tuned proofs;
authorwenzelm
Tue Aug 27 22:40:39 2013 +0200 (2013-08-27)
changeset 5323801ef0a103fc9
parent 53237 6bfe54791591
child 53239 2f21813cf2f0
tuned proofs;
src/HOL/Library/Permutation.thy
     1.1 --- a/src/HOL/Library/Permutation.thy	Tue Aug 27 22:23:40 2013 +0200
     1.2 +++ b/src/HOL/Library/Permutation.thy	Tue Aug 27 22:40:39 2013 +0200
     1.3 @@ -8,13 +8,12 @@
     1.4  imports Multiset
     1.5  begin
     1.6  
     1.7 -inductive
     1.8 -  perm :: "'a list => 'a list => bool"  ("_ <~~> _"  [50, 50] 50)
     1.9 -  where
    1.10 -    Nil  [intro!]: "[] <~~> []"
    1.11 -  | swap [intro!]: "y # x # l <~~> x # y # l"
    1.12 -  | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"
    1.13 -  | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"
    1.14 +inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)
    1.15 +where
    1.16 +  Nil [intro!]: "[] <~~> []"
    1.17 +| swap [intro!]: "y # x # l <~~> x # y # l"
    1.18 +| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
    1.19 +| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
    1.20  
    1.21  lemma perm_refl [iff]: "l <~~> l"
    1.22    by (induct l) auto
    1.23 @@ -22,7 +21,7 @@
    1.24  
    1.25  subsection {* Some examples of rule induction on permutations *}
    1.26  
    1.27 -lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"
    1.28 +lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
    1.29    by (induct xs == "[]::'a list" ys pred: perm) simp_all
    1.30  
    1.31  
    1.32 @@ -30,13 +29,13 @@
    1.33    \medskip This more general theorem is easier to understand!
    1.34    *}
    1.35  
    1.36 -lemma perm_length: "xs <~~> ys ==> length xs = length ys"
    1.37 +lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
    1.38    by (induct pred: perm) simp_all
    1.39  
    1.40 -lemma perm_empty_imp: "[] <~~> xs ==> xs = []"
    1.41 +lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
    1.42    by (drule perm_length) auto
    1.43  
    1.44 -lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"
    1.45 +lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
    1.46    by (induct pred: perm) auto
    1.47  
    1.48  
    1.49 @@ -64,10 +63,10 @@
    1.50    apply (blast intro!: perm_append_single intro: perm_sym)
    1.51    done
    1.52  
    1.53 -lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"
    1.54 +lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
    1.55    by (induct l) auto
    1.56  
    1.57 -lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"
    1.58 +lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
    1.59    by (blast intro!: perm_append_swap perm_append1)
    1.60  
    1.61  
    1.62 @@ -81,7 +80,7 @@
    1.63    apply (erule perm_sym [THEN perm_empty_imp])
    1.64    done
    1.65  
    1.66 -lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"
    1.67 +lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
    1.68    by (induct pred: perm) auto
    1.69  
    1.70  lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
    1.71 @@ -93,29 +92,26 @@
    1.72  
    1.73  subsection {* Removing elements *}
    1.74  
    1.75 -lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"
    1.76 +lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
    1.77    by (induct ys) auto
    1.78  
    1.79  
    1.80  text {* \medskip Congruence rule *}
    1.81  
    1.82 -lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"
    1.83 +lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
    1.84    by (induct pred: perm) auto
    1.85  
    1.86  lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
    1.87    by auto
    1.88  
    1.89 -lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"
    1.90 +lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
    1.91    by (drule_tac z = z in perm_remove_perm) auto
    1.92  
    1.93  lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
    1.94    by (blast intro: cons_perm_imp_perm)
    1.95  
    1.96 -lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"
    1.97 -  apply (induct zs arbitrary: xs ys rule: rev_induct)
    1.98 -   apply (simp_all (no_asm_use))
    1.99 -  apply blast
   1.100 -  done
   1.101 +lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
   1.102 +  by (induct zs arbitrary: xs ys rule: rev_induct) auto
   1.103  
   1.104  lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   1.105    by (blast intro: append_perm_imp_perm perm_append1)
   1.106 @@ -135,38 +131,38 @@
   1.107    apply (induct_tac xs, auto)
   1.108    apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
   1.109    apply (subgoal_tac "a \<in> set x")
   1.110 -  apply (drule_tac z=a in perm.Cons)
   1.111 +  apply (drule_tac z = a in perm.Cons)
   1.112    apply (erule perm.trans, rule perm_sym, erule perm_remove)
   1.113    apply (drule_tac f=set_of in arg_cong, simp)
   1.114    done
   1.115  
   1.116 -lemma multiset_of_le_perm_append:
   1.117 -    "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   1.118 +lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   1.119    apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   1.120    apply (insert surj_multiset_of, drule surjD)
   1.121    apply (blast intro: sym)+
   1.122    done
   1.123  
   1.124 -lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"
   1.125 +lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
   1.126    by (metis multiset_of_eq_perm multiset_of_eq_setD)
   1.127  
   1.128 -lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"
   1.129 +lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
   1.130    apply (induct pred: perm)
   1.131       apply simp_all
   1.132     apply fastforce
   1.133    apply (metis perm_set_eq)
   1.134    done
   1.135  
   1.136 -lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"
   1.137 +lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
   1.138    apply (induct xs arbitrary: ys rule: length_induct)
   1.139 -  apply (case_tac "remdups xs", simp, simp)
   1.140 -  apply (subgoal_tac "a : set (remdups ys)")
   1.141 +  apply (case_tac "remdups xs")
   1.142 +   apply simp_all
   1.143 +  apply (subgoal_tac "a \<in> set (remdups ys)")
   1.144     prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
   1.145    apply (drule split_list) apply(elim exE conjE)
   1.146    apply (drule_tac x=list in spec) apply(erule impE) prefer 2
   1.147     apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
   1.148      apply simp
   1.149 -    apply (subgoal_tac "a#list <~~> a#ysa@zs")
   1.150 +    apply (subgoal_tac "a # list <~~> a # ysa @ zs")
   1.151       apply (metis Cons_eq_appendI perm_append_Cons trans)
   1.152      apply (metis Cons Cons_eq_appendI distinct.simps(2)
   1.153        distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
   1.154 @@ -180,21 +176,23 @@
   1.155     apply (rule length_remdups_leq)
   1.156    done
   1.157  
   1.158 -lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"
   1.159 +lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
   1.160    by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
   1.161  
   1.162  lemma permutation_Ex_bij:
   1.163    assumes "xs <~~> ys"
   1.164    shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
   1.165  using assms proof induct
   1.166 -  case Nil then show ?case unfolding bij_betw_def by simp
   1.167 +  case Nil
   1.168 +  then show ?case unfolding bij_betw_def by simp
   1.169  next
   1.170    case (swap y x l)
   1.171    show ?case
   1.172    proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
   1.173      show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
   1.174        by (auto simp: bij_betw_def)
   1.175 -    fix i assume "i < length(y#x#l)"
   1.176 +    fix i
   1.177 +    assume "i < length(y#x#l)"
   1.178      show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
   1.179        by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
   1.180    qed
   1.181 @@ -202,19 +200,21 @@
   1.182    case (Cons xs ys z)
   1.183    then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
   1.184      perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
   1.185 -  let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
   1.186 +  let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
   1.187    show ?case
   1.188    proof (intro exI[of _ ?f] allI conjI impI)
   1.189      have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
   1.190              "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
   1.191        by (simp_all add: lessThan_Suc_eq_insert_0)
   1.192 -    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *
   1.193 +    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
   1.194 +      unfolding *
   1.195      proof (rule bij_betw_combine)
   1.196        show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
   1.197          using bij unfolding bij_betw_def
   1.198          by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
   1.199      qed (auto simp: bij_betw_def)
   1.200 -    fix i assume "i < length (z#xs)"
   1.201 +    fix i
   1.202 +    assume "i < length (z#xs)"
   1.203      then show "(z # xs) ! i = (z # ys) ! (?f i)"
   1.204        using perm by (cases i) auto
   1.205    qed
   1.206 @@ -224,13 +224,13 @@
   1.207      bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
   1.208      perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
   1.209    show ?case
   1.210 -  proof (intro exI[of _ "g\<circ>f"] conjI allI impI)
   1.211 +  proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
   1.212      show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
   1.213        using bij by (rule bij_betw_trans)
   1.214      fix i assume "i < length xs"
   1.215      with bij have "f i < length ys" unfolding bij_betw_def by force
   1.216      with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
   1.217 -      using trans(1,3)[THEN perm_length] perm by force
   1.218 +      using trans(1,3)[THEN perm_length] perm by auto
   1.219    qed
   1.220  qed
   1.221