src/HOL/Library/Permutation.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 15140 322485b816ac child 17200 3a4d03d1a31b permissions -rw-r--r--
Constant "If" is now local
1 (*  Title:      HOL/Library/Permutation.thy
2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
3 *)
5 header {* Permutations *}
7 theory Permutation
8 imports Multiset
9 begin
11 consts
12   perm :: "('a list * 'a list) set"
14 syntax
15   "_perm" :: "'a list => 'a list => bool"    ("_ <~~> _"  [50, 50] 50)
16 translations
17   "x <~~> y" == "(x, y) \<in> perm"
19 inductive perm
20   intros
21     Nil  [intro!]: "[] <~~> []"
22     swap [intro!]: "y # x # l <~~> x # y # l"
23     Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
24     trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
26 lemma perm_refl [iff]: "l <~~> l"
27 by (induct l, auto)
30 subsection {* Some examples of rule induction on permutations *}
32 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"
33     -- {*the form of the premise lets the induction bind @{term xs}
34          and @{term ys} *}
35   apply (erule perm.induct)
36      apply (simp_all (no_asm_simp))
37   done
39 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
40 by (insert xperm_empty_imp_aux, blast)
43 text {*
44   \medskip This more general theorem is easier to understand!
45   *}
47 lemma perm_length: "xs <~~> ys ==> length xs = length ys"
48 by (erule perm.induct, simp_all)
50 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
51 by (drule perm_length, auto)
53 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
54 by (erule perm.induct, auto)
56 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"
57 by (erule perm.induct, auto)
60 subsection {* Ways of making new permutations *}
62 text {*
63   We can insert the head anywhere in the list.
64 *}
66 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
67 by (induct xs, auto)
69 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
70   apply (induct xs, simp_all)
71   apply (blast intro: perm_append_Cons)
72   done
74 lemma perm_append_single: "a # xs <~~> xs @ [a]"
75   by (rule perm.trans [OF _ perm_append_swap], simp)
77 lemma perm_rev: "rev xs <~~> xs"
78   apply (induct xs, simp_all)
79   apply (blast intro!: perm_append_single intro: perm_sym)
80   done
82 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
83 by (induct l, auto)
85 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
86 by (blast intro!: perm_append_swap perm_append1)
89 subsection {* Further results *}
91 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
92 by (blast intro: perm_empty_imp)
94 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
95   apply auto
96   apply (erule perm_sym [THEN perm_empty_imp])
97   done
99 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"
100 by (erule perm.induct, auto)
102 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
103 by (blast intro: perm_sing_imp)
105 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
106 by (blast dest: perm_sym)
109 subsection {* Removing elements *}
111 consts
112   remove :: "'a => 'a list => 'a list"
113 primrec
114   "remove x [] = []"
115   "remove x (y # ys) = (if x = y then ys else y # remove x ys)"
117 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"
118 by (induct ys, auto)
120 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"
121 by (induct l, auto)
123 lemma multiset_of_remove[simp]:
124   "multiset_of (remove a x) = multiset_of x - {#a#}"
125   by (induct_tac x, auto simp: multiset_eq_conv_count_eq)
128 text {* \medskip Congruence rule *}
130 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"
131 by (erule perm.induct, auto)
133 lemma remove_hd [simp]: "remove z (z # xs) = xs"
134   by auto
136 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
137 by (drule_tac z = z in perm_remove_perm, auto)
139 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
140 by (blast intro: cons_perm_imp_perm)
142 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"
143   apply (induct zs rule: rev_induct)
144    apply (simp_all (no_asm_use))
145   apply blast
146   done
148 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
149 by (blast intro: append_perm_imp_perm perm_append1)
151 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
152   apply (safe intro!: perm_append2)
153   apply (rule append_perm_imp_perm)
154   apply (rule perm_append_swap [THEN perm.trans])
155     -- {* the previous step helps this @{text blast} call succeed quickly *}
156   apply (blast intro: perm_append_swap)
157   done
159 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
160   apply (rule iffI)
161   apply (erule_tac  perm.induct, simp_all add: union_ac)
162   apply (erule rev_mp, rule_tac x=ys in spec)
163   apply (induct_tac xs, auto)
164   apply (erule_tac x = "remove a x" in allE, drule sym, simp)
165   apply (subgoal_tac "a \<in> set x")
166   apply (drule_tac z=a in perm.Cons)
167   apply (erule perm.trans, rule perm_sym, erule perm_remove)
168   apply (drule_tac f=set_of in arg_cong, simp)
169   done
171 lemma multiset_of_le_perm_append:
172   "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)";
173   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
174   apply (insert surj_multiset_of, drule surjD)
175   apply (blast intro: sym)+
176   done
178 end