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src/HOL/Library/Permutation.thy

changeset 17200 | 3a4d03d1a31b |

parent 15140 | 322485b816ac |

child 19380 | b808efaa5828 |

1.1 --- a/src/HOL/Library/Permutation.thy Wed Aug 31 15:46:36 2005 +0200 1.2 +++ b/src/HOL/Library/Permutation.thy Wed Aug 31 15:46:37 2005 +0200 1.3 @@ -24,20 +24,20 @@ 1.4 trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" 1.5 1.6 lemma perm_refl [iff]: "l <~~> l" 1.7 -by (induct l, auto) 1.8 + by (induct l) auto 1.9 1.10 1.11 subsection {* Some examples of rule induction on permutations *} 1.12 1.13 lemma xperm_empty_imp_aux: "xs <~~> ys ==> xs = [] --> ys = []" 1.14 - -- {*the form of the premise lets the induction bind @{term xs} 1.15 + -- {*the form of the premise lets the induction bind @{term xs} 1.16 and @{term ys} *} 1.17 apply (erule perm.induct) 1.18 apply (simp_all (no_asm_simp)) 1.19 done 1.20 1.21 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" 1.22 -by (insert xperm_empty_imp_aux, blast) 1.23 + using xperm_empty_imp_aux by blast 1.24 1.25 1.26 text {* 1.27 @@ -45,16 +45,16 @@ 1.28 *} 1.29 1.30 lemma perm_length: "xs <~~> ys ==> length xs = length ys" 1.31 -by (erule perm.induct, simp_all) 1.32 + by (erule perm.induct) simp_all 1.33 1.34 lemma perm_empty_imp: "[] <~~> xs ==> xs = []" 1.35 -by (drule perm_length, auto) 1.36 + by (drule perm_length) auto 1.37 1.38 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" 1.39 -by (erule perm.induct, auto) 1.40 + by (erule perm.induct) auto 1.41 1.42 lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys" 1.43 -by (erule perm.induct, auto) 1.44 + by (erule perm.induct) auto 1.45 1.46 1.47 subsection {* Ways of making new permutations *} 1.48 @@ -64,32 +64,34 @@ 1.49 *} 1.50 1.51 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" 1.52 -by (induct xs, auto) 1.53 + by (induct xs) auto 1.54 1.55 lemma perm_append_swap: "xs @ ys <~~> ys @ xs" 1.56 - apply (induct xs, simp_all) 1.57 + apply (induct xs) 1.58 + apply simp_all 1.59 apply (blast intro: perm_append_Cons) 1.60 done 1.61 1.62 lemma perm_append_single: "a # xs <~~> xs @ [a]" 1.63 - by (rule perm.trans [OF _ perm_append_swap], simp) 1.64 + by (rule perm.trans [OF _ perm_append_swap]) simp 1.65 1.66 lemma perm_rev: "rev xs <~~> xs" 1.67 - apply (induct xs, simp_all) 1.68 + apply (induct xs) 1.69 + apply simp_all 1.70 apply (blast intro!: perm_append_single intro: perm_sym) 1.71 done 1.72 1.73 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" 1.74 -by (induct l, auto) 1.75 + by (induct l) auto 1.76 1.77 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" 1.78 -by (blast intro!: perm_append_swap perm_append1) 1.79 + by (blast intro!: perm_append_swap perm_append1) 1.80 1.81 1.82 subsection {* Further results *} 1.83 1.84 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])" 1.85 -by (blast intro: perm_empty_imp) 1.86 + by (blast intro: perm_empty_imp) 1.87 1.88 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])" 1.89 apply auto 1.90 @@ -97,13 +99,13 @@ 1.91 done 1.92 1.93 lemma perm_sing_imp [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]" 1.94 -by (erule perm.induct, auto) 1.95 + by (erule perm.induct) auto 1.96 1.97 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])" 1.98 -by (blast intro: perm_sing_imp) 1.99 + by (blast intro: perm_sing_imp) 1.100 1.101 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])" 1.102 -by (blast dest: perm_sym) 1.103 + by (blast dest: perm_sym) 1.104 1.105 1.106 subsection {* Removing elements *} 1.107 @@ -115,29 +117,31 @@ 1.108 "remove x (y # ys) = (if x = y then ys else y # remove x ys)" 1.109 1.110 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys" 1.111 -by (induct ys, auto) 1.112 + by (induct ys) auto 1.113 1.114 lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" 1.115 -by (induct l, auto) 1.116 + by (induct l) auto 1.117 1.118 -lemma multiset_of_remove[simp]: 1.119 - "multiset_of (remove a x) = multiset_of x - {#a#}" 1.120 - by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 1.121 +lemma multiset_of_remove[simp]: 1.122 + "multiset_of (remove a x) = multiset_of x - {#a#}" 1.123 + apply (induct x) 1.124 + apply (auto simp: multiset_eq_conv_count_eq) 1.125 + done 1.126 1.127 1.128 text {* \medskip Congruence rule *} 1.129 1.130 lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" 1.131 -by (erule perm.induct, auto) 1.132 + by (erule perm.induct) auto 1.133 1.134 lemma remove_hd [simp]: "remove z (z # xs) = xs" 1.135 by auto 1.136 1.137 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" 1.138 -by (drule_tac z = z in perm_remove_perm, auto) 1.139 + by (drule_tac z = z in perm_remove_perm) auto 1.140 1.141 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)" 1.142 -by (blast intro: cons_perm_imp_perm) 1.143 + by (blast intro: cons_perm_imp_perm) 1.144 1.145 lemma append_perm_imp_perm: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys" 1.146 apply (induct zs rule: rev_induct) 1.147 @@ -146,7 +150,7 @@ 1.148 done 1.149 1.150 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)" 1.151 -by (blast intro: append_perm_imp_perm perm_append1) 1.152 + by (blast intro: append_perm_imp_perm perm_append1) 1.153 1.154 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)" 1.155 apply (safe intro!: perm_append2) 1.156 @@ -157,20 +161,20 @@ 1.157 done 1.158 1.159 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) " 1.160 - apply (rule iffI) 1.161 - apply (erule_tac [2] perm.induct, simp_all add: union_ac) 1.162 - apply (erule rev_mp, rule_tac x=ys in spec) 1.163 - apply (induct_tac xs, auto) 1.164 - apply (erule_tac x = "remove a x" in allE, drule sym, simp) 1.165 - apply (subgoal_tac "a \<in> set x") 1.166 - apply (drule_tac z=a in perm.Cons) 1.167 - apply (erule perm.trans, rule perm_sym, erule perm_remove) 1.168 + apply (rule iffI) 1.169 + apply (erule_tac [2] perm.induct, simp_all add: union_ac) 1.170 + apply (erule rev_mp, rule_tac x=ys in spec) 1.171 + apply (induct_tac xs, auto) 1.172 + apply (erule_tac x = "remove a x" in allE, drule sym, simp) 1.173 + apply (subgoal_tac "a \<in> set x") 1.174 + apply (drule_tac z=a in perm.Cons) 1.175 + apply (erule perm.trans, rule perm_sym, erule perm_remove) 1.176 apply (drule_tac f=set_of in arg_cong, simp) 1.177 done 1.178 1.179 -lemma multiset_of_le_perm_append: 1.180 - "(multiset_of xs \<le># multiset_of ys) = (\<exists> zs. xs @ zs <~~> ys)"; 1.181 - apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) 1.182 +lemma multiset_of_le_perm_append: 1.183 + "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)"; 1.184 + apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) 1.185 apply (insert surj_multiset_of, drule surjD) 1.186 apply (blast intro: sym)+ 1.187 done