src/HOL/Library/Permutation.thy
 changeset 56796 9f84219715a7 parent 56154 f0a927235162 child 57816 d8bbb97689d3
```     1.1 --- a/src/HOL/Library/Permutation.thy	Tue Apr 29 21:54:26 2014 +0200
1.2 +++ b/src/HOL/Library/Permutation.thy	Tue Apr 29 22:50:55 2014 +0200
1.3 @@ -22,12 +22,10 @@
1.4  subsection {* Some examples of rule induction on permutations *}
1.5
1.6  lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
1.7 -  by (induct xs == "[]::'a list" ys pred: perm) simp_all
1.8 +  by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
1.9
1.10
1.11 -text {*
1.12 -  \medskip This more general theorem is easier to understand!
1.13 -  *}
1.14 +text {* \medskip This more general theorem is easier to understand! *}
1.15
1.16  lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
1.17    by (induct pred: perm) simp_all
1.18 @@ -41,9 +39,7 @@
1.19
1.20  subsection {* Ways of making new permutations *}
1.21
1.22 -text {*
1.23 -  We can insert the head anywhere in the list.
1.24 -*}
1.25 +text {* We can insert the head anywhere in the list. *}
1.26
1.27  lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
1.28    by (induct xs) auto
1.29 @@ -72,10 +68,10 @@
1.30
1.31  subsection {* Further results *}
1.32
1.33 -lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
1.34 +lemma perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
1.35    by (blast intro: perm_empty_imp)
1.36
1.37 -lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
1.38 +lemma perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
1.39    apply auto
1.40    apply (erule perm_sym [THEN perm_empty_imp])
1.41    done
1.42 @@ -83,10 +79,10 @@
1.43  lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
1.44    by (induct pred: perm) auto
1.45
1.46 -lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
1.47 +lemma perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
1.48    by (blast intro: perm_sing_imp)
1.49
1.50 -lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
1.51 +lemma perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
1.52    by (blast dest: perm_sym)
1.53
1.54
1.55 @@ -107,16 +103,16 @@
1.56  lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
1.57    by (drule_tac z = z in perm_remove_perm) auto
1.58
1.59 -lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
1.60 +lemma cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
1.61    by (blast intro: cons_perm_imp_perm)
1.62
1.63  lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
1.64    by (induct zs arbitrary: xs ys rule: rev_induct) auto
1.65
1.66 -lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
1.67 +lemma perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
1.68    by (blast intro: append_perm_imp_perm perm_append1)
1.69
1.70 -lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
1.71 +lemma perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
1.72    apply (safe intro!: perm_append2)
1.73    apply (rule append_perm_imp_perm)
1.74    apply (rule perm_append_swap [THEN perm.trans])
1.75 @@ -124,21 +120,30 @@
1.76    apply (blast intro: perm_append_swap)
1.77    done
1.78
1.79 -lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
1.80 +lemma multiset_of_eq_perm: "multiset_of xs = multiset_of ys \<longleftrightarrow> xs <~~> ys"
1.81    apply (rule iffI)
1.82 -  apply (erule_tac  perm.induct, simp_all add: union_ac)
1.83 -  apply (erule rev_mp, rule_tac x=ys in spec)
1.84 -  apply (induct_tac xs, auto)
1.85 -  apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
1.86 +  apply (erule_tac  perm.induct)
1.87 +  apply (simp_all add: union_ac)
1.88 +  apply (erule rev_mp)
1.89 +  apply (rule_tac x=ys in spec)
1.90 +  apply (induct_tac xs)
1.91 +  apply auto
1.92 +  apply (erule_tac x = "remove1 a x" in allE)
1.93 +  apply (drule sym)
1.94 +  apply simp
1.95    apply (subgoal_tac "a \<in> set x")
1.96    apply (drule_tac z = a in perm.Cons)
1.97 -  apply (erule perm.trans, rule perm_sym, erule perm_remove)
1.98 -  apply (drule_tac f=set_of in arg_cong, simp)
1.99 +  apply (erule perm.trans)
1.100 +  apply (rule perm_sym)
1.101 +  apply (erule perm_remove)
1.102 +  apply (drule_tac f=set_of in arg_cong)
1.103 +  apply simp
1.104    done
1.105
1.106  lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
1.107    apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
1.108 -  apply (insert surj_multiset_of, drule surjD)
1.109 +  apply (insert surj_multiset_of)
1.110 +  apply (drule surjD)
1.111    apply (blast intro: sym)+
1.112    done
1.113
1.114 @@ -158,15 +163,16 @@
1.115     apply simp_all
1.116    apply (subgoal_tac "a \<in> set (remdups ys)")
1.117     prefer 2 apply (metis set_simps(2) insert_iff set_remdups)
1.118 -  apply (drule split_list) apply(elim exE conjE)
1.119 -  apply (drule_tac x=list in spec) apply(erule impE) prefer 2
1.120 -   apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
1.121 +  apply (drule split_list) apply (elim exE conjE)
1.122 +  apply (drule_tac x = list in spec) apply (erule impE) prefer 2
1.123 +   apply (drule_tac x = "ysa @ zs" in spec) apply (erule impE) prefer 2
1.124      apply simp
1.125      apply (subgoal_tac "a # list <~~> a # ysa @ zs")
1.126       apply (metis Cons_eq_appendI perm_append_Cons trans)
1.127      apply (metis Cons Cons_eq_appendI distinct.simps(2)
1.128        distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
1.129 -   apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
1.130 +   apply (subgoal_tac "set (a # list) =
1.131 +      set (ysa @ a # zs) \<and> distinct (a # list) \<and> distinct (ysa @ a # zs)")
1.132      apply (fastforce simp add: insert_ident)
1.133     apply (metis distinct_remdups set_remdups)
1.134     apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
1.135 @@ -176,15 +182,17 @@
1.136     apply (rule length_remdups_leq)
1.137    done
1.138
1.139 -lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
1.140 +lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
1.141    by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
1.142
1.143  lemma permutation_Ex_bij:
1.144    assumes "xs <~~> ys"
1.145    shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
1.146 -using assms proof induct
1.147 +  using assms
1.148 +proof induct
1.149    case Nil
1.150 -  then show ?case unfolding bij_betw_def by simp
1.151 +  then show ?case
1.152 +    unfolding bij_betw_def by simp
1.153  next
1.154    case (swap y x l)
1.155    show ?case
1.156 @@ -192,14 +200,15 @@
1.157      show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
1.158        by (auto simp: bij_betw_def)
1.159      fix i
1.160 -    assume "i < length(y#x#l)"
1.161 +    assume "i < length (y # x # l)"
1.162      show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
1.163        by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
1.164    qed
1.165  next
1.166    case (Cons xs ys z)
1.167 -  then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
1.168 -    perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
1.169 +  then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
1.170 +    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
1.171 +    by blast
1.172    let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
1.173    show ?case
1.174    proof (intro exI[of _ ?f] allI conjI impI)
1.175 @@ -214,21 +223,24 @@
1.176          by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
1.177      qed (auto simp: bij_betw_def)
1.178      fix i
1.179 -    assume "i < length (z#xs)"
1.180 +    assume "i < length (z # xs)"
1.181      then show "(z # xs) ! i = (z # ys) ! (?f i)"
1.182        using perm by (cases i) auto
1.183    qed
1.184  next
1.185    case (trans xs ys zs)
1.186 -  then obtain f g where
1.187 -    bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
1.188 -    perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
1.189 +  then obtain f g
1.190 +    where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
1.191 +    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
1.192 +    by blast
1.193    show ?case
1.194    proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
1.195      show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
1.196        using bij by (rule bij_betw_trans)
1.197 -    fix i assume "i < length xs"
1.198 -    with bij have "f i < length ys" unfolding bij_betw_def by force
1.199 +    fix i
1.200 +    assume "i < length xs"
1.201 +    with bij have "f i < length ys"
1.202 +      unfolding bij_betw_def by force
1.203      with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
1.204        using trans(1,3)[THEN perm_length] perm by auto
1.205    qed
```