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src/HOL/Library/Permutation.thy

author | nipkow |

Wed Aug 18 11:09:40 2004 +0200 (2004-08-18) | |

changeset 15140 | 322485b816ac |

parent 15131 | c69542757a4d |

child 17200 | 3a4d03d1a31b |

permissions | -rw-r--r-- |

import -> imports

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 [2] 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