src/HOL/Library/Permutation.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (21 months ago) changeset 67003 49850a679c2c parent 64587 8355a6e2df79 child 69597 ff784d5a5bfb permissions -rw-r--r--
more robust sorted_entries;
```     1 (*  Title:      HOL/Library/Permutation.thy
```
```     2     Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
```
```     3 *)
```
```     4
```
```     5 section \<open>Permutations\<close>
```
```     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 proposition perm_refl [iff]: "l <~~> l"
```
```    19   by (induct l) auto
```
```    20
```
```    21
```
```    22 subsection \<open>Some examples of rule induction on permutations\<close>
```
```    23
```
```    24 proposition xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
```
```    25   by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
```
```    26
```
```    27
```
```    28 text \<open>\medskip This more general theorem is easier to understand!\<close>
```
```    29
```
```    30 proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
```
```    31   by (induct pred: perm) simp_all
```
```    32
```
```    33 proposition perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
```
```    34   by (drule perm_length) auto
```
```    35
```
```    36 proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
```
```    37   by (induct pred: perm) auto
```
```    38
```
```    39
```
```    40 subsection \<open>Ways of making new permutations\<close>
```
```    41
```
```    42 text \<open>We can insert the head anywhere in the list.\<close>
```
```    43
```
```    44 proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
```
```    45   by (induct xs) auto
```
```    46
```
```    47 proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
```
```    48   by (induct xs) (auto intro: perm_append_Cons)
```
```    49
```
```    50 proposition perm_append_single: "a # xs <~~> xs @ [a]"
```
```    51   by (rule perm.trans [OF _ perm_append_swap]) simp
```
```    52
```
```    53 proposition perm_rev: "rev xs <~~> xs"
```
```    54   by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
```
```    55
```
```    56 proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
```
```    57   by (induct l) auto
```
```    58
```
```    59 proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
```
```    60   by (blast intro!: perm_append_swap perm_append1)
```
```    61
```
```    62
```
```    63 subsection \<open>Further results\<close>
```
```    64
```
```    65 proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
```
```    66   by (blast intro: perm_empty_imp)
```
```    67
```
```    68 proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
```
```    69   apply auto
```
```    70   apply (erule perm_sym [THEN perm_empty_imp])
```
```    71   done
```
```    72
```
```    73 proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
```
```    74   by (induct pred: perm) auto
```
```    75
```
```    76 proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
```
```    77   by (blast intro: perm_sing_imp)
```
```    78
```
```    79 proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
```
```    80   by (blast dest: perm_sym)
```
```    81
```
```    82
```
```    83 subsection \<open>Removing elements\<close>
```
```    84
```
```    85 proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
```
```    86   by (induct ys) auto
```
```    87
```
```    88
```
```    89 text \<open>\medskip Congruence rule\<close>
```
```    90
```
```    91 proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
```
```    92   by (induct pred: perm) auto
```
```    93
```
```    94 proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
```
```    95   by auto
```
```    96
```
```    97 proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
```
```    98   by (drule perm_remove_perm [where z = z]) auto
```
```    99
```
```   100 proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
```
```   101   by (blast intro: cons_perm_imp_perm)
```
```   102
```
```   103 proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
```
```   104   by (induct zs arbitrary: xs ys rule: rev_induct) auto
```
```   105
```
```   106 proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
```
```   107   by (blast intro: append_perm_imp_perm perm_append1)
```
```   108
```
```   109 proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
```
```   110   apply (safe intro!: perm_append2)
```
```   111   apply (rule append_perm_imp_perm)
```
```   112   apply (rule perm_append_swap [THEN perm.trans])
```
```   113     \<comment> \<open>the previous step helps this \<open>blast\<close> call succeed quickly\<close>
```
```   114   apply (blast intro: perm_append_swap)
```
```   115   done
```
```   116
```
```   117 theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
```
```   118   apply (rule iffI)
```
```   119   apply (erule_tac  perm.induct)
```
```   120   apply (simp_all add: union_ac)
```
```   121   apply (erule rev_mp)
```
```   122   apply (rule_tac x=ys in spec)
```
```   123   apply (induct_tac xs)
```
```   124   apply auto
```
```   125   apply (erule_tac x = "remove1 a x" in allE)
```
```   126   apply (drule sym)
```
```   127   apply simp
```
```   128   apply (subgoal_tac "a \<in> set x")
```
```   129   apply (drule_tac z = a in perm.Cons)
```
```   130   apply (erule perm.trans)
```
```   131   apply (rule perm_sym)
```
```   132   apply (erule perm_remove)
```
```   133   apply (drule_tac f=set_mset in arg_cong)
```
```   134   apply simp
```
```   135   done
```
```   136
```
```   137 proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
```
```   138   apply (auto simp: mset_eq_perm[THEN sym] mset_subset_eq_exists_conv)
```
```   139   apply (insert surj_mset)
```
```   140   apply (drule surjD)
```
```   141   apply (blast intro: sym)+
```
```   142   done
```
```   143
```
```   144 proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
```
```   145   by (metis mset_eq_perm mset_eq_setD)
```
```   146
```
```   147 proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
```
```   148   apply (induct pred: perm)
```
```   149      apply simp_all
```
```   150    apply fastforce
```
```   151   apply (metis perm_set_eq)
```
```   152   done
```
```   153
```
```   154 theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
```
```   155   apply (induct xs arbitrary: ys rule: length_induct)
```
```   156   apply (case_tac "remdups xs")
```
```   157    apply simp_all
```
```   158   apply (subgoal_tac "a \<in> set (remdups ys)")
```
```   159    prefer 2 apply (metis list.set(2) insert_iff set_remdups)
```
```   160   apply (drule split_list) apply (elim exE conjE)
```
```   161   apply (drule_tac x = list in spec) apply (erule impE) prefer 2
```
```   162    apply (drule_tac x = "ysa @ zs" in spec) apply (erule impE) prefer 2
```
```   163     apply simp
```
```   164     apply (subgoal_tac "a # list <~~> a # ysa @ zs")
```
```   165      apply (metis Cons_eq_appendI perm_append_Cons trans)
```
```   166     apply (metis Cons Cons_eq_appendI distinct.simps(2)
```
```   167       distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
```
```   168    apply (subgoal_tac "set (a # list) =
```
```   169       set (ysa @ a # zs) \<and> distinct (a # list) \<and> 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 proposition 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 theorem 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
```
```   186 proof induct
```
```   187   case Nil
```
```   188   then show ?case
```
```   189     unfolding bij_betw_def by simp
```
```   190 next
```
```   191   case (swap y x l)
```
```   192   show ?case
```
```   193   proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
```
```   194     show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
```
```   195       by (auto simp: bij_betw_def)
```
```   196     fix i
```
```   197     assume "i < length (y # x # l)"
```
```   198     show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
```
```   199       by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
```
```   200   qed
```
```   201 next
```
```   202   case (Cons xs ys z)
```
```   203   then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
```
```   204     and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
```
```   205     by blast
```
```   206   let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
```
```   207   show ?case
```
```   208   proof (intro exI[of _ ?f] allI conjI impI)
```
```   209     have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
```
```   210             "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
```
```   211       by (simp_all add: lessThan_Suc_eq_insert_0)
```
```   212     show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
```
```   213       unfolding *
```
```   214     proof (rule bij_betw_combine)
```
```   215       show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
```
```   216         using bij unfolding bij_betw_def
```
```   217         by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
```
```   218     qed (auto simp: bij_betw_def)
```
```   219     fix i
```
```   220     assume "i < length (z # xs)"
```
```   221     then show "(z # xs) ! i = (z # ys) ! (?f i)"
```
```   222       using perm by (cases i) auto
```
```   223   qed
```
```   224 next
```
```   225   case (trans xs ys zs)
```
```   226   then obtain f g
```
```   227     where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
```
```   228     and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
```
```   229     by blast
```
```   230   show ?case
```
```   231   proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
```
```   232     show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
```
```   233       using bij by (rule bij_betw_trans)
```
```   234     fix i
```
```   235     assume "i < length xs"
```
```   236     with bij have "f i < length ys"
```
```   237       unfolding bij_betw_def by force
```
```   238     with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i"
```
```   239       using trans(1,3)[THEN perm_length] perm by auto
```
```   240   qed
```
```   241 qed
```
```   242
```
```   243 proposition perm_finite: "finite {B. B <~~> A}"
```
```   244 proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
```
```   245  show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
```
```   246    apply (cases A, simp)
```
```   247    apply (rule card_ge_0_finite)
```
```   248    apply (auto simp: card_lists_length_le)
```
```   249    done
```
```   250 next
```
```   251  show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
```
```   252    by (clarsimp simp add: perm_length perm_set_eq)
```
```   253 qed
```
```   254
```
```   255 proposition perm_swap:
```
```   256     assumes "i < length xs" "j < length xs"
```
```   257     shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
```
```   258   using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
```
```   259
```
```   260 end
```