src/HOL/Library/Permutation.thy
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(*  Title:      HOL/Library/Permutation.thy
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    Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
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*)
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section \<open>Permutations\<close>
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theory Permutation
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imports Multiset
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begin
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inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"  ("_ <~~> _"  [50, 50] 50)  (* FIXME proper infix, without ambiguity!? *)
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where
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  Nil [intro!]: "[] <~~> []"
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| swap [intro!]: "y # x # l <~~> x # y # l"
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| Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys"
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| trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs"
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proposition perm_refl [iff]: "l <~~> l"
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  by (induct l) auto
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subsection \<open>Some examples of rule induction on permutations\<close>
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proposition xperm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []"
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  by (induct xs == "[] :: 'a list" ys pred: perm) simp_all
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text \<open>\medskip This more general theorem is easier to understand!\<close>
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proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys"
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  by (induct pred: perm) simp_all
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proposition perm_empty_imp: "[] <~~> xs \<Longrightarrow> xs = []"
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  by (drule perm_length) auto
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proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs"
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  by (induct pred: perm) auto
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subsection \<open>Ways of making new permutations\<close>
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text \<open>We can insert the head anywhere in the list.\<close>
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proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
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  by (induct xs) auto
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proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
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  by (induct xs) (auto intro: perm_append_Cons)
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proposition perm_append_single: "a # xs <~~> xs @ [a]"
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  by (rule perm.trans [OF _ perm_append_swap]) simp
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proposition perm_rev: "rev xs <~~> xs"
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  by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
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proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys"
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  by (induct l) auto
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proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l"
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  by (blast intro!: perm_append_swap perm_append1)
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subsection \<open>Further results\<close>
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proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []"
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  by (blast intro: perm_empty_imp)
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proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []"
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  apply auto
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  apply (erule perm_sym [THEN perm_empty_imp])
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  done
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proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]"
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  by (induct pred: perm) auto
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proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]"
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  by (blast intro: perm_sing_imp)
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proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]"
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  by (blast dest: perm_sym)
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subsection \<open>Removing elements\<close>
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proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys"
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  by (induct ys) auto
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text \<open>\medskip Congruence rule\<close>
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proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys"
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  by (induct pred: perm) auto
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proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
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  by auto
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proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys"
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  by (drule perm_remove_perm [where z = z]) auto
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proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys"
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  by (blast intro: cons_perm_imp_perm)
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proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys"
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  by (induct zs arbitrary: xs ys rule: rev_induct) auto
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proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys"
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  by (blast intro: append_perm_imp_perm perm_append1)
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proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys"
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  apply (safe intro!: perm_append2)
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  apply (rule append_perm_imp_perm)
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  apply (rule perm_append_swap [THEN perm.trans])
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    \<comment> \<open>the previous step helps this \<open>blast\<close> call succeed quickly\<close>
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  apply (blast intro: perm_append_swap)
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  done
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theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys"
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  apply (rule iffI)
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  apply (erule_tac [2] perm.induct)
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  apply (simp_all add: union_ac)
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  apply (erule rev_mp)
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  apply (rule_tac x=ys in spec)
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  apply (induct_tac xs)
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  apply auto
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  apply (erule_tac x = "remove1 a x" in allE)
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  apply (drule sym)
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  apply simp
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  apply (subgoal_tac "a \<in> set x")
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  apply (drule_tac z = a in perm.Cons)
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  apply (erule perm.trans)
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  apply (rule perm_sym)
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  apply (erule perm_remove)
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  apply (drule_tac f=set_mset in arg_cong)
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  apply simp
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  done
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proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"
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  apply (auto simp: mset_eq_perm[THEN sym] mset_subset_eq_exists_conv)
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  apply (insert surj_mset)
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  apply (drule surjD)
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  apply (blast intro: sym)+
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  done
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proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys"
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  by (metis mset_eq_perm mset_eq_setD)
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proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys"
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  apply (induct pred: perm)
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     apply simp_all
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   apply fastforce
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diff changeset
   151
  apply (metis perm_set_eq)
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   152
  done
25277
95128fcdd7e8 added lemmas
nipkow
parents: 23755
diff changeset
   153
61699
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   154
theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   155
  apply (induct xs arbitrary: ys rule: length_induct)
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   156
  apply (case_tac "remdups xs")
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   157
   apply simp_all
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   158
  apply (subgoal_tac "a \<in> set (remdups ys)")
57816
d8bbb97689d3 no need for 'set_simps' now that 'datatype_new' generates the desired 'set' property
blanchet
parents: 56796
diff changeset
   159
   prefer 2 apply (metis list.set(2) insert_iff set_remdups)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   160
  apply (drule split_list) apply (elim exE conjE)
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   161
  apply (drule_tac x = list in spec) apply (erule impE) prefer 2
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   162
   apply (drule_tac x = "ysa @ zs" in spec) apply (erule impE) prefer 2
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   163
    apply simp
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   164
    apply (subgoal_tac "a # list <~~> a # ysa @ zs")
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   165
     apply (metis Cons_eq_appendI perm_append_Cons trans)
40122
1d8ad2ff3e01 dropped (almost) redundant distinct.induct rule; distinct_simps again named distinct.simps
haftmann
parents: 39916
diff changeset
   166
    apply (metis Cons Cons_eq_appendI distinct.simps(2)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   167
      distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   168
   apply (subgoal_tac "set (a # list) =
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   169
      set (ysa @ a # zs) \<and> distinct (a # list) \<and> distinct (ysa @ a # zs)")
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 40122
diff changeset
   170
    apply (fastforce simp add: insert_ident)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   171
   apply (metis distinct_remdups set_remdups)
30742
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   172
   apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   173
   apply simp
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   174
   apply (subgoal_tac "length (remdups xs) \<le> length xs")
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   175
   apply simp
3e89ac3905b9 tuned notoriously slow metis proof
haftmann
parents: 30738
diff changeset
   176
   apply (rule length_remdups_leq)
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   177
  done
25287
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   178
61699
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   179
proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y"
25379
12bcf37252b1 tuned proofs;
wenzelm
parents: 25287
diff changeset
   180
  by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)
25287
094dab519ff5 added lemmas
nipkow
parents: 25277
diff changeset
   181
61699
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   182
theorem permutation_Ex_bij:
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   183
  assumes "xs <~~> ys"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   184
  shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   185
  using assms
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   186
proof induct
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   187
  case Nil
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   188
  then show ?case
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   189
    unfolding bij_betw_def by simp
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   190
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   191
  case (swap y x l)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   192
  show ?case
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   193
  proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   194
    show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"
50037
f2a32197a33a tuned proofs
bulwahn
parents: 44890
diff changeset
   195
      by (auto simp: bij_betw_def)
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   196
    fix i
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   197
    assume "i < length (y # x # l)"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   198
    show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   199
      by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   200
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   201
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   202
  case (Cons xs ys z)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   203
  then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   204
    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   205
    by blast
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   206
  let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   207
  show ?case
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   208
  proof (intro exI[of _ ?f] allI conjI impI)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   209
    have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   210
            "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"
39078
39f8f6d1eb74 Fixes lemma names
hoelzl
parents: 39075
diff changeset
   211
      by (simp_all add: lessThan_Suc_eq_insert_0)
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   212
    show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   213
      unfolding *
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   214
    proof (rule bij_betw_combine)
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   215
      show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   216
        using bij unfolding bij_betw_def
56154
f0a927235162 more complete set of lemmas wrt. image and composition
haftmann
parents: 55584
diff changeset
   217
        by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def)
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   218
    qed (auto simp: bij_betw_def)
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   219
    fix i
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   220
    assume "i < length (z # xs)"
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   221
    then show "(z # xs) ! i = (z # ys) ! (?f i)"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   222
      using perm by (cases i) auto
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   223
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   224
next
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   225
  case (trans xs ys zs)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   226
  then obtain f g
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   227
    where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   228
    and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   229
    by blast
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   230
  show ?case
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   231
  proof (intro exI[of _ "g \<circ> f"] conjI allI impI)
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   232
    show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   233
      using bij by (rule bij_betw_trans)
56796
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   234
    fix i
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   235
    assume "i < length xs"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   236
    with bij have "f i < length ys"
9f84219715a7 tuned proofs;
wenzelm
parents: 56154
diff changeset
   237
      unfolding bij_betw_def by force
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 60397
diff changeset
   238
    with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i"
53238
01ef0a103fc9 tuned proofs;
wenzelm
parents: 51542
diff changeset
   239
      using trans(1,3)[THEN perm_length] perm by auto
39075
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   240
  qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   241
qed
a18e5946d63c Permutation implies bij function
hoelzl
parents: 36903
diff changeset
   242
61699
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   243
proposition perm_finite: "finite {B. B <~~> A}"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   244
proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"])
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   245
 show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   246
   apply (cases A, simp)
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   247
   apply (rule card_ge_0_finite)
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   248
   apply (auto simp: card_lists_length_le)
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   249
   done
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   250
next
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   251
 show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   252
   by (clarsimp simp add: perm_length perm_set_eq)
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   253
qed
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   254
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   255
proposition perm_swap:
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   256
    assumes "i < length xs" "j < length xs"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   257
    shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   258
  using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
a81dc5c4d6a9 New theorems mostly from Peter Gammie
paulson <lp15@cam.ac.uk>
parents: 61585
diff changeset
   259
11054
a5404c70982f moved from Induct/ to Library/
wenzelm
parents:
diff changeset
   260
end