src/HOL/Library/Permutations.thy
 author haftmann Wed Jul 18 20:51:21 2018 +0200 (11 months ago) changeset 68658 16cc1161ad7f parent 67673 c8caefb20564 child 69895 6b03a8cf092d permissions -rw-r--r--
tuned equation
```     1 (*  Title:      HOL/Library/Permutations.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3 *)
```
```     4
```
```     5 section \<open>Permutations, both general and specifically on finite sets.\<close>
```
```     6
```
```     7 theory Permutations
```
```     8   imports Multiset Disjoint_Sets
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Transpositions\<close>
```
```    12
```
```    13 lemma swap_id_idempotent [simp]: "Fun.swap a b id \<circ> Fun.swap a b id = id"
```
```    14   by (rule ext) (auto simp add: Fun.swap_def)
```
```    15
```
```    16 lemma inv_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id"
```
```    17   by (rule inv_unique_comp) simp_all
```
```    18
```
```    19 lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
```
```    20   by (simp add: Fun.swap_def)
```
```    21
```
```    22 lemma bij_swap_comp:
```
```    23   assumes "bij p"
```
```    24   shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
```
```    25   using surj_f_inv_f[OF bij_is_surj[OF \<open>bij p\<close>]]
```
```    26   by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF \<open>bij p\<close>])
```
```    27
```
```    28 lemma bij_swap_compose_bij:
```
```    29   assumes "bij p"
```
```    30   shows "bij (Fun.swap a b id \<circ> p)"
```
```    31   by (simp only: bij_swap_comp[OF \<open>bij p\<close>] bij_swap_iff \<open>bij p\<close>)
```
```    32
```
```    33
```
```    34 subsection \<open>Basic consequences of the definition\<close>
```
```    35
```
```    36 definition permutes  (infixr "permutes" 41)
```
```    37   where "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"
```
```    38
```
```    39 lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
```
```    40   unfolding permutes_def by metis
```
```    41
```
```    42 lemma permutes_not_in: "f permutes S \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = x"
```
```    43   by (auto simp: permutes_def)
```
```    44
```
```    45 lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S"
```
```    46   unfolding permutes_def
```
```    47   apply (rule set_eqI)
```
```    48   apply (simp add: image_iff)
```
```    49   apply metis
```
```    50   done
```
```    51
```
```    52 lemma permutes_inj: "p permutes S \<Longrightarrow> inj p"
```
```    53   unfolding permutes_def inj_def by blast
```
```    54
```
```    55 lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A"
```
```    56   by (auto simp: permutes_def inj_on_def)
```
```    57
```
```    58 lemma permutes_surj: "p permutes s \<Longrightarrow> surj p"
```
```    59   unfolding permutes_def surj_def by metis
```
```    60
```
```    61 lemma permutes_bij: "p permutes s \<Longrightarrow> bij p"
```
```    62   unfolding bij_def by (metis permutes_inj permutes_surj)
```
```    63
```
```    64 lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S"
```
```    65   by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI)
```
```    66
```
```    67 lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S"
```
```    68   unfolding permutes_def bij_betw_def inj_on_def
```
```    69   by auto (metis image_iff)+
```
```    70
```
```    71 lemma permutes_inv_o:
```
```    72   assumes permutes: "p permutes S"
```
```    73   shows "p \<circ> inv p = id"
```
```    74     and "inv p \<circ> p = id"
```
```    75   using permutes_inj[OF permutes] permutes_surj[OF permutes]
```
```    76   unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
```
```    77
```
```    78 lemma permutes_inverses:
```
```    79   fixes p :: "'a \<Rightarrow> 'a"
```
```    80   assumes permutes: "p permutes S"
```
```    81   shows "p (inv p x) = x"
```
```    82     and "inv p (p x) = x"
```
```    83   using permutes_inv_o[OF permutes, unfolded fun_eq_iff o_def] by auto
```
```    84
```
```    85 lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T"
```
```    86   unfolding permutes_def by blast
```
```    87
```
```    88 lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
```
```    89   by (auto simp add: fun_eq_iff permutes_def)
```
```    90
```
```    91 lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
```
```    92   by (simp add: fun_eq_iff permutes_def) metis  (*somewhat slow*)
```
```    93
```
```    94 lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
```
```    95   by (simp add: permutes_def)
```
```    96
```
```    97 lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y"
```
```    98   unfolding permutes_def inv_def
```
```    99   apply auto
```
```   100   apply (erule allE[where x=y])
```
```   101   apply (erule allE[where x=y])
```
```   102   apply (rule someI_ex)
```
```   103   apply blast
```
```   104   apply (rule some1_equality)
```
```   105   apply blast
```
```   106   apply blast
```
```   107   done
```
```   108
```
```   109 lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S"
```
```   110   unfolding permutes_def Fun.swap_def fun_upd_def by auto metis
```
```   111
```
```   112 lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
```
```   113   by (simp add: Ball_def permutes_def) metis
```
```   114
```
```   115 (* Next three lemmas contributed by Lukas Bulwahn *)
```
```   116 lemma permutes_bij_inv_into:
```
```   117   fixes A :: "'a set"
```
```   118     and B :: "'b set"
```
```   119   assumes "p permutes A"
```
```   120     and "bij_betw f A B"
```
```   121   shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
```
```   122 proof (rule bij_imp_permutes)
```
```   123   from assms have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
```
```   124     by (auto simp add: permutes_imp_bij bij_betw_inv_into)
```
```   125   then have "bij_betw (f \<circ> p \<circ> inv_into A f) B B"
```
```   126     by (simp add: bij_betw_trans)
```
```   127   then show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B"
```
```   128     by (subst bij_betw_cong[where g="f \<circ> p \<circ> inv_into A f"]) auto
```
```   129 next
```
```   130   fix x
```
```   131   assume "x \<notin> B"
```
```   132   then show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto
```
```   133 qed
```
```   134
```
```   135 lemma permutes_image_mset:
```
```   136   assumes "p permutes A"
```
```   137   shows "image_mset p (mset_set A) = mset_set A"
```
```   138   using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image)
```
```   139
```
```   140 lemma permutes_implies_image_mset_eq:
```
```   141   assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)"
```
```   142   shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
```
```   143 proof -
```
```   144   have "f x = f' (p x)" if "x \<in># mset_set A" for x
```
```   145     using assms(2)[of x] that by (cases "finite A") auto
```
```   146   with assms have "image_mset f (mset_set A) = image_mset (f' \<circ> p) (mset_set A)"
```
```   147     by (auto intro!: image_mset_cong)
```
```   148   also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
```
```   149     by (simp add: image_mset.compositionality)
```
```   150   also have "\<dots> = image_mset f' (mset_set A)"
```
```   151   proof -
```
```   152     from assms permutes_image_mset have "image_mset p (mset_set A) = mset_set A"
```
```   153       by blast
```
```   154     then show ?thesis by simp
```
```   155   qed
```
```   156   finally show ?thesis ..
```
```   157 qed
```
```   158
```
```   159
```
```   160 subsection \<open>Group properties\<close>
```
```   161
```
```   162 lemma permutes_id: "id permutes S"
```
```   163   by (simp add: permutes_def)
```
```   164
```
```   165 lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S"
```
```   166   unfolding permutes_def o_def by metis
```
```   167
```
```   168 lemma permutes_inv:
```
```   169   assumes "p permutes S"
```
```   170   shows "inv p permutes S"
```
```   171   using assms unfolding permutes_def permutes_inv_eq[OF assms] by metis
```
```   172
```
```   173 lemma permutes_inv_inv:
```
```   174   assumes "p permutes S"
```
```   175   shows "inv (inv p) = p"
```
```   176   unfolding fun_eq_iff permutes_inv_eq[OF assms] permutes_inv_eq[OF permutes_inv[OF assms]]
```
```   177   by blast
```
```   178
```
```   179 lemma permutes_invI:
```
```   180   assumes perm: "p permutes S"
```
```   181     and inv: "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
```
```   182     and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
```
```   183   shows "inv p = p'"
```
```   184 proof
```
```   185   show "inv p x = p' x" for x
```
```   186   proof (cases "x \<in> S")
```
```   187     case True
```
```   188     from assms have "p' x = p' (p (inv p x))"
```
```   189       by (simp add: permutes_inverses)
```
```   190     also from permutes_inv[OF perm] True have "\<dots> = inv p x"
```
```   191       by (subst inv) (simp_all add: permutes_in_image)
```
```   192     finally show ?thesis ..
```
```   193   next
```
```   194     case False
```
```   195     with permutes_inv[OF perm] show ?thesis
```
```   196       by (simp_all add: outside permutes_not_in)
```
```   197   qed
```
```   198 qed
```
```   199
```
```   200 lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
```
```   201   by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
```
```   202
```
```   203
```
```   204 subsection \<open>Mapping permutations with bijections\<close>
```
```   205
```
```   206 lemma bij_betw_permutations:
```
```   207   assumes "bij_betw f A B"
```
```   208   shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x)
```
```   209              {\<pi>. \<pi> permutes A} {\<pi>. \<pi> permutes B}" (is "bij_betw ?f _ _")
```
```   210 proof -
```
```   211   let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
```
```   212   show ?thesis
```
```   213   proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
```
```   214     case 3
```
```   215     show ?case using permutes_bij_inv_into[OF _ assms] by auto
```
```   216   next
```
```   217     case 4
```
```   218     have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
```
```   219     {
```
```   220       fix \<pi> assume "\<pi> permutes B"
```
```   221       from permutes_bij_inv_into[OF this bij_inv] and assms
```
```   222         have "(\<lambda>x. if x \<in> A then inv_into A f (\<pi> (f x)) else x) permutes A"
```
```   223         by (simp add: inv_into_inv_into_eq cong: if_cong)
```
```   224     }
```
```   225     from this show ?case by (auto simp: permutes_inv)
```
```   226   next
```
```   227     case 1
```
```   228     thus ?case using assms
```
```   229       by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
```
```   230                dest: bij_betwE)
```
```   231   next
```
```   232     case 2
```
```   233     moreover have "bij_betw (inv_into A f) B A"
```
```   234       by (intro bij_betw_inv_into assms)
```
```   235     ultimately show ?case using assms
```
```   236       by (auto simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right
```
```   237                dest: bij_betwE)
```
```   238   qed
```
```   239 qed
```
```   240
```
```   241 lemma bij_betw_derangements:
```
```   242   assumes "bij_betw f A B"
```
```   243   shows   "bij_betw (\<lambda>\<pi> x. if x \<in> B then f (\<pi> (inv_into A f x)) else x)
```
```   244              {\<pi>. \<pi> permutes A \<and> (\<forall>x\<in>A. \<pi> x \<noteq> x)} {\<pi>. \<pi> permutes B \<and> (\<forall>x\<in>B. \<pi> x \<noteq> x)}"
```
```   245            (is "bij_betw ?f _ _")
```
```   246 proof -
```
```   247   let ?g = "(\<lambda>\<pi> x. if x \<in> A then inv_into A f (\<pi> (f x)) else x)"
```
```   248   show ?thesis
```
```   249   proof (rule bij_betw_byWitness [of _ ?g], goal_cases)
```
```   250     case 3
```
```   251     have "?f \<pi> x \<noteq> x" if "\<pi> permutes A" "\<And>x. x \<in> A \<Longrightarrow> \<pi> x \<noteq> x" "x \<in> B" for \<pi> x
```
```   252       using that and assms by (metis bij_betwE bij_betw_imp_inj_on bij_betw_imp_surj_on
```
```   253                                      inv_into_f_f inv_into_into permutes_imp_bij)
```
```   254     with permutes_bij_inv_into[OF _ assms] show ?case by auto
```
```   255   next
```
```   256     case 4
```
```   257     have bij_inv: "bij_betw (inv_into A f) B A" by (intro bij_betw_inv_into assms)
```
```   258     have "?g \<pi> permutes A" if "\<pi> permutes B" for \<pi>
```
```   259       using permutes_bij_inv_into[OF that bij_inv] and assms
```
```   260       by (simp add: inv_into_inv_into_eq cong: if_cong)
```
```   261     moreover have "?g \<pi> x \<noteq> x" if "\<pi> permutes B" "\<And>x. x \<in> B \<Longrightarrow> \<pi> x \<noteq> x" "x \<in> A" for \<pi> x
```
```   262       using that and assms by (metis bij_betwE bij_betw_imp_surj_on f_inv_into_f permutes_imp_bij)
```
```   263     ultimately show ?case by auto
```
```   264   next
```
```   265     case 1
```
```   266     thus ?case using assms
```
```   267       by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_left
```
```   268                 dest: bij_betwE)
```
```   269   next
```
```   270     case 2
```
```   271     moreover have "bij_betw (inv_into A f) B A"
```
```   272       by (intro bij_betw_inv_into assms)
```
```   273     ultimately show ?case using assms
```
```   274       by (force simp: fun_eq_iff permutes_not_in permutes_in_image bij_betw_inv_into_right
```
```   275                 dest: bij_betwE)
```
```   276   qed
```
```   277 qed
```
```   278
```
```   279
```
```   280 subsection \<open>The number of permutations on a finite set\<close>
```
```   281
```
```   282 lemma permutes_insert_lemma:
```
```   283   assumes "p permutes (insert a S)"
```
```   284   shows "Fun.swap a (p a) id \<circ> p permutes S"
```
```   285   apply (rule permutes_superset[where S = "insert a S"])
```
```   286   apply (rule permutes_compose[OF assms])
```
```   287   apply (rule permutes_swap_id, simp)
```
```   288   using permutes_in_image[OF assms, of a]
```
```   289   apply simp
```
```   290   apply (auto simp add: Ball_def Fun.swap_def)
```
```   291   done
```
```   292
```
```   293 lemma permutes_insert: "{p. p permutes (insert a S)} =
```
```   294   (\<lambda>(b, p). Fun.swap a b id \<circ> p) ` {(b, p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
```
```   295 proof -
```
```   296   have "p permutes insert a S \<longleftrightarrow>
```
```   297     (\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)" for p
```
```   298   proof -
```
```   299     have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S"
```
```   300       if p: "p permutes insert a S"
```
```   301     proof -
```
```   302       let ?b = "p a"
```
```   303       let ?q = "Fun.swap a (p a) id \<circ> p"
```
```   304       have *: "p = Fun.swap a ?b id \<circ> ?q"
```
```   305         by (simp add: fun_eq_iff o_assoc)
```
```   306       have **: "?b \<in> insert a S"
```
```   307         unfolding permutes_in_image[OF p] by simp
```
```   308       from permutes_insert_lemma[OF p] * ** show ?thesis
```
```   309        by blast
```
```   310     qed
```
```   311     moreover have "p permutes insert a S"
```
```   312       if bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" for b q
```
```   313     proof -
```
```   314       from permutes_subset[OF bq(3), of "insert a S"] have q: "q permutes insert a S"
```
```   315         by auto
```
```   316       have a: "a \<in> insert a S"
```
```   317         by simp
```
```   318       from bq(1) permutes_compose[OF q permutes_swap_id[OF a bq(2)]] show ?thesis
```
```   319         by simp
```
```   320     qed
```
```   321     ultimately show ?thesis by blast
```
```   322   qed
```
```   323   then show ?thesis by auto
```
```   324 qed
```
```   325
```
```   326 lemma card_permutations:
```
```   327   assumes "card S = n"
```
```   328     and "finite S"
```
```   329   shows "card {p. p permutes S} = fact n"
```
```   330   using assms(2,1)
```
```   331 proof (induct arbitrary: n)
```
```   332   case empty
```
```   333   then show ?case by simp
```
```   334 next
```
```   335   case (insert x F)
```
```   336   {
```
```   337     fix n
```
```   338     assume card_insert: "card (insert x F) = n"
```
```   339     let ?xF = "{p. p permutes insert x F}"
```
```   340     let ?pF = "{p. p permutes F}"
```
```   341     let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
```
```   342     let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
```
```   343     have xfgpF': "?xF = ?g ` ?pF'"
```
```   344       by (rule permutes_insert[of x F])
```
```   345     from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have Fs: "card F = n - 1"
```
```   346       by auto
```
```   347     from \<open>finite F\<close> insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
```
```   348       by auto
```
```   349     then have "finite ?pF"
```
```   350       by (auto intro: card_ge_0_finite)
```
```   351     with \<open>finite F\<close> card_insert have pF'f: "finite ?pF'"
```
```   352       apply (simp only: Collect_case_prod Collect_mem_eq)
```
```   353       apply (rule finite_cartesian_product)
```
```   354       apply simp_all
```
```   355       done
```
```   356
```
```   357     have ginj: "inj_on ?g ?pF'"
```
```   358     proof -
```
```   359       {
```
```   360         fix b p c q
```
```   361         assume bp: "(b, p) \<in> ?pF'"
```
```   362         assume cq: "(c, q) \<in> ?pF'"
```
```   363         assume eq: "?g (b, p) = ?g (c, q)"
```
```   364         from bp cq have pF: "p permutes F" and qF: "q permutes F"
```
```   365           by auto
```
```   366         from pF \<open>x \<notin> F\<close> eq have "b = ?g (b, p) x"
```
```   367           by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
```
```   368         also from qF \<open>x \<notin> F\<close> eq have "\<dots> = ?g (c, q) x"
```
```   369           by (auto simp: swap_def fun_upd_def fun_eq_iff)
```
```   370         also from qF \<open>x \<notin> F\<close> have "\<dots> = c"
```
```   371           by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
```
```   372         finally have "b = c" .
```
```   373         then have "Fun.swap x b id = Fun.swap x c id"
```
```   374           by simp
```
```   375         with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q"
```
```   376           by simp
```
```   377         then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) = Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)"
```
```   378           by simp
```
```   379         then have "p = q"
```
```   380           by (simp add: o_assoc)
```
```   381         with \<open>b = c\<close> have "(b, p) = (c, q)"
```
```   382           by simp
```
```   383       }
```
```   384       then show ?thesis
```
```   385         unfolding inj_on_def by blast
```
```   386     qed
```
```   387     from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have "n \<noteq> 0"
```
```   388       by auto
```
```   389     then have "\<exists>m. n = Suc m"
```
```   390       by presburger
```
```   391     then obtain m where n: "n = Suc m"
```
```   392       by blast
```
```   393     from pFs card_insert have *: "card ?xF = fact n"
```
```   394       unfolding xfgpF' card_image[OF ginj]
```
```   395       using \<open>finite F\<close> \<open>finite ?pF\<close>
```
```   396       by (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) (simp add: n)
```
```   397     from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF"
```
```   398       by (simp add: xfgpF' n)
```
```   399     from * have "card ?xF = fact n"
```
```   400       unfolding xFf by blast
```
```   401   }
```
```   402   with insert show ?case by simp
```
```   403 qed
```
```   404
```
```   405 lemma finite_permutations:
```
```   406   assumes "finite S"
```
```   407   shows "finite {p. p permutes S}"
```
```   408   using card_permutations[OF refl assms] by (auto intro: card_ge_0_finite)
```
```   409
```
```   410
```
```   411 subsection \<open>Permutations of index set for iterated operations\<close>
```
```   412
```
```   413 lemma (in comm_monoid_set) permute:
```
```   414   assumes "p permutes S"
```
```   415   shows "F g S = F (g \<circ> p) S"
```
```   416 proof -
```
```   417   from \<open>p permutes S\<close> have "inj p"
```
```   418     by (rule permutes_inj)
```
```   419   then have "inj_on p S"
```
```   420     by (auto intro: subset_inj_on)
```
```   421   then have "F g (p ` S) = F (g \<circ> p) S"
```
```   422     by (rule reindex)
```
```   423   moreover from \<open>p permutes S\<close> have "p ` S = S"
```
```   424     by (rule permutes_image)
```
```   425   ultimately show ?thesis
```
```   426     by simp
```
```   427 qed
```
```   428
```
```   429
```
```   430 subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close>
```
```   431
```
```   432 lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>
```
```   433   Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
```
```   434   by (simp add: fun_eq_iff Fun.swap_def)
```
```   435
```
```   436 lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow>
```
```   437   Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"
```
```   438   by (simp add: fun_eq_iff Fun.swap_def)
```
```   439
```
```   440 lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow>
```
```   441   Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id"
```
```   442   by (simp add: fun_eq_iff Fun.swap_def)
```
```   443
```
```   444
```
```   445 subsection \<open>Permutations as transposition sequences\<close>
```
```   446
```
```   447 inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
```
```   448   where
```
```   449     id[simp]: "swapidseq 0 id"
```
```   450   | comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
```
```   451
```
```   452 declare id[unfolded id_def, simp]
```
```   453
```
```   454 definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
```
```   455
```
```   456
```
```   457 subsection \<open>Some closure properties of the set of permutations, with lengths\<close>
```
```   458
```
```   459 lemma permutation_id[simp]: "permutation id"
```
```   460   unfolding permutation_def by (rule exI[where x=0]) simp
```
```   461
```
```   462 declare permutation_id[unfolded id_def, simp]
```
```   463
```
```   464 lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
```
```   465   apply clarsimp
```
```   466   using comp_Suc[of 0 id a b]
```
```   467   apply simp
```
```   468   done
```
```   469
```
```   470 lemma permutation_swap_id: "permutation (Fun.swap a b id)"
```
```   471 proof (cases "a = b")
```
```   472   case True
```
```   473   then show ?thesis by simp
```
```   474 next
```
```   475   case False
```
```   476   then show ?thesis
```
```   477     unfolding permutation_def
```
```   478     using swapidseq_swap[of a b] by blast
```
```   479 qed
```
```   480
```
```   481
```
```   482 lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)"
```
```   483 proof (induct n p arbitrary: m q rule: swapidseq.induct)
```
```   484   case (id m q)
```
```   485   then show ?case by simp
```
```   486 next
```
```   487   case (comp_Suc n p a b m q)
```
```   488   have eq: "Suc n + m = Suc (n + m)"
```
```   489     by arith
```
```   490   show ?case
```
```   491     apply (simp only: eq comp_assoc)
```
```   492     apply (rule swapidseq.comp_Suc)
```
```   493     using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
```
```   494      apply blast+
```
```   495     done
```
```   496 qed
```
```   497
```
```   498 lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)"
```
```   499   unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
```
```   500
```
```   501 lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
```
```   502   by (induct n p rule: swapidseq.induct)
```
```   503     (use swapidseq_swap[of a b] in \<open>auto simp add: comp_assoc intro: swapidseq.comp_Suc\<close>)
```
```   504
```
```   505 lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
```
```   506 proof (induct n p rule: swapidseq.induct)
```
```   507   case id
```
```   508   then show ?case
```
```   509     by (rule exI[where x=id]) simp
```
```   510 next
```
```   511   case (comp_Suc n p a b)
```
```   512   from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
```
```   513     by blast
```
```   514   let ?q = "q \<circ> Fun.swap a b id"
```
```   515   note H = comp_Suc.hyps
```
```   516   from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (Fun.swap a b id)"
```
```   517     by simp
```
```   518   from swapidseq_comp_add[OF q(1) *] have **: "swapidseq (Suc n) ?q"
```
```   519     by simp
```
```   520   have "Fun.swap a b id \<circ> p \<circ> ?q = Fun.swap a b id \<circ> (p \<circ> q) \<circ> Fun.swap a b id"
```
```   521     by (simp add: o_assoc)
```
```   522   also have "\<dots> = id"
```
```   523     by (simp add: q(2))
```
```   524   finally have ***: "Fun.swap a b id \<circ> p \<circ> ?q = id" .
```
```   525   have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id \<circ> Fun.swap a b id) \<circ> p"
```
```   526     by (simp only: o_assoc)
```
```   527   then have "?q \<circ> (Fun.swap a b id \<circ> p) = id"
```
```   528     by (simp add: q(3))
```
```   529   with ** *** show ?case
```
```   530     by blast
```
```   531 qed
```
```   532
```
```   533 lemma swapidseq_inverse:
```
```   534   assumes "swapidseq n p"
```
```   535   shows "swapidseq n (inv p)"
```
```   536   using swapidseq_inverse_exists[OF assms] inv_unique_comp[of p] by auto
```
```   537
```
```   538 lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)"
```
```   539   using permutation_def swapidseq_inverse by blast
```
```   540
```
```   541
```
```   542 subsection \<open>The identity map only has even transposition sequences\<close>
```
```   543
```
```   544 lemma symmetry_lemma:
```
```   545   assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"
```
```   546     and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
```
```   547       a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<Longrightarrow>
```
```   548       P a b c d"
```
```   549   shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>  P a b c d"
```
```   550   using assms by metis
```
```   551
```
```   552 lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
```
```   553   Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
```
```   554   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
```
```   555     Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)"
```
```   556 proof -
```
```   557   assume neq: "a \<noteq> b" "c \<noteq> d"
```
```   558   have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
```
```   559     (Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
```
```   560       (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
```
```   561         Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))"
```
```   562     apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
```
```   563      apply (simp_all only: swap_commute)
```
```   564     apply (case_tac "a = c \<and> b = d")
```
```   565      apply (clarsimp simp only: swap_commute swap_id_idempotent)
```
```   566     apply (case_tac "a = c \<and> b \<noteq> d")
```
```   567      apply (rule disjI2)
```
```   568      apply (rule_tac x="b" in exI)
```
```   569      apply (rule_tac x="d" in exI)
```
```   570      apply (rule_tac x="b" in exI)
```
```   571      apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
```
```   572     apply (case_tac "a \<noteq> c \<and> b = d")
```
```   573      apply (rule disjI2)
```
```   574      apply (rule_tac x="c" in exI)
```
```   575      apply (rule_tac x="d" in exI)
```
```   576      apply (rule_tac x="c" in exI)
```
```   577      apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
```
```   578     apply (rule disjI2)
```
```   579     apply (rule_tac x="c" in exI)
```
```   580     apply (rule_tac x="d" in exI)
```
```   581     apply (rule_tac x="b" in exI)
```
```   582     apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
```
```   583     done
```
```   584   with neq show ?thesis by metis
```
```   585 qed
```
```   586
```
```   587 lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
```
```   588   using swapidseq.cases[of 0 p "p = id"] by auto
```
```   589
```
```   590 lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow>
```
```   591     n = 0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id \<circ> q \<and> swapidseq m q \<and> a \<noteq> b)"
```
```   592   apply (rule iffI)
```
```   593    apply (erule swapidseq.cases[of n p])
```
```   594     apply simp
```
```   595    apply (rule disjI2)
```
```   596    apply (rule_tac x= "a" in exI)
```
```   597    apply (rule_tac x= "b" in exI)
```
```   598    apply (rule_tac x= "pa" in exI)
```
```   599    apply (rule_tac x= "na" in exI)
```
```   600    apply simp
```
```   601   apply auto
```
```   602   apply (rule comp_Suc, simp_all)
```
```   603   done
```
```   604
```
```   605 lemma fixing_swapidseq_decrease:
```
```   606   assumes "swapidseq n p"
```
```   607     and "a \<noteq> b"
```
```   608     and "(Fun.swap a b id \<circ> p) a = a"
```
```   609   shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
```
```   610   using assms
```
```   611 proof (induct n arbitrary: p a b)
```
```   612   case 0
```
```   613   then show ?case
```
```   614     by (auto simp add: Fun.swap_def fun_upd_def)
```
```   615 next
```
```   616   case (Suc n p a b)
```
```   617   from Suc.prems(1) swapidseq_cases[of "Suc n" p]
```
```   618   obtain c d q m where
```
```   619     cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m"
```
```   620     by auto
```
```   621   consider "Fun.swap a b id \<circ> Fun.swap c d id = id"
```
```   622     | x y z where "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y"
```
```   623       "Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id"
```
```   624     using swap_general[OF Suc.prems(2) cdqm(4)] by metis
```
```   625   then show ?case
```
```   626   proof cases
```
```   627     case 1
```
```   628     then show ?thesis
```
```   629       by (simp only: cdqm o_assoc) (simp add: cdqm)
```
```   630   next
```
```   631     case prems: 2
```
```   632     then have az: "a \<noteq> z"
```
```   633       by simp
```
```   634     from prems have *: "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" for h
```
```   635       by (simp add: Fun.swap_def)
```
```   636     from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)"
```
```   637       by simp
```
```   638     then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)"
```
```   639       by (simp add: o_assoc prems)
```
```   640     then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a"
```
```   641       by simp
```
```   642     then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a"
```
```   643       unfolding Suc by metis
```
```   644     then have "(Fun.swap a z id \<circ> q) a = a"
```
```   645       by (simp only: *)
```
```   646     from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this]
```
```   647     have **: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0"
```
```   648       by blast+
```
```   649     from \<open>n \<noteq> 0\<close> have ***: "Suc n - 1 = Suc (n - 1)"
```
```   650       by auto
```
```   651     show ?thesis
```
```   652       apply (simp only: cdqm(2) prems o_assoc ***)
```
```   653       apply (simp only: Suc_not_Zero simp_thms comp_assoc)
```
```   654       apply (rule comp_Suc)
```
```   655       using ** prems
```
```   656        apply blast+
```
```   657       done
```
```   658   qed
```
```   659 qed
```
```   660
```
```   661 lemma swapidseq_identity_even:
```
```   662   assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)"
```
```   663   shows "even n"
```
```   664   using \<open>swapidseq n id\<close>
```
```   665 proof (induct n rule: nat_less_induct)
```
```   666   case H: (1 n)
```
```   667   consider "n = 0"
```
```   668     | a b :: 'a and q m where "n = Suc m" "id = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
```
```   669     using H(2)[unfolded swapidseq_cases[of n id]] by auto
```
```   670   then show ?case
```
```   671   proof cases
```
```   672     case 1
```
```   673     then show ?thesis by presburger
```
```   674   next
```
```   675     case h: 2
```
```   676     from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
```
```   677     have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)"
```
```   678       by auto
```
```   679     from h m have mn: "m - 1 < n"
```
```   680       by arith
```
```   681     from H(1)[rule_format, OF mn m(2)] h(1) m(1) show ?thesis
```
```   682       by presburger
```
```   683   qed
```
```   684 qed
```
```   685
```
```   686
```
```   687 subsection \<open>Therefore we have a welldefined notion of parity\<close>
```
```   688
```
```   689 definition "evenperm p = even (SOME n. swapidseq n p)"
```
```   690
```
```   691 lemma swapidseq_even_even:
```
```   692   assumes m: "swapidseq m p"
```
```   693     and n: "swapidseq n p"
```
```   694   shows "even m \<longleftrightarrow> even n"
```
```   695 proof -
```
```   696   from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
```
```   697     by blast
```
```   698   from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] show ?thesis
```
```   699     by arith
```
```   700 qed
```
```   701
```
```   702 lemma evenperm_unique:
```
```   703   assumes p: "swapidseq n p"
```
```   704     and n:"even n = b"
```
```   705   shows "evenperm p = b"
```
```   706   unfolding n[symmetric] evenperm_def
```
```   707   apply (rule swapidseq_even_even[where p = p])
```
```   708    apply (rule someI[where x = n])
```
```   709   using p
```
```   710    apply blast+
```
```   711   done
```
```   712
```
```   713
```
```   714 subsection \<open>And it has the expected composition properties\<close>
```
```   715
```
```   716 lemma evenperm_id[simp]: "evenperm id = True"
```
```   717   by (rule evenperm_unique[where n = 0]) simp_all
```
```   718
```
```   719 lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
```
```   720   by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap)
```
```   721
```
```   722 lemma evenperm_comp:
```
```   723   assumes "permutation p" "permutation q"
```
```   724   shows "evenperm (p \<circ> q) \<longleftrightarrow> evenperm p = evenperm q"
```
```   725 proof -
```
```   726   from assms obtain n m where n: "swapidseq n p" and m: "swapidseq m q"
```
```   727     unfolding permutation_def by blast
```
```   728   have "even (n + m) \<longleftrightarrow> (even n \<longleftrightarrow> even m)"
```
```   729     by arith
```
```   730   from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
```
```   731     and evenperm_unique[OF swapidseq_comp_add[OF n m] this] show ?thesis
```
```   732     by blast
```
```   733 qed
```
```   734
```
```   735 lemma evenperm_inv:
```
```   736   assumes "permutation p"
```
```   737   shows "evenperm (inv p) = evenperm p"
```
```   738 proof -
```
```   739   from assms obtain n where n: "swapidseq n p"
```
```   740     unfolding permutation_def by blast
```
```   741   show ?thesis
```
```   742     by (rule evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]])
```
```   743 qed
```
```   744
```
```   745
```
```   746 subsection \<open>A more abstract characterization of permutations\<close>
```
```   747
```
```   748 lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
```
```   749   unfolding bij_def inj_def surj_def
```
```   750   apply auto
```
```   751    apply metis
```
```   752   apply metis
```
```   753   done
```
```   754
```
```   755 lemma permutation_bijective:
```
```   756   assumes "permutation p"
```
```   757   shows "bij p"
```
```   758 proof -
```
```   759   from assms obtain n where n: "swapidseq n p"
```
```   760     unfolding permutation_def by blast
```
```   761   from swapidseq_inverse_exists[OF n] obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
```
```   762     by blast
```
```   763   then show ?thesis
```
```   764     unfolding bij_iff
```
```   765     apply (auto simp add: fun_eq_iff)
```
```   766     apply metis
```
```   767     done
```
```   768 qed
```
```   769
```
```   770 lemma permutation_finite_support:
```
```   771   assumes "permutation p"
```
```   772   shows "finite {x. p x \<noteq> x}"
```
```   773 proof -
```
```   774   from assms obtain n where "swapidseq n p"
```
```   775     unfolding permutation_def by blast
```
```   776   then show ?thesis
```
```   777   proof (induct n p rule: swapidseq.induct)
```
```   778     case id
```
```   779     then show ?case by simp
```
```   780   next
```
```   781     case (comp_Suc n p a b)
```
```   782     let ?S = "insert a (insert b {x. p x \<noteq> x})"
```
```   783     from comp_Suc.hyps(2) have *: "finite ?S"
```
```   784       by simp
```
```   785     from \<open>a \<noteq> b\<close> have **: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
```
```   786       by (auto simp: Fun.swap_def)
```
```   787     show ?case
```
```   788       by (rule finite_subset[OF ** *])
```
```   789   qed
```
```   790 qed
```
```   791
```
```   792 lemma permutation_lemma:
```
```   793   assumes "finite S"
```
```   794     and "bij p"
```
```   795     and "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
```
```   796   shows "permutation p"
```
```   797   using assms
```
```   798 proof (induct S arbitrary: p rule: finite_induct)
```
```   799   case empty
```
```   800   then show ?case
```
```   801     by simp
```
```   802 next
```
```   803   case (insert a F p)
```
```   804   let ?r = "Fun.swap a (p a) id \<circ> p"
```
```   805   let ?q = "Fun.swap a (p a) id \<circ> ?r"
```
```   806   have *: "?r a = a"
```
```   807     by (simp add: Fun.swap_def)
```
```   808   from insert * have **: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
```
```   809     by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
```
```   810   have "bij ?r"
```
```   811     by (rule bij_swap_compose_bij[OF insert(4)])
```
```   812   have "permutation ?r"
```
```   813     by (rule insert(3)[OF \<open>bij ?r\<close> **])
```
```   814   then have "permutation ?q"
```
```   815     by (simp add: permutation_compose permutation_swap_id)
```
```   816   then show ?case
```
```   817     by (simp add: o_assoc)
```
```   818 qed
```
```   819
```
```   820 lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
```
```   821   (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
```
```   822 proof
```
```   823   assume ?lhs
```
```   824   with permutation_bijective permutation_finite_support show "?b \<and> ?f"
```
```   825     by auto
```
```   826 next
```
```   827   assume "?b \<and> ?f"
```
```   828   then have "?f" "?b" by blast+
```
```   829   from permutation_lemma[OF this] show ?lhs
```
```   830     by blast
```
```   831 qed
```
```   832
```
```   833 lemma permutation_inverse_works:
```
```   834   assumes "permutation p"
```
```   835   shows "inv p \<circ> p = id"
```
```   836     and "p \<circ> inv p = id"
```
```   837   using permutation_bijective [OF assms] by (auto simp: bij_def inj_iff surj_iff)
```
```   838
```
```   839 lemma permutation_inverse_compose:
```
```   840   assumes p: "permutation p"
```
```   841     and q: "permutation q"
```
```   842   shows "inv (p \<circ> q) = inv q \<circ> inv p"
```
```   843 proof -
```
```   844   note ps = permutation_inverse_works[OF p]
```
```   845   note qs = permutation_inverse_works[OF q]
```
```   846   have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p"
```
```   847     by (simp add: o_assoc)
```
```   848   also have "\<dots> = id"
```
```   849     by (simp add: ps qs)
```
```   850   finally have *: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" .
```
```   851   have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q"
```
```   852     by (simp add: o_assoc)
```
```   853   also have "\<dots> = id"
```
```   854     by (simp add: ps qs)
```
```   855   finally have **: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
```
```   856   show ?thesis
```
```   857     by (rule inv_unique_comp[OF * **])
```
```   858 qed
```
```   859
```
```   860
```
```   861 subsection \<open>Relation to \<open>permutes\<close>\<close>
```
```   862
```
```   863 lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
```
```   864   unfolding permutation permutes_def bij_iff[symmetric]
```
```   865   apply (rule iffI, clarify)
```
```   866    apply (rule exI[where x="{x. p x \<noteq> x}"])
```
```   867    apply simp
```
```   868   apply clarsimp
```
```   869   apply (rule_tac B="S" in finite_subset)
```
```   870    apply auto
```
```   871   done
```
```   872
```
```   873
```
```   874 subsection \<open>Hence a sort of induction principle composing by swaps\<close>
```
```   875
```
```   876 lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow>
```
```   877   (\<And>a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p \<Longrightarrow> P (Fun.swap a b id \<circ> p)) \<Longrightarrow>
```
```   878   (\<And>p. p permutes S \<Longrightarrow> P p)"
```
```   879 proof (induct S rule: finite_induct)
```
```   880   case empty
```
```   881   then show ?case by auto
```
```   882 next
```
```   883   case (insert x F p)
```
```   884   let ?r = "Fun.swap x (p x) id \<circ> p"
```
```   885   let ?q = "Fun.swap x (p x) id \<circ> ?r"
```
```   886   have qp: "?q = p"
```
```   887     by (simp add: o_assoc)
```
```   888   from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r"
```
```   889     by blast
```
```   890   from permutes_in_image[OF insert.prems(3), of x]
```
```   891   have pxF: "p x \<in> insert x F"
```
```   892     by simp
```
```   893   have xF: "x \<in> insert x F"
```
```   894     by simp
```
```   895   have rp: "permutation ?r"
```
```   896     unfolding permutation_permutes
```
```   897     using insert.hyps(1) permutes_insert_lemma[OF insert.prems(3)]
```
```   898     by blast
```
```   899   from insert.prems(2)[OF xF pxF Pr Pr rp] qp show ?case
```
```   900     by (simp only:)
```
```   901 qed
```
```   902
```
```   903
```
```   904 subsection \<open>Sign of a permutation as a real number\<close>
```
```   905
```
```   906 definition "sign p = (if evenperm p then (1::int) else -1)"
```
```   907
```
```   908 lemma sign_nz: "sign p \<noteq> 0"
```
```   909   by (simp add: sign_def)
```
```   910
```
```   911 lemma sign_id: "sign id = 1"
```
```   912   by (simp add: sign_def)
```
```   913
```
```   914 lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p"
```
```   915   by (simp add: sign_def evenperm_inv)
```
```   916
```
```   917 lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q"
```
```   918   by (simp add: sign_def evenperm_comp)
```
```   919
```
```   920 lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
```
```   921   by (simp add: sign_def evenperm_swap)
```
```   922
```
```   923 lemma sign_idempotent: "sign p * sign p = 1"
```
```   924   by (simp add: sign_def)
```
```   925
```
```   926
```
```   927 subsection \<open>Permuting a list\<close>
```
```   928
```
```   929 text \<open>This function permutes a list by applying a permutation to the indices.\<close>
```
```   930
```
```   931 definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list"
```
```   932   where "permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"
```
```   933
```
```   934 lemma permute_list_map:
```
```   935   assumes "f permutes {..<length xs}"
```
```   936   shows "permute_list f (map g xs) = map g (permute_list f xs)"
```
```   937   using permutes_in_image[OF assms] by (auto simp: permute_list_def)
```
```   938
```
```   939 lemma permute_list_nth:
```
```   940   assumes "f permutes {..<length xs}" "i < length xs"
```
```   941   shows "permute_list f xs ! i = xs ! f i"
```
```   942   using permutes_in_image[OF assms(1)] assms(2)
```
```   943   by (simp add: permute_list_def)
```
```   944
```
```   945 lemma permute_list_Nil [simp]: "permute_list f [] = []"
```
```   946   by (simp add: permute_list_def)
```
```   947
```
```   948 lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"
```
```   949   by (simp add: permute_list_def)
```
```   950
```
```   951 lemma permute_list_compose:
```
```   952   assumes "g permutes {..<length xs}"
```
```   953   shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
```
```   954   using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)
```
```   955
```
```   956 lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs"
```
```   957   by (simp add: permute_list_def map_nth)
```
```   958
```
```   959 lemma permute_list_id [simp]: "permute_list id xs = xs"
```
```   960   by (simp add: id_def)
```
```   961
```
```   962 lemma mset_permute_list [simp]:
```
```   963   fixes xs :: "'a list"
```
```   964   assumes "f permutes {..<length xs}"
```
```   965   shows "mset (permute_list f xs) = mset xs"
```
```   966 proof (rule multiset_eqI)
```
```   967   fix y :: 'a
```
```   968   from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
```
```   969     using permutes_in_image[OF assms] by auto
```
```   970   have "count (mset (permute_list f xs)) y = card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})"
```
```   971     by (simp add: permute_list_def count_image_mset atLeast0LessThan)
```
```   972   also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}"
```
```   973     by auto
```
```   974   also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}"
```
```   975     by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
```
```   976   also have "\<dots> = count (mset xs) y"
```
```   977     by (simp add: count_mset length_filter_conv_card)
```
```   978   finally show "count (mset (permute_list f xs)) y = count (mset xs) y"
```
```   979     by simp
```
```   980 qed
```
```   981
```
```   982 lemma set_permute_list [simp]:
```
```   983   assumes "f permutes {..<length xs}"
```
```   984   shows "set (permute_list f xs) = set xs"
```
```   985   by (rule mset_eq_setD[OF mset_permute_list]) fact
```
```   986
```
```   987 lemma distinct_permute_list [simp]:
```
```   988   assumes "f permutes {..<length xs}"
```
```   989   shows "distinct (permute_list f xs) = distinct xs"
```
```   990   by (simp add: distinct_count_atmost_1 assms)
```
```   991
```
```   992 lemma permute_list_zip:
```
```   993   assumes "f permutes A" "A = {..<length xs}"
```
```   994   assumes [simp]: "length xs = length ys"
```
```   995   shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
```
```   996 proof -
```
```   997   from permutes_in_image[OF assms(1)] assms(2) have *: "f i < length ys \<longleftrightarrow> i < length ys" for i
```
```   998     by simp
```
```   999   have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]"
```
```  1000     by (simp_all add: permute_list_def zip_map_map)
```
```  1001   also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
```
```  1002     by (intro nth_equalityI) (simp_all add: *)
```
```  1003   also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
```
```  1004     by (simp_all add: permute_list_def zip_map_map)
```
```  1005   finally show ?thesis .
```
```  1006 qed
```
```  1007
```
```  1008 lemma map_of_permute:
```
```  1009   assumes "\<sigma> permutes fst ` set xs"
```
```  1010   shows "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)"
```
```  1011     (is "_ = map_of (map ?f _)")
```
```  1012 proof
```
```  1013   from assms have "inj \<sigma>" "surj \<sigma>"
```
```  1014     by (simp_all add: permutes_inj permutes_surj)
```
```  1015   then show "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x" for x
```
```  1016     by (induct xs) (auto simp: inv_f_f surj_f_inv_f)
```
```  1017 qed
```
```  1018
```
```  1019
```
```  1020 subsection \<open>More lemmas about permutations\<close>
```
```  1021
```
```  1022 text \<open>The following few lemmas were contributed by Lukas Bulwahn.\<close>
```
```  1023
```
```  1024 lemma count_image_mset_eq_card_vimage:
```
```  1025   assumes "finite A"
```
```  1026   shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
```
```  1027   using assms
```
```  1028 proof (induct A)
```
```  1029   case empty
```
```  1030   show ?case by simp
```
```  1031 next
```
```  1032   case (insert x F)
```
```  1033   show ?case
```
```  1034   proof (cases "f x = b")
```
```  1035     case True
```
```  1036     with insert.hyps
```
```  1037     have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
```
```  1038       by auto
```
```  1039     also from insert.hyps(1,2) have "\<dots> = card (insert x {a \<in> F. f a = f x})"
```
```  1040       by simp
```
```  1041     also from \<open>f x = b\<close> have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
```
```  1042       by (auto intro: arg_cong[where f="card"])
```
```  1043     finally show ?thesis
```
```  1044       using insert by auto
```
```  1045   next
```
```  1046     case False
```
```  1047     then have "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}"
```
```  1048       by auto
```
```  1049     with insert False show ?thesis
```
```  1050       by simp
```
```  1051   qed
```
```  1052 qed
```
```  1053
```
```  1054 \<comment> \<open>Prove \<open>image_mset_eq_implies_permutes\<close> ...\<close>
```
```  1055 lemma image_mset_eq_implies_permutes:
```
```  1056   fixes f :: "'a \<Rightarrow> 'b"
```
```  1057   assumes "finite A"
```
```  1058     and mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)"
```
```  1059   obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
```
```  1060 proof -
```
```  1061   from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto
```
```  1062   have "f ` A = f' ` A"
```
```  1063   proof -
```
```  1064     from \<open>finite A\<close> have "f ` A = f ` (set_mset (mset_set A))"
```
```  1065       by simp
```
```  1066     also have "\<dots> = f' ` set_mset (mset_set A)"
```
```  1067       by (metis mset_eq multiset.set_map)
```
```  1068     also from \<open>finite A\<close> have "\<dots> = f' ` A"
```
```  1069       by simp
```
```  1070     finally show ?thesis .
```
```  1071   qed
```
```  1072   have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}"
```
```  1073   proof
```
```  1074     fix b
```
```  1075     from mset_eq have "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b"
```
```  1076       by simp
```
```  1077     with \<open>finite A\<close> have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
```
```  1078       by (simp add: count_image_mset_eq_card_vimage)
```
```  1079     then show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
```
```  1080       by (intro finite_same_card_bij) simp_all
```
```  1081   qed
```
```  1082   then have "\<exists>p. \<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}"
```
```  1083     by (rule bchoice)
```
```  1084   then obtain p where p: "\<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}" ..
```
```  1085   define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
```
```  1086   have "p' permutes A"
```
```  1087   proof (rule bij_imp_permutes)
```
```  1088     have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)"
```
```  1089       by (auto simp: disjoint_family_on_def)
```
```  1090     moreover
```
```  1091     have "bij_betw (\<lambda>a. p (f a) a) {a \<in> A. f a = b} {a \<in> A. f' a = b}" if "b \<in> f ` A" for b
```
```  1092       using p that by (subst bij_betw_cong[where g="p b"]) auto
```
```  1093     ultimately
```
```  1094     have "bij_betw (\<lambda>a. p (f a) a) (\<Union>b\<in>f ` A. {a \<in> A. f a = b}) (\<Union>b\<in>f ` A. {a \<in> A. f' a = b})"
```
```  1095       by (rule bij_betw_UNION_disjoint)
```
```  1096     moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A"
```
```  1097       by auto
```
```  1098     moreover from \<open>f ` A = f' ` A\<close> have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A"
```
```  1099       by auto
```
```  1100     ultimately show "bij_betw p' A A"
```
```  1101       unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto
```
```  1102   next
```
```  1103     show "\<And>x. x \<notin> A \<Longrightarrow> p' x = x"
```
```  1104       by (simp add: p'_def)
```
```  1105   qed
```
```  1106   moreover from p have "\<forall>x\<in>A. f x = f' (p' x)"
```
```  1107     unfolding p'_def using bij_betwE by fastforce
```
```  1108   ultimately show ?thesis ..
```
```  1109 qed
```
```  1110
```
```  1111 lemma mset_set_upto_eq_mset_upto: "mset_set {..<n} = mset [0..<n]"
```
```  1112   by (induct n) (auto simp: add.commute lessThan_Suc)
```
```  1113
```
```  1114 \<comment> \<open>... and derive the existing property:\<close>
```
```  1115 lemma mset_eq_permutation:
```
```  1116   fixes xs ys :: "'a list"
```
```  1117   assumes mset_eq: "mset xs = mset ys"
```
```  1118   obtains p where "p permutes {..<length ys}" "permute_list p ys = xs"
```
```  1119 proof -
```
```  1120   from mset_eq have length_eq: "length xs = length ys"
```
```  1121     by (rule mset_eq_length)
```
```  1122   have "mset_set {..<length ys} = mset [0..<length ys]"
```
```  1123     by (rule mset_set_upto_eq_mset_upto)
```
```  1124   with mset_eq length_eq have "image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) =
```
```  1125     image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})"
```
```  1126     by (metis map_nth mset_map)
```
```  1127   from image_mset_eq_implies_permutes[OF _ this]
```
```  1128   obtain p where p: "p permutes {..<length ys}" and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)"
```
```  1129     by auto
```
```  1130   with length_eq have "permute_list p ys = xs"
```
```  1131     by (auto intro!: nth_equalityI simp: permute_list_nth)
```
```  1132   with p show thesis ..
```
```  1133 qed
```
```  1134
```
```  1135 lemma permutes_natset_le:
```
```  1136   fixes S :: "'a::wellorder set"
```
```  1137   assumes "p permutes S"
```
```  1138     and "\<forall>i \<in> S. p i \<le> i"
```
```  1139   shows "p = id"
```
```  1140 proof -
```
```  1141   have "p n = n" for n
```
```  1142     using assms
```
```  1143   proof (induct n arbitrary: S rule: less_induct)
```
```  1144     case (less n)
```
```  1145     show ?case
```
```  1146     proof (cases "n \<in> S")
```
```  1147       case False
```
```  1148       with less(2) show ?thesis
```
```  1149         unfolding permutes_def by metis
```
```  1150     next
```
```  1151       case True
```
```  1152       with less(3) have "p n < n \<or> p n = n"
```
```  1153         by auto
```
```  1154       then show ?thesis
```
```  1155       proof
```
```  1156         assume "p n < n"
```
```  1157         with less have "p (p n) = p n"
```
```  1158           by metis
```
```  1159         with permutes_inj[OF less(2)] have "p n = n"
```
```  1160           unfolding inj_def by blast
```
```  1161         with \<open>p n < n\<close> have False
```
```  1162           by simp
```
```  1163         then show ?thesis ..
```
```  1164       qed
```
```  1165     qed
```
```  1166   qed
```
```  1167   then show ?thesis by (auto simp: fun_eq_iff)
```
```  1168 qed
```
```  1169
```
```  1170 lemma permutes_natset_ge:
```
```  1171   fixes S :: "'a::wellorder set"
```
```  1172   assumes p: "p permutes S"
```
```  1173     and le: "\<forall>i \<in> S. p i \<ge> i"
```
```  1174   shows "p = id"
```
```  1175 proof -
```
```  1176   have "i \<ge> inv p i" if "i \<in> S" for i
```
```  1177   proof -
```
```  1178     from that permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S"
```
```  1179       by simp
```
```  1180     with le have "p (inv p i) \<ge> inv p i"
```
```  1181       by blast
```
```  1182     with permutes_inverses[OF p] show ?thesis
```
```  1183       by simp
```
```  1184   qed
```
```  1185   then have "\<forall>i\<in>S. inv p i \<le> i"
```
```  1186     by blast
```
```  1187   from permutes_natset_le[OF permutes_inv[OF p] this] have "inv p = inv id"
```
```  1188     by simp
```
```  1189   then show ?thesis
```
```  1190     apply (subst permutes_inv_inv[OF p, symmetric])
```
```  1191     apply (rule inv_unique_comp)
```
```  1192      apply simp_all
```
```  1193     done
```
```  1194 qed
```
```  1195
```
```  1196 lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
```
```  1197   apply (rule set_eqI)
```
```  1198   apply auto
```
```  1199   using permutes_inv_inv permutes_inv
```
```  1200    apply auto
```
```  1201   apply (rule_tac x="inv x" in exI)
```
```  1202   apply auto
```
```  1203   done
```
```  1204
```
```  1205 lemma image_compose_permutations_left:
```
```  1206   assumes "q permutes S"
```
```  1207   shows "{q \<circ> p |p. p permutes S} = {p. p permutes S}"
```
```  1208   apply (rule set_eqI)
```
```  1209   apply auto
```
```  1210    apply (rule permutes_compose)
```
```  1211   using assms
```
```  1212     apply auto
```
```  1213   apply (rule_tac x = "inv q \<circ> x" in exI)
```
```  1214   apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
```
```  1215   done
```
```  1216
```
```  1217 lemma image_compose_permutations_right:
```
```  1218   assumes "q permutes S"
```
```  1219   shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}"
```
```  1220   apply (rule set_eqI)
```
```  1221   apply auto
```
```  1222    apply (rule permutes_compose)
```
```  1223   using assms
```
```  1224     apply auto
```
```  1225   apply (rule_tac x = "x \<circ> inv q" in exI)
```
```  1226   apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc)
```
```  1227   done
```
```  1228
```
```  1229 lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n"
```
```  1230   by (simp add: permutes_def) metis
```
```  1231
```
```  1232 lemma sum_permutations_inverse: "sum f {p. p permutes S} = sum (\<lambda>p. f(inv p)) {p. p permutes S}"
```
```  1233   (is "?lhs = ?rhs")
```
```  1234 proof -
```
```  1235   let ?S = "{p . p permutes S}"
```
```  1236   have *: "inj_on inv ?S"
```
```  1237   proof (auto simp add: inj_on_def)
```
```  1238     fix q r
```
```  1239     assume q: "q permutes S"
```
```  1240       and r: "r permutes S"
```
```  1241       and qr: "inv q = inv r"
```
```  1242     then have "inv (inv q) = inv (inv r)"
```
```  1243       by simp
```
```  1244     with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r"
```
```  1245       by metis
```
```  1246   qed
```
```  1247   have **: "inv ` ?S = ?S"
```
```  1248     using image_inverse_permutations by blast
```
```  1249   have ***: "?rhs = sum (f \<circ> inv) ?S"
```
```  1250     by (simp add: o_def)
```
```  1251   from sum.reindex[OF *, of f] show ?thesis
```
```  1252     by (simp only: ** ***)
```
```  1253 qed
```
```  1254
```
```  1255 lemma setum_permutations_compose_left:
```
```  1256   assumes q: "q permutes S"
```
```  1257   shows "sum f {p. p permutes S} = sum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
```
```  1258   (is "?lhs = ?rhs")
```
```  1259 proof -
```
```  1260   let ?S = "{p. p permutes S}"
```
```  1261   have *: "?rhs = sum (f \<circ> ((\<circ>) q)) ?S"
```
```  1262     by (simp add: o_def)
```
```  1263   have **: "inj_on ((\<circ>) q) ?S"
```
```  1264   proof (auto simp add: inj_on_def)
```
```  1265     fix p r
```
```  1266     assume "p permutes S"
```
```  1267       and r: "r permutes S"
```
```  1268       and rp: "q \<circ> p = q \<circ> r"
```
```  1269     then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r"
```
```  1270       by (simp add: comp_assoc)
```
```  1271     with permutes_inj[OF q, unfolded inj_iff] show "p = r"
```
```  1272       by simp
```
```  1273   qed
```
```  1274   have "((\<circ>) q) ` ?S = ?S"
```
```  1275     using image_compose_permutations_left[OF q] by auto
```
```  1276   with * sum.reindex[OF **, of f] show ?thesis
```
```  1277     by (simp only:)
```
```  1278 qed
```
```  1279
```
```  1280 lemma sum_permutations_compose_right:
```
```  1281   assumes q: "q permutes S"
```
```  1282   shows "sum f {p. p permutes S} = sum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
```
```  1283   (is "?lhs = ?rhs")
```
```  1284 proof -
```
```  1285   let ?S = "{p. p permutes S}"
```
```  1286   have *: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
```
```  1287     by (simp add: o_def)
```
```  1288   have **: "inj_on (\<lambda>p. p \<circ> q) ?S"
```
```  1289   proof (auto simp add: inj_on_def)
```
```  1290     fix p r
```
```  1291     assume "p permutes S"
```
```  1292       and r: "r permutes S"
```
```  1293       and rp: "p \<circ> q = r \<circ> q"
```
```  1294     then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)"
```
```  1295       by (simp add: o_assoc)
```
```  1296     with permutes_surj[OF q, unfolded surj_iff] show "p = r"
```
```  1297       by simp
```
```  1298   qed
```
```  1299   from image_compose_permutations_right[OF q] have "(\<lambda>p. p \<circ> q) ` ?S = ?S"
```
```  1300     by auto
```
```  1301   with * sum.reindex[OF **, of f] show ?thesis
```
```  1302     by (simp only:)
```
```  1303 qed
```
```  1304
```
```  1305
```
```  1306 subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close>
```
```  1307
```
```  1308 lemma sum_over_permutations_insert:
```
```  1309   assumes fS: "finite S"
```
```  1310     and aS: "a \<notin> S"
```
```  1311   shows "sum f {p. p permutes (insert a S)} =
```
```  1312     sum (\<lambda>b. sum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)"
```
```  1313 proof -
```
```  1314   have *: "\<And>f a b. (\<lambda>(b, p). f (Fun.swap a b id \<circ> p)) = f \<circ> (\<lambda>(b,p). Fun.swap a b id \<circ> p)"
```
```  1315     by (simp add: fun_eq_iff)
```
```  1316   have **: "\<And>P Q. {(a, b). a \<in> P \<and> b \<in> Q} = P \<times> Q"
```
```  1317     by blast
```
```  1318   show ?thesis
```
```  1319     unfolding * ** sum.cartesian_product permutes_insert
```
```  1320   proof (rule sum.reindex)
```
```  1321     let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
```
```  1322     let ?P = "{p. p permutes S}"
```
```  1323     {
```
```  1324       fix b c p q
```
```  1325       assume b: "b \<in> insert a S"
```
```  1326       assume c: "c \<in> insert a S"
```
```  1327       assume p: "p permutes S"
```
```  1328       assume q: "q permutes S"
```
```  1329       assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> q"
```
```  1330       from p q aS have pa: "p a = a" and qa: "q a = a"
```
```  1331         unfolding permutes_def by metis+
```
```  1332       from eq have "(Fun.swap a b id \<circ> p) a  = (Fun.swap a c id \<circ> q) a"
```
```  1333         by simp
```
```  1334       then have bc: "b = c"
```
```  1335         by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def
```
```  1336             cong del: if_weak_cong split: if_split_asm)
```
```  1337       from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) =
```
```  1338         (\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp
```
```  1339       then have "p = q"
```
```  1340         unfolding o_assoc swap_id_idempotent by simp
```
```  1341       with bc have "b = c \<and> p = q"
```
```  1342         by blast
```
```  1343     }
```
```  1344     then show "inj_on ?f (insert a S \<times> ?P)"
```
```  1345       unfolding inj_on_def by clarify metis
```
```  1346   qed
```
```  1347 qed
```
```  1348
```
```  1349
```
```  1350 subsection \<open>Constructing permutations from association lists\<close>
```
```  1351
```
```  1352 definition list_permutes :: "('a \<times> 'a) list \<Rightarrow> 'a set \<Rightarrow> bool"
```
```  1353   where "list_permutes xs A \<longleftrightarrow>
```
```  1354     set (map fst xs) \<subseteq> A \<and>
```
```  1355     set (map snd xs) = set (map fst xs) \<and>
```
```  1356     distinct (map fst xs) \<and>
```
```  1357     distinct (map snd xs)"
```
```  1358
```
```  1359 lemma list_permutesI [simp]:
```
```  1360   assumes "set (map fst xs) \<subseteq> A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)"
```
```  1361   shows "list_permutes xs A"
```
```  1362 proof -
```
```  1363   from assms(2,3) have "distinct (map snd xs)"
```
```  1364     by (intro card_distinct) (simp_all add: distinct_card del: set_map)
```
```  1365   with assms show ?thesis
```
```  1366     by (simp add: list_permutes_def)
```
```  1367 qed
```
```  1368
```
```  1369 definition permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  1370   where "permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)"
```
```  1371
```
```  1372 lemma permutation_of_list_Cons:
```
```  1373   "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"
```
```  1374   by (simp add: permutation_of_list_def)
```
```  1375
```
```  1376 fun inverse_permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
```
```  1377   where
```
```  1378     "inverse_permutation_of_list [] x = x"
```
```  1379   | "inverse_permutation_of_list ((y, x') # xs) x =
```
```  1380       (if x = x' then y else inverse_permutation_of_list xs x)"
```
```  1381
```
```  1382 declare inverse_permutation_of_list.simps [simp del]
```
```  1383
```
```  1384 lemma inj_on_map_of:
```
```  1385   assumes "distinct (map snd xs)"
```
```  1386   shows "inj_on (map_of xs) (set (map fst xs))"
```
```  1387 proof (rule inj_onI)
```
```  1388   fix x y
```
```  1389   assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)"
```
```  1390   assume eq: "map_of xs x = map_of xs y"
```
```  1391   from xy obtain x' y' where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
```
```  1392     by (cases "map_of xs x"; cases "map_of xs y") (simp_all add: map_of_eq_None_iff)
```
```  1393   moreover from x'y' have *: "(x, x') \<in> set xs" "(y, y') \<in> set xs"
```
```  1394     by (force dest: map_of_SomeD)+
```
```  1395   moreover from * eq x'y' have "x' = y'"
```
```  1396     by simp
```
```  1397   ultimately show "x = y"
```
```  1398     using assms by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"])
```
```  1399 qed
```
```  1400
```
```  1401 lemma inj_on_the: "None \<notin> A \<Longrightarrow> inj_on the A"
```
```  1402   by (auto simp: inj_on_def option.the_def split: option.splits)
```
```  1403
```
```  1404 lemma inj_on_map_of':
```
```  1405   assumes "distinct (map snd xs)"
```
```  1406   shows "inj_on (the \<circ> map_of xs) (set (map fst xs))"
```
```  1407   by (intro comp_inj_on inj_on_map_of assms inj_on_the)
```
```  1408     (force simp: eq_commute[of None] map_of_eq_None_iff)
```
```  1409
```
```  1410 lemma image_map_of:
```
```  1411   assumes "distinct (map fst xs)"
```
```  1412   shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)"
```
```  1413   using assms by (auto simp: rev_image_eqI)
```
```  1414
```
```  1415 lemma the_Some_image [simp]: "the ` Some ` A = A"
```
```  1416   by (subst image_image) simp
```
```  1417
```
```  1418 lemma image_map_of':
```
```  1419   assumes "distinct (map fst xs)"
```
```  1420   shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)"
```
```  1421   by (simp only: image_comp [symmetric] image_map_of assms the_Some_image)
```
```  1422
```
```  1423 lemma permutation_of_list_permutes [simp]:
```
```  1424   assumes "list_permutes xs A"
```
```  1425   shows "permutation_of_list xs permutes A"
```
```  1426     (is "?f permutes _")
```
```  1427 proof (rule permutes_subset[OF bij_imp_permutes])
```
```  1428   from assms show "set (map fst xs) \<subseteq> A"
```
```  1429     by (simp add: list_permutes_def)
```
```  1430   from assms have "inj_on (the \<circ> map_of xs) (set (map fst xs))" (is ?P)
```
```  1431     by (intro inj_on_map_of') (simp_all add: list_permutes_def)
```
```  1432   also have "?P \<longleftrightarrow> inj_on ?f (set (map fst xs))"
```
```  1433     by (intro inj_on_cong)
```
```  1434       (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
```
```  1435   finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))"
```
```  1436     by (rule inj_on_imp_bij_betw)
```
```  1437   also from assms have "?f ` set (map fst xs) = (the \<circ> map_of xs) ` set (map fst xs)"
```
```  1438     by (intro image_cong refl)
```
```  1439       (auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
```
```  1440   also from assms have "\<dots> = set (map fst xs)"
```
```  1441     by (subst image_map_of') (simp_all add: list_permutes_def)
```
```  1442   finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" .
```
```  1443 qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+
```
```  1444
```
```  1445 lemma eval_permutation_of_list [simp]:
```
```  1446   "permutation_of_list [] x = x"
```
```  1447   "x = x' \<Longrightarrow> permutation_of_list ((x',y)#xs) x = y"
```
```  1448   "x \<noteq> x' \<Longrightarrow> permutation_of_list ((x',y')#xs) x = permutation_of_list xs x"
```
```  1449   by (simp_all add: permutation_of_list_def)
```
```  1450
```
```  1451 lemma eval_inverse_permutation_of_list [simp]:
```
```  1452   "inverse_permutation_of_list [] x = x"
```
```  1453   "x = x' \<Longrightarrow> inverse_permutation_of_list ((y,x')#xs) x = y"
```
```  1454   "x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x"
```
```  1455   by (simp_all add: inverse_permutation_of_list.simps)
```
```  1456
```
```  1457 lemma permutation_of_list_id: "x \<notin> set (map fst xs) \<Longrightarrow> permutation_of_list xs x = x"
```
```  1458   by (induct xs) (auto simp: permutation_of_list_Cons)
```
```  1459
```
```  1460 lemma permutation_of_list_unique':
```
```  1461   "distinct (map fst xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
```
```  1462   by (induct xs) (force simp: permutation_of_list_Cons)+
```
```  1463
```
```  1464 lemma permutation_of_list_unique:
```
```  1465   "list_permutes xs A \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
```
```  1466   by (intro permutation_of_list_unique') (simp_all add: list_permutes_def)
```
```  1467
```
```  1468 lemma inverse_permutation_of_list_id:
```
```  1469   "x \<notin> set (map snd xs) \<Longrightarrow> inverse_permutation_of_list xs x = x"
```
```  1470   by (induct xs) auto
```
```  1471
```
```  1472 lemma inverse_permutation_of_list_unique':
```
```  1473   "distinct (map snd xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
```
```  1474   by (induct xs) (force simp: inverse_permutation_of_list.simps)+
```
```  1475
```
```  1476 lemma inverse_permutation_of_list_unique:
```
```  1477   "list_permutes xs A \<Longrightarrow> (x,y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
```
```  1478   by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def)
```
```  1479
```
```  1480 lemma inverse_permutation_of_list_correct:
```
```  1481   fixes A :: "'a set"
```
```  1482   assumes "list_permutes xs A"
```
```  1483   shows "inverse_permutation_of_list xs = inv (permutation_of_list xs)"
```
```  1484 proof (rule ext, rule sym, subst permutes_inv_eq)
```
```  1485   from assms show "permutation_of_list xs permutes A"
```
```  1486     by simp
```
```  1487   show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" for x
```
```  1488   proof (cases "x \<in> set (map snd xs)")
```
```  1489     case True
```
```  1490     then obtain y where "(y, x) \<in> set xs" by auto
```
```  1491     with assms show ?thesis
```
```  1492       by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique)
```
```  1493   next
```
```  1494     case False
```
```  1495     with assms show ?thesis
```
```  1496       by (auto simp: list_permutes_def inverse_permutation_of_list_id permutation_of_list_id)
```
```  1497   qed
```
```  1498 qed
```
```  1499
```
```  1500 end
```