--- /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