by (induct_tac x, auto simp: multiset_eq_conv_count_eq)
+
+lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
+ apply (rule iffI)
+ apply (erule_tac [2] perm.induct, simp_all add: union_ac)
+ apply (erule rev_mp, rule_tac x=ys in spec, induct_tac xs, auto)
+ apply (erule_tac x = "remove a x" in allE, drule sym, simp)
+ apply (subgoal_tac "a \<in> set x")
+ apply (drule_tac z=a in perm.Cons)
+ apply (erule perm.trans, rule perm_sym, erule perm_remove)
+ apply (drule_tac f=set_of in arg_cong, simp)
+ done
+
+lemma set_count_multiset_of: "set x = {a. 0 < count (multiset_of x) a}"
+ by (induct_tac x, auto)
+
+lemma distinct_count_multiset_of:
+ "distinct x \<Longrightarrow> count (multiset_of x) a = (if a \<in> set x then 1 else 0)"
+ by (erule rev_mp, induct_tac x, auto)
+
+lemma distinct_set_eq_iff_multiset_of_eq:
+ "\<lbrakk>distinct x; distinct y\<rbrakk> \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)"
+ by (auto simp: multiset_eq_conv_count_eq distinct_count_multiset_of)
+
end