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src/HOL/Library/Permutation.thy

author | wenzelm |

Thu May 06 14:14:18 2004 +0200 (2004-05-06) | |

changeset 14706 | 71590b7733b7 |

parent 11153 | 950ede59c05a |

child 15005 | 546c8e7e28d4 |

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

tuned document;

1 (* Title: HOL/Library/Permutation.thy

2 ID: $Id$

3 Author: Lawrence C Paulson and Thomas M Rasmussen

4 Copyright 1995 University of Cambridge

6 TODO: it would be nice to prove (for "multiset", defined on

7 HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x)

8 *)

10 header {* Permutations *}

12 theory Permutation = Main:

14 consts

15 perm :: "('a list * 'a list) set"

17 syntax

18 "_perm" :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50)

19 translations

20 "x <~~> y" == "(x, y) \<in> perm"

22 inductive perm

23 intros

24 Nil [intro!]: "[] <~~> []"

25 swap [intro!]: "y # x # l <~~> x # y # l"

26 Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"

27 trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"

29 lemma perm_refl [iff]: "l <~~> l"

30 apply (induct l)

31 apply auto

32 done

35 subsection {* Some examples of rule induction on permutations *}

37 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []"

38 -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *}

39 apply (erule perm.induct)

40 apply (simp_all (no_asm_simp))

41 done

43 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"

44 apply (insert xperm_empty_imp_aux)

45 apply blast

46 done

49 text {*

50 \medskip This more general theorem is easier to understand!

51 *}

53 lemma perm_length: "xs <~~> ys ==> length xs = length ys"

54 apply (erule perm.induct)

55 apply simp_all

56 done

58 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"

59 apply (drule perm_length)

60 apply auto

61 done

63 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"

64 apply (erule perm.induct)

65 apply auto

66 done

68 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys"

69 apply (erule perm.induct)

70 apply auto

71 done

74 subsection {* Ways of making new permutations *}

76 text {*

77 We can insert the head anywhere in the list.

78 *}

80 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"

81 apply (induct xs)

82 apply auto

83 done

85 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"

86 apply (induct xs)

87 apply simp_all

88 apply (blast intro: perm_append_Cons)

89 done

91 lemma perm_append_single: "a # xs <~~> xs @ [a]"

92 apply (rule perm.trans)

93 prefer 2

94 apply (rule perm_append_swap)

95 apply simp

96 done

98 lemma perm_rev: "rev xs <~~> xs"

99 apply (induct xs)

100 apply simp_all

101 apply (blast intro!: perm_append_single intro: perm_sym)

102 done

104 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"

105 apply (induct l)

106 apply auto

107 done

109 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"

110 apply (blast intro!: perm_append_swap perm_append1)

111 done

114 subsection {* Further results *}

116 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"

117 apply (blast intro: perm_empty_imp)

118 done

120 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"

121 apply auto

122 apply (erule perm_sym [THEN perm_empty_imp])

123 done

125 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]"

126 apply (erule perm.induct)

127 apply auto

128 done

130 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"

131 apply (blast intro: perm_sing_imp)

132 done

134 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"

135 apply (blast dest: perm_sym)

136 done

139 subsection {* Removing elements *}

141 consts

142 remove :: "'a => 'a list => 'a list"

143 primrec

144 "remove x [] = []"

145 "remove x (y # ys) = (if x = y then ys else y # remove x ys)"

147 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys"

148 apply (induct ys)

149 apply auto

150 done

152 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)"

153 apply (induct l)

154 apply auto

155 done

158 text {* \medskip Congruence rule *}

160 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys"

161 apply (erule perm.induct)

162 apply auto

163 done

165 lemma remove_hd [simp]: "remove z (z # xs) = xs"

166 apply auto

167 done

169 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"

170 apply (drule_tac z = z in perm_remove_perm)

171 apply auto

172 done

174 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"

175 apply (blast intro: cons_perm_imp_perm)

176 done

178 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys"

179 apply (induct zs rule: rev_induct)

180 apply (simp_all (no_asm_use))

181 apply blast

182 done

184 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"

185 apply (blast intro: append_perm_imp_perm perm_append1)

186 done

188 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"

189 apply (safe intro!: perm_append2)

190 apply (rule append_perm_imp_perm)

191 apply (rule perm_append_swap [THEN perm.trans])

192 -- {* the previous step helps this @{text blast} call succeed quickly *}

193 apply (blast intro: perm_append_swap)

194 done

196 end