src/HOL/Library/Permutation.thy
author wenzelm
Tue Aug 27 22:40:39 2013 +0200 (2013-08-27)
changeset 53238 01ef0a103fc9
parent 51542 738598beeb26
child 55584 a879f14b6f95
permissions -rw-r--r--
tuned proofs;
     1 (*  Title:      HOL/Library/Permutation.thy
     2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
     3 *)
     4 
     5 header {* Permutations *}
     6 
     7 theory Permutation
     8 imports Multiset
     9 begin
    10 
    11 inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)
    12 where
    13   Nil [intro!]: "[] <~~> []"
    14 | swap [intro!]: "y # x # l <~~> x # y # l"
    15 | Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
    16 | trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
    17 
    18 lemma perm_refl [iff]: "l <~~> l"
    19   by (induct l) auto
    20 
    21 
    22 subsection {* Some examples of rule induction on permutations *}
    23 
    24 lemma xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
    25   by (induct xs == "[]::'a list" ys pred: perm) simp_all
    26 
    27 
    28 text {*
    29   \medskip This more general theorem is easier to understand!
    30   *}
    31 
    32 lemma perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
    33   by (induct pred: perm) simp_all
    34 
    35 lemma perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
    36   by (drule perm_length) auto
    37 
    38 lemma perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
    39   by (induct pred: perm) auto
    40 
    41 
    42 subsection {* Ways of making new permutations *}
    43 
    44 text {*
    45   We can insert the head anywhere in the list.
    46 *}
    47 
    48 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
    49   by (induct xs) auto
    50 
    51 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"
    52   apply (induct xs)
    53     apply simp_all
    54   apply (blast intro: perm_append_Cons)
    55   done
    56 
    57 lemma perm_append_single: "a # xs <~~> xs @ [a]"
    58   by (rule perm.trans [OF _ perm_append_swap]) simp
    59 
    60 lemma perm_rev: "rev xs <~~> xs"
    61   apply (induct xs)
    62    apply simp_all
    63   apply (blast intro!: perm_append_single intro: perm_sym)
    64   done
    65 
    66 lemma perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
    67   by (induct l) auto
    68 
    69 lemma perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
    70   by (blast intro!: perm_append_swap perm_append1)
    71 
    72 
    73 subsection {* Further results *}
    74 
    75 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"
    76   by (blast intro: perm_empty_imp)
    77 
    78 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"
    79   apply auto
    80   apply (erule perm_sym [THEN perm_empty_imp])
    81   done
    82 
    83 lemma perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
    84   by (induct pred: perm) auto
    85 
    86 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"
    87   by (blast intro: perm_sing_imp)
    88 
    89 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"
    90   by (blast dest: perm_sym)
    91 
    92 
    93 subsection {* Removing elements *}
    94 
    95 lemma perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
    96   by (induct ys) auto
    97 
    98 
    99 text {* \medskip Congruence rule *}
   100 
   101 lemma perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
   102   by (induct pred: perm) auto
   103 
   104 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"
   105   by auto
   106 
   107 lemma cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
   108   by (drule_tac z = z in perm_remove_perm) auto
   109 
   110 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"
   111   by (blast intro: cons_perm_imp_perm)
   112 
   113 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
   114   by (induct zs arbitrary: xs ys rule: rev_induct) auto
   115 
   116 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
   117   by (blast intro: append_perm_imp_perm perm_append1)
   118 
   119 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"
   120   apply (safe intro!: perm_append2)
   121   apply (rule append_perm_imp_perm)
   122   apply (rule perm_append_swap [THEN perm.trans])
   123     -- {* the previous step helps this @{text blast} call succeed quickly *}
   124   apply (blast intro: perm_append_swap)
   125   done
   126 
   127 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "
   128   apply (rule iffI)
   129   apply (erule_tac [2] perm.induct, simp_all add: union_ac)
   130   apply (erule rev_mp, rule_tac x=ys in spec)
   131   apply (induct_tac xs, auto)
   132   apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)
   133   apply (subgoal_tac "a \<in> set x")
   134   apply (drule_tac z = a in perm.Cons)
   135   apply (erule perm.trans, rule perm_sym, erule perm_remove)
   136   apply (drule_tac f=set_of in arg_cong, simp)
   137   done
   138 
   139 lemma multiset_of_le_perm_append: "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
   140   apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)
   141   apply (insert surj_multiset_of, drule surjD)
   142   apply (blast intro: sym)+
   143   done
   144 
   145 lemma perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
   146   by (metis multiset_of_eq_perm multiset_of_eq_setD)
   147 
   148 lemma perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
   149   apply (induct pred: perm)
   150      apply simp_all
   151    apply fastforce
   152   apply (metis perm_set_eq)
   153   done
   154 
   155 lemma eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
   156   apply (induct xs arbitrary: ys rule: length_induct)
   157   apply (case_tac "remdups xs")
   158    apply simp_all
   159   apply (subgoal_tac "a \<in> set (remdups ys)")
   160    prefer 2 apply (metis set.simps(2) insert_iff set_remdups)
   161   apply (drule split_list) apply(elim exE conjE)
   162   apply (drule_tac x=list in spec) apply(erule impE) prefer 2
   163    apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2
   164     apply simp
   165     apply (subgoal_tac "a # list <~~> a # ysa @ zs")
   166      apply (metis Cons_eq_appendI perm_append_Cons trans)
   167     apply (metis Cons Cons_eq_appendI distinct.simps(2)
   168       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
   169    apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")
   170     apply (fastforce simp add: insert_ident)
   171    apply (metis distinct_remdups set_remdups)
   172    apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
   173    apply simp
   174    apply (subgoal_tac "length (remdups xs) \<le> length xs")
   175    apply simp
   176    apply (rule length_remdups_leq)
   177   done
   178 
   179 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> (set x = set y)"
   180   by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
   181 
   182 lemma permutation_Ex_bij:
   183   assumes "xs <~~> ys"
   184   shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
   185 using assms proof induct
   186   case Nil
   187   then show ?case unfolding bij_betw_def by simp
   188 next
   189   case (swap y x l)
   190   show ?case
   191   proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
   192     show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
   193       by (auto simp: bij_betw_def)
   194     fix i
   195     assume "i < length(y#x#l)"
   196     show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
   197       by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
   198   qed
   199 next
   200   case (Cons xs ys z)
   201   then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and
   202     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast
   203   let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
   204   show ?case
   205   proof (intro exI[of _ ?f] allI conjI impI)
   206     have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
   207             "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
   208       by (simp_all add: lessThan_Suc_eq_insert_0)
   209     show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
   210       unfolding *
   211     proof (rule bij_betw_combine)
   212       show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
   213         using bij unfolding bij_betw_def
   214         by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)
   215     qed (auto simp: bij_betw_def)
   216     fix i
   217     assume "i < length (z#xs)"
   218     then show "(z # xs) ! i = (z # ys) ! (?f i)"
   219       using perm by (cases i) auto
   220   qed
   221 next
   222   case (trans xs ys zs)
   223   then obtain f g where
   224     bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and
   225     perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast
   226   show ?case
   227   proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
   228     show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
   229       using bij by (rule bij_betw_trans)
   230     fix i assume "i < length xs"
   231     with bij have "f i < length ys" unfolding bij_betw_def by force
   232     with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"
   233       using trans(1,3)[THEN perm_length] perm by auto
   234   qed
   235 qed
   236 
   237 end