# HG changeset patch
# User chaieb
# Date 1234197735 0
# Node ID cfab6a76aa13fbfb346d215eaa97e4e93f89e2e6
# Parent  018ac1fa1ed31914fc46d15a9b071cb201326d0d
Permutations, both general and specifically on finite sets.

diff -r 018ac1fa1ed3 -r cfab6a76aa13 src/HOL/Library/Permutations.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Permutations.thy	Mon Feb 09 16:42:15 2009 +0000
@@ -0,0 +1,862 @@
+(* Title:      Library/Permutations
+   ID:         $Id: 
+   Author:     Amine Chaieb, University of Cambridge
+*)
+
+header {* Permutations, both general and specifically on finite sets.*}
+
+theory Permutations
+imports Main Finite_Cartesian_Product Parity 
+begin
+
+  (* Why should I import Main just to solve the Typerep problem! *)
+
+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_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  apply auto apply metis done
+
+lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
+apply (simp add: Ball_def permutes_def Diff_iff) by 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:: nat 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: nat_less_induct)
+      fix n S assume H: "\<forall> m< n. \<forall>S. p permutes S \<longrightarrow> (\<forall>i\<in>S. p i \<le> i) \<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 arith}
+	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:: nat 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 {m..n}} = setsum (\<lambda>p. f(inv p)) {p. p permutes {m..n}}" (is "?lhs = ?rhs")
+proof-
+  let ?S = "{p . p permutes {m .. n}}"
+have th0: "inj_on inv ?S" 
+proof(auto simp add: inj_on_def)
+  fix q r
+  assume q: "q permutes {m .. n}" and r: "r permutes {m .. n}" 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 {m..n}"
+  shows "setsum f {p. p permutes {m..n}} =
+            setsum (\<lambda>p. f(q o p)) {p. p permutes {m..n}}" (is "?lhs = ?rhs")
+proof-
+  let ?S = "{p. p permutes {m..n}}"
+  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 {m..n}" and r:"r permutes {m..n}" 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 {m..n}"
+  shows "setsum f {p. p permutes {m..n}} =
+            setsum (\<lambda>p. f(p o q)) {p. p permutes {m..n}}" (is "?lhs = ?rhs")
+proof-
+  let ?S = "{p. p permutes {m..n}}"
+  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 {m..n}" and r:"r permutes {m..n}" 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"
+	apply (simp add: permutes_def pa qa o_def fun_upd_def swap_def id_def cong del: if_weak_cong)
+	apply (cases "a = b", auto)
+	by (cases "b = c", auto)
+      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