(* Title:      Library/Permutations

    Author:     Amine Chaieb, University of Cambridge

 *)

 header {* Permutations, both general and specifically on finite sets.*}

 theory Permutations

 imports Finite_Cartesian_Product Parity Fact

 begin

 definition permutes (infixr "permutes" 41) where

   "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"

 (* ------------------------------------------------------------------------- *)

 (* Transpositions.                                                           *)

 (* ------------------------------------------------------------------------- *)

 declare swap_self[simp]

 lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id"

   by (auto simp add: expand_fun_eq swap_def fun_upd_def)

 lemma swap_id_refl: "Fun.swap a a id = id" by simp

 lemma swap_id_sym: "Fun.swap a b id = Fun.swap b a id"

   by (rule ext, simp add: swap_def)

 lemma swap_id_idempotent[simp]: "Fun.swap a b id o Fun.swap a b id = id"

   by (rule ext, auto simp add: swap_def)

 lemma inv_unique_comp: assumes fg: "f o g = id" and gf: "g o f = id"

   shows "inv f = g"

   using fg gf inv_equality[of g f] by (auto simp add: expand_fun_eq)

 lemma inverse_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id"

   by (rule inv_unique_comp, simp_all)

 lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"

   by (simp add: swap_def)

 (* ------------------------------------------------------------------------- *)

 (* Basic consequences of the definition.                                     *)

 (* ------------------------------------------------------------------------- *)

 lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"

   unfolding permutes_def by metis

 lemma permutes_image: assumes pS: "p permutes S" shows "p ` S = S"

   using pS

   unfolding permutes_def

   apply -

   apply (rule set_ext)

   apply (simp add: image_iff)

   apply metis

   done

 lemma permutes_inj: "p permutes S ==> inj p "

   unfolding permutes_def inj_on_def by blast

 lemma permutes_surj: "p permutes s ==> surj p"

   unfolding permutes_def surj_def by metis

 lemma permutes_inv_o: assumes pS: "p permutes S"

   shows " p o inv p = id"

   and "inv p o p = id"

   using permutes_inj[OF pS] permutes_surj[OF pS]

   unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+

 lemma permutes_inverses:

   fixes p :: "'a \<Rightarrow> 'a"

   assumes pS: "p permutes S"

   shows "p (inv p x) = x"

   and "inv p (p x) = x"

   using permutes_inv_o[OF pS, unfolded expand_fun_eq o_def] by auto

 lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T ==> p permutes T"

   unfolding permutes_def by blast

 lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"

   unfolding expand_fun_eq permutes_def apply simp by metis

 lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"

   unfolding expand_fun_eq permutes_def apply simp by metis

 lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"

   unfolding permutes_def by simp

 lemma permutes_inv_eq: "p permutes S ==> inv p y = x \<longleftrightarrow> p x = y"

   unfolding permutes_def inv_onto_def apply auto

   apply (erule allE[where x=y])

   apply (erule allE[where x=y])

   apply (rule someI_ex) apply blast

   apply (rule some1_equality)

   apply blast

   apply blast

   done

 lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S ==> Fun.swap a b id permutes S"

   unfolding permutes_def swap_def fun_upd_def  by auto metis

 lemma permutes_superset:

   "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"

 by (simp add: Ball_def permutes_def Diff_iff) metis

 (* ------------------------------------------------------------------------- *)

 (* Group properties.                                                         *)

 (* ------------------------------------------------------------------------- *)

 lemma permutes_id: "id permutes S" unfolding permutes_def by simp

 lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S ==> q o p permutes S"

   unfolding permutes_def o_def by metis

 lemma permutes_inv: assumes pS: "p permutes S" shows "inv p permutes S"

   using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis

 lemma permutes_inv_inv: assumes pS: "p permutes S" shows "inv (inv p) = p"

   unfolding expand_fun_eq permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]

   by blast

 (* ------------------------------------------------------------------------- *)

 (* The number of permutations on a finite set.                               *)

 (* ------------------------------------------------------------------------- *)

 lemma permutes_insert_lemma:

   assumes pS: "p permutes (insert a S)"

   shows "Fun.swap a (p a) id o p permutes S"

   apply (rule permutes_superset[where S = "insert a S"])

   apply (rule permutes_compose[OF pS])

   apply (rule permutes_swap_id, simp)

   using permutes_in_image[OF pS, of a] apply simp

   apply (auto simp add: Ball_def Diff_iff swap_def)

   done

 lemma permutes_insert: "{p. p permutes (insert a S)} =

         (\<lambda>(b,p). Fun.swap a b id o p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"

 proof-

   {fix p

     {assume pS: "p permutes insert a S"

       let ?b = "p a"

       let ?q = "Fun.swap a (p a) id o p"

       have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp

       have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp

       from permutes_insert_lemma[OF pS] th0 th1

       have "\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S" by blast}

     moreover

     {fix b q assume bq: "p = Fun.swap a b id o q" "b \<in> insert a S" "q permutes S"

       from permutes_subset[OF bq(3), of "insert a S"]

       have qS: "q permutes insert a S" by auto

       have aS: "a \<in> insert a S" by simp

       from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]]

       have "p permutes insert a S"  by simp }

     ultimately have "p permutes insert a S \<longleftrightarrow> (\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S)" by blast}

   thus ?thesis by auto

 qed

 lemma hassize_insert: "a \<notin> F \<Longrightarrow> insert a F hassize n \<Longrightarrow> F hassize (n - 1)"

   by (auto simp add: hassize_def)

 lemma hassize_permutations: assumes Sn: "S hassize n"

   shows "{p. p permutes S} hassize (fact n)"

 proof-

   from Sn have fS:"finite S" by (simp add: hassize_def)

   have "\<forall>n. (S hassize n) \<longrightarrow> ({p. p permutes S} hassize (fact n))"

   proof(rule finite_induct[where F = S])

     from fS show "finite S" .

   next

     show "\<forall>n. ({} hassize n) \<longrightarrow> ({p. p permutes {}} hassize fact n)"

       by (simp add: hassize_def permutes_empty)

   next

     fix x F

     assume fF: "finite F" and xF: "x \<notin> F"

       and H: "\<forall>n. (F hassize n) \<longrightarrow> ({p. p permutes F} hassize fact n)"

     {fix n assume H0: "insert x F hassize n"

       let ?xF = "{p. p permutes insert x F}"

       let ?pF = "{p. p permutes F}"

       let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"

       let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"

       from permutes_insert[of x F]

       have xfgpF': "?xF = ?g ` ?pF'" .

       from hassize_insert[OF xF H0] have Fs: "F hassize (n - 1)" .

       from H Fs have pFs: "?pF hassize fact (n - 1)" by blast

       hence pF'f: "finite ?pF'" using H0 unfolding hassize_def

         apply (simp only: Collect_split Collect_mem_eq)

         apply (rule finite_cartesian_product)

         apply simp_all

         done

       have ginj: "inj_on ?g ?pF'"

       proof-

         {

           fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'"

             and eq: "?g (b,p) = ?g (c,q)"

           from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" "p permutes F" "q permutes F" by auto

           from ths(4) xF eq have "b = ?g (b,p) x" unfolding permutes_def

             by (auto simp add: swap_def fun_upd_def expand_fun_eq)

           also have "\<dots> = ?g (c,q) x" using ths(5) xF eq

             by (auto simp add: swap_def fun_upd_def expand_fun_eq)

           also have "\<dots> = c"using ths(5) xF unfolding permutes_def

             by (auto simp add: swap_def fun_upd_def expand_fun_eq)

           finally have bc: "b = c" .

           hence "Fun.swap x b id = Fun.swap x c id" by simp

           with eq have "Fun.swap x b id o p = Fun.swap x b id o q" by simp

           hence "Fun.swap x b id o (Fun.swap x b id o p) = Fun.swap x b id o (Fun.swap x b id o q)" by simp

           hence "p = q" by (simp add: o_assoc)

           with bc have "(b,p) = (c,q)" by simp }

         thus ?thesis  unfolding inj_on_def by blast

       qed

       from xF H0 have n0: "n \<noteq> 0 " by (auto simp add: hassize_def)

       hence "\<exists>m. n = Suc m" by arith

       then obtain m where n[simp]: "n = Suc m" by blast

       from pFs H0 have xFc: "card ?xF = fact n"

         unfolding xfgpF' card_image[OF ginj] hassize_def

         apply (simp only: Collect_split Collect_mem_eq card_cartesian_product)

         by simp

       from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp

       have "?xF hassize fact n"

         using xFf xFc

         unfolding hassize_def  xFf by blast }

     thus "\<forall>n. (insert x F hassize n) \<longrightarrow> ({p. p permutes insert x F} hassize fact n)"

       by blast

   qed

   with Sn show ?thesis by blast

 qed

 lemma finite_permutations: "finite S ==> finite {p. p permutes S}"

   using hassize_permutations[of S] unfolding hassize_def by blast

 (* ------------------------------------------------------------------------- *)

 (* Permutations of index set for iterated operations.                        *)

 (* ------------------------------------------------------------------------- *)

 lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"

   shows "fold_image times f z S = fold_image times (f o p) z S"

   using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]

   unfolding permutes_image[OF pS] .

 lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"

   shows "fold_image plus f z S = fold_image plus (f o p) z S"

 proof-

   interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc)

     apply (simp add: add_commute) done

   from fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]

   show ?thesis

   unfolding permutes_image[OF pS] .

 qed

 lemma setsum_permute: assumes pS: "p permutes S"

   shows "setsum f S = setsum (f o p) S"

   unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp

 lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}"

   shows "setsum f {m .. n} = setsum (f o p) {m .. n}"

   using setsum_permute[OF pS, of f ] pS by blast

 lemma setprod_permute: assumes pS: "p permutes S"

   shows "setprod f S = setprod (f o p) S"

   unfolding setprod_def

   using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp

 lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}"

   shows "setprod f {m .. n} = setprod (f o p) {m .. n}"

   using setprod_permute[OF pS, of f ] pS by blast

 (* ------------------------------------------------------------------------- *)

 (* Various combinations of transpositions with 2, 1 and 0 common elements.   *)

 (* ------------------------------------------------------------------------- *)

 lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>  Fun.swap a b id o Fun.swap a c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)

 lemma swap_id_common': "~(a = b) \<Longrightarrow> ~(a = c) \<Longrightarrow> Fun.swap a c id o Fun.swap b c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)

 lemma swap_id_independent: "~(a = c) \<Longrightarrow> ~(a = d) \<Longrightarrow> ~(b = c) \<Longrightarrow> ~(b = d) ==> Fun.swap a b id o Fun.swap c d id = Fun.swap c d id o Fun.swap a b id"

   by (simp add: swap_def expand_fun_eq)

 (* ------------------------------------------------------------------------- *)

 (* Permutations as transposition sequences.                                  *)

 (* ------------------------------------------------------------------------- *)

 inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where

   id[simp]: "swapidseq 0 id"

 | comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id o p)"

 declare id[unfolded id_def, simp]

 definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"

 (* ------------------------------------------------------------------------- *)

 (* Some closure properties of the set of permutations, with lengths.         *)

 (* ------------------------------------------------------------------------- *)

 lemma permutation_id[simp]: "permutation id"unfolding permutation_def

   by (rule exI[where x=0], simp)

 declare permutation_id[unfolded id_def, simp]

 lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"

   apply clarsimp

   using comp_Suc[of 0 id a b] by simp

 lemma permutation_swap_id: "permutation (Fun.swap a b id)"

   apply (cases "a=b", simp_all)

   unfolding permutation_def using swapidseq_swap[of a b] by blast

 lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q ==> swapidseq (n + m) (p o q)"

   proof (induct n p arbitrary: m q rule: swapidseq.induct)

     case (id m q) thus ?case by simp

   next

     case (comp_Suc n p a b m q)

     have th: "Suc n + m = Suc (n + m)" by arith

     show ?case unfolding th o_assoc[symmetric]

       apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems]  comp_Suc.hyps(3) by blast+

 qed

 lemma permutation_compose: "permutation p \<Longrightarrow> permutation q ==> permutation(p o q)"

   unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis

 lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b ==> swapidseq (Suc n) (p o Fun.swap a b id)"

   apply (induct n p rule: swapidseq.induct)

   using swapidseq_swap[of a b]

   by (auto simp add: o_assoc[symmetric] intro: swapidseq.comp_Suc)

 lemma swapidseq_inverse_exists: "swapidseq n p ==> \<exists>q. swapidseq n q \<and> p o q = id \<and> q o p = id"

 proof(induct n p rule: swapidseq.induct)

   case id  thus ?case by (rule exI[where x=id], simp)

 next

   case (comp_Suc n p a b)

   from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast

   let ?q = "q o Fun.swap a b id"

   note H = comp_Suc.hyps

   from swapidseq_swap[of a b] H(3)  have th0: "swapidseq 1 (Fun.swap a b id)" by simp

   from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp

   have "Fun.swap a b id o p o ?q = Fun.swap a b id o (p o q) o Fun.swap a b id" by (simp add: o_assoc)

   also have "\<dots> = id" by (simp add: q(2))

   finally have th2: "Fun.swap a b id o p o ?q = id" .

   have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc)

   hence "?q \<circ> (Fun.swap a b id \<circ> p) = id" by (simp add: q(3))

   with th1 th2 show ?case by blast

 qed

 lemma swapidseq_inverse: assumes H: "swapidseq n p" shows "swapidseq n (inv p)"

   using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto

 lemma permutation_inverse: "permutation p ==> permutation (inv p)"

   using permutation_def swapidseq_inverse by blast

 (* ------------------------------------------------------------------------- *)

 (* The identity map only has even transposition sequences.                   *)

 (* ------------------------------------------------------------------------- *)

 lemma symmetry_lemma:"(\<And>a b c d. P a b c d ==> P a b d c) \<Longrightarrow>

    (\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> (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) ==> P a b c d)

    ==> (\<And>a b c d. a \<noteq> b --> c \<noteq> d \<longrightarrow>  P a b c d)" by metis

 lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or>

   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)"

 proof-

   assume H: "a\<noteq>b" "c\<noteq>d"

 have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>

 (  Fun.swap a b id o Fun.swap c d id = id \<or>

   (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))"

   apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])

   apply (simp_all only: swapid_sym)

   apply (case_tac "a = c \<and> b = d", clarsimp simp only: swapid_sym swap_id_idempotent)

   apply (case_tac "a = c \<and> b \<noteq> d")

   apply (rule disjI2)

   apply (rule_tac x="b" in exI)

   apply (rule_tac x="d" in exI)

   apply (rule_tac x="b" in exI)

   apply (clarsimp simp add: expand_fun_eq swap_def)

   apply (case_tac "a \<noteq> c \<and> b = d")

   apply (rule disjI2)

   apply (rule_tac x="c" in exI)

   apply (rule_tac x="d" in exI)

   apply (rule_tac x="c" in exI)

   apply (clarsimp simp add: expand_fun_eq swap_def)

   apply (rule disjI2)

   apply (rule_tac x="c" in exI)

   apply (rule_tac x="d" in exI)

   apply (rule_tac x="b" in exI)

   apply (clarsimp simp add: expand_fun_eq swap_def)

   done

 with H show ?thesis by metis

 qed

 lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"

   using swapidseq.cases[of 0 p "p = id"]

   by auto

 lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> (n=0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id o q \<and> swapidseq m q \<and> a\<noteq> b))"

   apply (rule iffI)

   apply (erule swapidseq.cases[of n p])

   apply simp

   apply (rule disjI2)

   apply (rule_tac x= "a" in exI)

   apply (rule_tac x= "b" in exI)

   apply (rule_tac x= "pa" in exI)

   apply (rule_tac x= "na" in exI)

   apply simp

   apply auto

   apply (rule comp_Suc, simp_all)

   done

 lemma fixing_swapidseq_decrease:

   assumes spn: "swapidseq n p" and ab: "a\<noteq>b" and pa: "(Fun.swap a b id o p) a = a"

   shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id o p)"

   using spn ab pa

 proof(induct n arbitrary: p a b)

   case 0 thus ?case by (auto simp add: swap_def fun_upd_def)

 next

   case (Suc n p a b)

   from Suc.prems(1) swapidseq_cases[of "Suc n" p] obtain

     c d q m where cdqm: "Suc n = Suc m" "p = Fun.swap c d id o q" "swapidseq m q" "c \<noteq> d" "n = m"

     by auto

   {assume H: "Fun.swap a b id o Fun.swap c d id = id"

     have ?case apply (simp only: cdqm o_assoc H)

       by (simp add: cdqm)}

   moreover

   { fix x y z

     assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y"

       "Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id"

     from H have az: "a \<noteq> z" by simp

     {fix h have "(Fun.swap x y id o h) a = a \<longleftrightarrow> h a = a"

       using H by (simp add: swap_def)}

     note th3 = this

     from cdqm(2) have "Fun.swap a b id o p = Fun.swap a b id o (Fun.swap c d id o q)" by simp

     hence "Fun.swap a b id o p = Fun.swap x y id o (Fun.swap a z id o q)" by (simp add: o_assoc H)

     hence "(Fun.swap a b id o p) a = (Fun.swap x y id o (Fun.swap a z id o q)) a" by simp

     hence "(Fun.swap x y id o (Fun.swap a z id o q)) a  = a" unfolding Suc by metis

     hence th1: "(Fun.swap a z id o q) a = a" unfolding th3 .

     from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1]

     have th2: "swapidseq (n - 1) (Fun.swap a z id o q)" "n \<noteq> 0" by blast+

     have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto

     have ?case unfolding cdqm(2) H o_assoc th

       apply (simp only: Suc_not_Zero simp_thms o_assoc[symmetric])

       apply (rule comp_Suc)

       using th2 H apply blast+

       done}

   ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis

 qed

 lemma swapidseq_identity_even:

   assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" shows "even n"

   using `swapidseq n id`

 proof(induct n rule: nat_less_induct)

   fix n

   assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)"

   {assume "n = 0" hence "even n" by arith}

   moreover

   {fix a b :: 'a and q m

     assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"

     from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]

     have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" by auto

     from h m have mn: "m - 1 < n" by arith

     from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" apply arith done}

   ultimately show "even n" using H(2)[unfolded swapidseq_cases[of n id]] by auto

 qed

 (* ------------------------------------------------------------------------- *)

 (* Therefore we have a welldefined notion of parity.                         *)

 (* ------------------------------------------------------------------------- *)

 definition "evenperm p = even (SOME n. swapidseq n p)"

 lemma swapidseq_even_even: assumes

   m: "swapidseq m p" and n: "swapidseq n p"

   shows "even m \<longleftrightarrow> even n"

 proof-

   from swapidseq_inverse_exists[OF n]

   obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast

   from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]]

   show ?thesis by arith

 qed

 lemma evenperm_unique: assumes p: "swapidseq n p" and n:"even n = b"

   shows "evenperm p = b"

   unfolding n[symmetric] evenperm_def

   apply (rule swapidseq_even_even[where p = p])

   apply (rule someI[where x = n])

   using p by blast+

 (* ------------------------------------------------------------------------- *)

 (* And it has the expected composition properties.                           *)

 (* ------------------------------------------------------------------------- *)

 lemma evenperm_id[simp]: "evenperm id = True"

   apply (rule evenperm_unique[where n = 0]) by simp_all

 lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"

 apply (rule evenperm_unique[where n="if a = b then 0 else 1"])

 by (simp_all add: swapidseq_swap)

 lemma evenperm_comp:

   assumes p: "permutation p" and q:"permutation q"

   shows "evenperm (p o q) = (evenperm p = evenperm q)"

 proof-

   from p q obtain

     n m where n: "swapidseq n p" and m: "swapidseq m q"

     unfolding permutation_def by blast

   note nm =  swapidseq_comp_add[OF n m]

   have th: "even (n + m) = (even n \<longleftrightarrow> even m)" by arith

   from evenperm_unique[OF n refl] evenperm_unique[OF m refl]

     evenperm_unique[OF nm th]

   show ?thesis by blast

 qed

 lemma evenperm_inv: assumes p: "permutation p"

   shows "evenperm (inv p) = evenperm p"

 proof-

   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast

   from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]]

   show ?thesis .

 qed

 (* ------------------------------------------------------------------------- *)

 (* A more abstract characterization of permutations.                         *)

 (* ------------------------------------------------------------------------- *)

 lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"

   unfolding bij_def inj_on_def surj_def

   apply auto

   apply metis

   apply metis

   done

 lemma permutation_bijective:

   assumes p: "permutation p"

   shows "bij p"

 proof-

   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast

   from swapidseq_inverse_exists[OF n] obtain q where

     q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast

   thus ?thesis unfolding bij_iff  apply (auto simp add: expand_fun_eq) apply metis done

 qed

 lemma permutation_finite_support: assumes p: "permutation p"

   shows "finite {x. p x \<noteq> x}"

 proof-

   from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast

   from n show ?thesis

   proof(induct n p rule: swapidseq.induct)

     case id thus ?case by simp

   next

     case (comp_Suc n p a b)

     let ?S = "insert a (insert b {x. p x \<noteq> x})"

     from comp_Suc.hyps(2) have fS: "finite ?S" by simp

     from `a \<noteq> b` have th: "{x. (Fun.swap a b id o p) x \<noteq> x} \<subseteq> ?S"

       by (auto simp add: swap_def)

     from finite_subset[OF th fS] show ?case  .

 qed

 qed

 lemma bij_inv_eq_iff: "bij p ==> x = inv p y \<longleftrightarrow> p x = y"

   using surj_f_inv_f[of p] inv_f_f[of f] by (auto simp add: bij_def)

 lemma bij_swap_comp:

   assumes bp: "bij p" shows "Fun.swap a b id o p = Fun.swap (inv p a) (inv p b) p"

   using surj_f_inv_f[OF bij_is_surj[OF bp]]

   by (simp add: expand_fun_eq swap_def bij_inv_eq_iff[OF bp])

 lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id o p)"

 proof-

   assume H: "bij p"

   show ?thesis

     unfolding bij_swap_comp[OF H] bij_swap_iff

     using H .

 qed

 lemma permutation_lemma:

   assumes fS: "finite S" and p: "bij p" and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"

   shows "permutation p"

 using fS p pS

 proof(induct S arbitrary: p rule: finite_induct)

   case (empty p) thus ?case by simp

 next

   case (insert a F p)

   let ?r = "Fun.swap a (p a) id o p"

   let ?q = "Fun.swap a (p a) id o ?r "

   have raa: "?r a = a" by (simp add: swap_def)

   from bij_swap_ompose_bij[OF insert(4)]

   have br: "bij ?r"  .

   from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"

     apply (clarsimp simp add: swap_def)

     apply (erule_tac x="x" in allE)

     apply auto

     unfolding bij_iff apply metis

     done

   from insert(3)[OF br th]

   have rp: "permutation ?r" .

   have "permutation ?q" by (simp add: permutation_compose permutation_swap_id rp)

   thus ?case by (simp add: o_assoc)

 qed

 lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"

   (is "?lhs \<longleftrightarrow> ?b \<and> ?f")

 proof

   assume p: ?lhs

   from p permutation_bijective permutation_finite_support show "?b \<and> ?f" by auto

 next

   assume bf: "?b \<and> ?f"

   hence bf: "?f" "?b" by blast+

   from permutation_lemma[OF bf] show ?lhs by blast

 qed

 lemma permutation_inverse_works: assumes p: "permutation p"

   shows "inv p o p = id" "p o inv p = id"

 using permutation_bijective[OF p] surj_iff bij_def inj_iff by auto

 lemma permutation_inverse_compose:

   assumes p: "permutation p" and q: "permutation q"

   shows "inv (p o q) = inv q o inv p"

 proof-

   note ps = permutation_inverse_works[OF p]

   note qs = permutation_inverse_works[OF q]

   have "p o q o (inv q o inv p) = p o (q o inv q) o inv p" by (simp add: o_assoc)

   also have "\<dots> = id" by (simp add: ps qs)

   finally have th0: "p o q o (inv q o inv p) = id" .

   have "inv q o inv p o (p o q) = inv q o (inv p o p) o q" by (simp add: o_assoc)

   also have "\<dots> = id" by (simp add: ps qs)

   finally have th1: "inv q o inv p o (p o q) = id" .

   from inv_unique_comp[OF th0 th1] show ?thesis .

 qed

 (* ------------------------------------------------------------------------- *)

 (* Relation to "permutes".                                                   *)

 (* ------------------------------------------------------------------------- *)

 lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"

 unfolding permutation permutes_def bij_iff[symmetric]

 apply (rule iffI, clarify)

 apply (rule exI[where x="{x. p x \<noteq> x}"])

 apply simp

 apply clarsimp

 apply (rule_tac B="S" in finite_subset)

 apply auto

 done

 (* ------------------------------------------------------------------------- *)

 (* Hence a sort of induction principle composing by swaps.                   *)

 (* ------------------------------------------------------------------------- *)

 lemma permutes_induct: "finite S \<Longrightarrow>  P id  \<Longrightarrow> (\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p ==> P (Fun.swap a b id o p))

          ==> (\<And>p. p permutes S ==> P p)"

 proof(induct S rule: finite_induct)

   case empty thus ?case by auto

 next

   case (insert x F p)

   let ?r = "Fun.swap x (p x) id o p"

   let ?q = "Fun.swap x (p x) id o ?r"

   have qp: "?q = p" by (simp add: o_assoc)

   from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast

   from permutes_in_image[OF insert.prems(3), of x]

   have pxF: "p x \<in> insert x F" by simp

   have xF: "x \<in> insert x F" by simp

   have rp: "permutation ?r"

     unfolding permutation_permutes using insert.hyps(1)

       permutes_insert_lemma[OF insert.prems(3)] by blast

   from insert.prems(2)[OF xF pxF Pr Pr rp]

   show ?case  unfolding qp .

 qed

 (* ------------------------------------------------------------------------- *)

 (* Sign of a permutation as a real number.                                   *)

 (* ------------------------------------------------------------------------- *)

 definition "sign p = (if evenperm p then (1::int) else -1)"

 lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def)

 lemma sign_id: "sign id = 1" by (simp add: sign_def)

 lemma sign_inverse: "permutation p ==> sign (inv p) = sign p"

   by (simp add: sign_def evenperm_inv)

 lemma sign_compose: "permutation p \<Longrightarrow> permutation q ==> sign (p o q) = sign(p) * sign(q)" by (simp add: sign_def evenperm_comp)

 lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"

   by (simp add: sign_def evenperm_swap)

 lemma sign_idempotent: "sign p * sign p = 1" by (simp add: sign_def)

 (* ------------------------------------------------------------------------- *)

 (* More lemmas about permutations.                                           *)

 (* ------------------------------------------------------------------------- *)

 lemma permutes_natset_le:

   assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i <= i" shows "p = id"

 proof-

   {fix n

     have "p n = n"

       using p le

     proof(induct n arbitrary: S rule: less_induct)

       fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m"

         "p permutes S" "\<forall>i \<in>S. p i \<le> i"

       {assume "n \<notin> S"

         with H(2) have "p n = n" unfolding permutes_def by metis}

       moreover

       {assume ns: "n \<in> S"

         from H(3)  ns have "p n < n \<or> p n = n" by auto

         moreover{assume h: "p n < n"

           from H h have "p (p n) = p n" by metis

           with permutes_inj[OF H(2)] have "p n = n" unfolding inj_on_def by blast

           with h have False by simp}

         ultimately have "p n = n" by blast }

       ultimately show "p n = n"  by blast

     qed}

   thus ?thesis by (auto simp add: expand_fun_eq)

 qed

 lemma permutes_natset_ge:

   assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i \<ge> i" shows "p = id"

 proof-

   {fix i assume i: "i \<in> S"

     from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" by simp

     with le have "p (inv p i) \<ge> inv p i" by blast

     with permutes_inverses[OF p] have "i \<ge> inv p i" by simp}

   then have th: "\<forall>i\<in>S. inv p i \<le> i"  by blast

   from permutes_natset_le[OF permutes_inv[OF p] th]

   have "inv p = inv id" by simp

   then show ?thesis

     apply (subst permutes_inv_inv[OF p, symmetric])

     apply (rule inv_unique_comp)

     apply simp_all

     done

 qed

 lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"

 apply (rule set_ext)

 apply auto

   using permutes_inv_inv permutes_inv apply auto

   apply (rule_tac x="inv x" in exI)

   apply auto

   done

 lemma image_compose_permutations_left:

   assumes q: "q permutes S" shows "{q o p | p. p permutes S} = {p . p permutes S}"

 apply (rule set_ext)

 apply auto

 apply (rule permutes_compose)

 using q apply auto

 apply (rule_tac x = "inv q o x" in exI)

 by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)

 lemma image_compose_permutations_right:

   assumes q: "q permutes S"

   shows "{p o q | p. p permutes S} = {p . p permutes S}"

 apply (rule set_ext)

 apply auto

 apply (rule permutes_compose)

 using q apply auto

 apply (rule_tac x = "x o inv q" in exI)

 by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o o_assoc[symmetric])

 lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} ==> 1 <= p i \<and> p i <= n"

 apply (simp add: permutes_def)

 apply metis

 done

 term setsum

 lemma setsum_permutations_inverse: "setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs")

 proof-

   let ?S = "{p . p permutes S}"

 have th0: "inj_on inv ?S"

 proof(auto simp add: inj_on_def)

   fix q r

   assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r"

   hence "inv (inv q) = inv (inv r)" by simp

   with permutes_inv_inv[OF q] permutes_inv_inv[OF r]

   show "q = r" by metis

 qed

   have th1: "inv ` ?S = ?S" using image_inverse_permutations by blast

   have th2: "?rhs = setsum (f o inv) ?S" by (simp add: o_def)

   from setsum_reindex[OF th0, of f]  show ?thesis unfolding th1 th2 .

 qed

 lemma setum_permutations_compose_left:

   assumes q: "q permutes S"

   shows "setsum f {p. p permutes S} =

             setsum (\<lambda>p. f(q o p)) {p. p permutes S}" (is "?lhs = ?rhs")

 proof-

   let ?S = "{p. p permutes S}"

   have th0: "?rhs = setsum (f o (op o q)) ?S" by (simp add: o_def)

   have th1: "inj_on (op o q) ?S"

     apply (auto simp add: inj_on_def)

   proof-

     fix p r

     assume "p permutes S" and r:"r permutes S" and rp: "q \<circ> p = q \<circ> r"

     hence "inv q o q o p = inv q o q o r" by (simp add: o_assoc[symmetric])

     with permutes_inj[OF q, unfolded inj_iff]

     show "p = r" by simp

   qed

   have th3: "(op o q) ` ?S = ?S" using image_compose_permutations_left[OF q] by auto

   from setsum_reindex[OF th1, of f]

   show ?thesis unfolding th0 th1 th3 .

 qed

 lemma sum_permutations_compose_right:

   assumes q: "q permutes S"

   shows "setsum f {p. p permutes S} =

             setsum (\<lambda>p. f(p o q)) {p. p permutes S}" (is "?lhs = ?rhs")

 proof-

   let ?S = "{p. p permutes S}"

   have th0: "?rhs = setsum (f o (\<lambda>p. p o q)) ?S" by (simp add: o_def)

   have th1: "inj_on (\<lambda>p. p o q) ?S"

     apply (auto simp add: inj_on_def)

   proof-

     fix p r

     assume "p permutes S" and r:"r permutes S" and rp: "p o q = r o q"

     hence "p o (q o inv q)  = r o (q o inv q)" by (simp add: o_assoc)

     with permutes_surj[OF q, unfolded surj_iff]

     show "p = r" by simp

   qed

   have th3: "(\<lambda>p. p o q) ` ?S = ?S" using image_compose_permutations_right[OF q] by auto

   from setsum_reindex[OF th1, of f]

   show ?thesis unfolding th0 th1 th3 .

 qed

 (* ------------------------------------------------------------------------- *)

 (* Sum over a set of permutations (could generalize to iteration).           *)

 (* ------------------------------------------------------------------------- *)

 lemma setsum_over_permutations_insert:

   assumes fS: "finite S" and aS: "a \<notin> S"

   shows "setsum f {p. p permutes (insert a S)} = setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id o q)) {p. p permutes S}) (insert a S)"

 proof-

   have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id o p)) = f o (\<lambda>(b,p). Fun.swap a b id o p)"

     by (simp add: expand_fun_eq)

   have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" by blast

   have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" by blast

   show ?thesis

     unfolding permutes_insert

     unfolding setsum_cartesian_product

     unfolding  th1[symmetric]

     unfolding th0

   proof(rule setsum_reindex)

     let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"

     let ?P = "{p. p permutes S}"

     {fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S"

       and p: "p permutes S" and q: "q permutes S"

       and eq: "Fun.swap a b id o p = Fun.swap a c id o q"

       from p q aS have pa: "p a = a" and qa: "q a = a"

         unfolding permutes_def by metis+

       from eq have "(Fun.swap a b id o p) a  = (Fun.swap a c id o q) a" by simp

       hence bc: "b = c"

         by (simp add: permutes_def pa qa o_def fun_upd_def swap_def id_def cong del: if_weak_cong split: split_if_asm)

       from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o p) = (\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o q)" by simp

       hence "p = q" unfolding o_assoc swap_id_idempotent

         by (simp add: o_def)

       with bc have "b = c \<and> p = q" by blast

     }

     then show "inj_on ?f (insert a S \<times> ?P)"

       unfolding inj_on_def

       apply clarify by metis

   qed

 qed

 end