src/HOL/Library/Permutations.thy
 author bulwahn Thu, 08 Dec 2016 17:22:51 +0100 changeset 64543 6b13586ef1a2 parent 64284 f3b905b2eee2 child 64966 d53d7ca3303e permissions -rw-r--r--
remove typo in bij_swap_compose_bij theorem name; tune proof
```
(*  Title:      HOL/Library/Permutations.thy
Author:     Amine Chaieb, University of Cambridge
*)

section \<open>Permutations, both general and specifically on finite sets.\<close>

theory Permutations
imports Binomial Multiset Disjoint_Sets
begin

subsection \<open>Transpositions\<close>

lemma swap_id_idempotent [simp]:
"Fun.swap a b id \<circ> Fun.swap a b id = id"
by (rule ext, auto simp add: Fun.swap_def)

lemma inv_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)"

lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y"
using surj_f_inv_f[of p] by (auto simp add: bij_def)

lemma bij_swap_comp:
assumes bp: "bij p"
shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
using surj_f_inv_f[OF bij_is_surj[OF bp]]
by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp])

lemma bij_swap_compose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)"
proof -
assume H: "bij p"
show ?thesis
unfolding bij_swap_comp[OF H] bij_swap_iff
using H .
qed

subsection \<open>Basic consequences of the definition\<close>

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)"

lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
unfolding permutes_def by metis

lemma permutes_not_in:
assumes "f permutes S" "x \<notin> S" shows "f x = x"
using assms by (auto simp: permutes_def)

lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S"
unfolding permutes_def
apply (rule set_eqI)
apply metis
done

lemma permutes_inj: "p permutes S \<Longrightarrow> inj p"
unfolding permutes_def inj_on_def by blast

lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A"
unfolding permutes_def inj_on_def by auto

lemma permutes_surj: "p permutes s \<Longrightarrow> surj p"
unfolding permutes_def surj_def by metis

lemma permutes_bij: "p permutes s \<Longrightarrow> bij p"
unfolding bij_def by (metis permutes_inj permutes_surj)

lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S"
by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI)

lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S"
unfolding permutes_def bij_betw_def inj_on_def
by auto (metis image_iff)+

lemma permutes_inv_o:
assumes pS: "p permutes S"
shows "p \<circ> inv p = id"
and "inv p \<circ> 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 fun_eq_iff o_def] by auto

lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T"
unfolding permutes_def by blast

lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
unfolding fun_eq_iff permutes_def by simp metis

lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
unfolding fun_eq_iff permutes_def by simp 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 \<Longrightarrow> 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 \<Longrightarrow> Fun.swap a b id permutes S"
unfolding permutes_def Fun.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) metis

(* Next three lemmas contributed by Lukas Bulwahn *)
lemma permutes_bij_inv_into:
fixes A :: "'a set" and B :: "'b set"
assumes "p permutes A"
assumes "bij_betw f A B"
shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
proof (rule bij_imp_permutes)
have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
using assms by (auto simp add: permutes_imp_bij bij_betw_inv_into)
from this have "bij_betw (f o p o inv_into A f) B B" by (simp add: bij_betw_trans)
from this show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B"
by (subst bij_betw_cong[where g="f o p o inv_into A f"]) auto
next
fix x
assume "x \<notin> B"
from this show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto
qed

lemma permutes_image_mset:
assumes "p permutes A"
shows "image_mset p (mset_set A) = mset_set A"
using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image)

lemma permutes_implies_image_mset_eq:
assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)"
shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
proof -
have "f x = f' (p x)" if x: "x \<in># mset_set A" for x
using assms(2)[of x] x by (cases "finite A") auto
from this have "image_mset f (mset_set A) = image_mset (f' o p) (mset_set A)"
using assms by (auto intro!: image_mset_cong)
also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
also have "\<dots> = image_mset f' (mset_set A)"
proof -
from assms have "image_mset p (mset_set A) = mset_set A"
using permutes_image_mset by blast
from this show ?thesis by simp
qed
finally show ?thesis ..
qed

subsection \<open>Group properties\<close>

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

lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> 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 fun_eq_iff permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
by blast

lemma permutes_invI:
assumes perm: "p permutes S"
and inv:  "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
shows   "inv p = p'"
proof
fix x show "inv p x = p' x"
proof (cases "x \<in> S")
assume [simp]: "x \<in> S"
from assms have "p' x = p' (p (inv p x))" by (simp add: permutes_inverses)
also from permutes_inv[OF perm]
have "\<dots> = inv p x" by (subst inv) (simp_all add: permutes_in_image)
finally show "inv p x = p' x" ..
qed (insert permutes_inv[OF perm], simp_all add: outside permutes_not_in)
qed

lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])

subsection \<open>The number of permutations on a finite set\<close>

lemma permutes_insert_lemma:
assumes pS: "p permutes (insert a S)"
shows "Fun.swap a (p a) id \<circ> 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 Fun.swap_def)
done

lemma permutes_insert: "{p. p permutes (insert a S)} =
(\<lambda>(b,p). Fun.swap a b id \<circ> 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 \<circ> p"
have th0: "p = Fun.swap a ?b id \<circ> ?q"
unfolding fun_eq_iff 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 \<circ> 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 \<circ> 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 \<circ> q \<and> b \<in> insert a S \<and> q permutes S)"
by blast
}
then show ?thesis
by auto
qed

lemma card_permutations:
assumes Sn: "card S = n"
and fS: "finite S"
shows "card {p. p permutes S} = fact n"
using fS Sn
proof (induct arbitrary: n)
case empty
then show ?case by simp
next
case (insert x F)
{
fix n
assume H0: "card (insert x F) = 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'" .
have Fs: "card F = n - 1"
using \<open>x \<notin> F\<close> H0 \<open>finite F\<close> by auto
from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
using \<open>finite F\<close> by auto
then have "finite ?pF"
by (auto intro: card_ge_0_finite)
then have pF'f: "finite ?pF'"
using H0 \<open>finite F\<close>
apply (simp only: Collect_case_prod 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'"
assume cq: "(c,q) \<in> ?pF'"
assume 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) \<open>x \<notin> F\<close> eq have "b = ?g (b,p) x"
unfolding permutes_def
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff)
also have "\<dots> = ?g (c,q) x"
using ths(5) \<open>x \<notin> F\<close> eq
by (auto simp add: swap_def fun_upd_def fun_eq_iff)
also have "\<dots> = c"
using ths(5) \<open>x \<notin> F\<close>
unfolding permutes_def
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff)
finally have bc: "b = c" .
then have "Fun.swap x b id = Fun.swap x c id"
by simp
with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q"
by simp
then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) =
Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)"
by simp
then have "p = q"
with bc have "(b, p) = (c, q)"
by simp
}
then show ?thesis
unfolding inj_on_def by blast
qed
from \<open>x \<notin> F\<close> H0 have n0: "n \<noteq> 0"
using \<open>finite F\<close> by auto
then have "\<exists>m. n = Suc m"
by presburger
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]
using \<open>finite F\<close> \<open>finite ?pF\<close>
apply (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product)
apply simp
done
from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF"
unfolding xfgpF' by simp
have "card ?xF = fact n"
using xFf xFc unfolding xFf by blast
}
then show ?case
using insert by simp
qed

lemma finite_permutations:
assumes fS: "finite S"
shows "finite {p. p permutes S}"
using card_permutations[OF refl fS]
by (auto intro: card_ge_0_finite)

subsection \<open>Permutations of index set for iterated operations\<close>

lemma (in comm_monoid_set) permute:
assumes "p permutes S"
shows "F g S = F (g \<circ> p) S"
proof -
from \<open>p permutes S\<close> have "inj p"
by (rule permutes_inj)
then have "inj_on p S"
by (auto intro: subset_inj_on)
then have "F g (p ` S) = F (g \<circ> p) S"
by (rule reindex)
moreover from \<open>p permutes S\<close> have "p ` S = S"
by (rule permutes_image)
ultimately show ?thesis
by simp
qed

subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close>

lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>
Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"

lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow>
Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"

lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow>
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id"

subsection \<open>Permutations as transposition sequences\<close>

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 \<circ> p)"

declare id[unfolded id_def, simp]

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

subsection \<open>Some closure properties of the set of permutations, with lengths\<close>

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]
apply simp
done

lemma permutation_swap_id: "permutation (Fun.swap a b id)"
apply (cases "a = b")
apply simp_all
unfolding permutation_def
using swapidseq_swap[of a b]
apply blast
done

lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)"
proof (induct n p arbitrary: m q rule: swapidseq.induct)
case (id m q)
then show ?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 comp_assoc
apply (rule swapidseq.comp_Suc)
using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
apply blast+
done
qed

lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)"
unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis

lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
apply (induct n p rule: swapidseq.induct)
using swapidseq_swap[of a b]
apply (auto simp add: comp_assoc intro: swapidseq.comp_Suc)
done

lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
proof (induct n p rule: swapidseq.induct)
case id
then show ?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 \<circ> 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 \<circ> p \<circ> ?q = Fun.swap a b id \<circ> (p \<circ> q) \<circ> Fun.swap a b id"
also have "\<dots> = id"
finally have th2: "Fun.swap a b id \<circ> p \<circ> ?q = id" .
have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id \<circ> Fun.swap a b id) \<circ> p"
by (simp only: o_assoc)
then have "?q \<circ> (Fun.swap a b id \<circ> p) = id"
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 \<Longrightarrow> permutation (inv p)"
using permutation_def swapidseq_inverse by blast

subsection \<open>The identity map only has even transposition sequences\<close>

lemma symmetry_lemma:
assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"
and "\<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 \<Longrightarrow>
P a b c d"
shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>  P a b c d"
using assms by metis

lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
Fun.swap a b id \<circ> 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 \<circ> Fun.swap c d id = Fun.swap x y id \<circ> 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 \<circ> 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 \<circ> Fun.swap c d id = Fun.swap x y id \<circ> 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: swap_commute)
apply (case_tac "a = c \<and> b = d")
apply (clarsimp simp only: swap_commute 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: fun_eq_iff Fun.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: fun_eq_iff Fun.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: fun_eq_iff Fun.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 \<circ> 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 \<circ> p) a = a"
shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
using spn ab pa
proof (induct n arbitrary: p a b)
case 0
then show ?case
by (auto simp add: Fun.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 \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m"
by auto
{
assume H: "Fun.swap a b id \<circ> Fun.swap c d id = id"
have ?case by (simp only: cdqm o_assoc H) (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 \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id"
from H have az: "a \<noteq> z"
by simp

{
fix h
have "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a"
using H by (simp add: Fun.swap_def)
}
note th3 = this
from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)"
by simp
then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)"
then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a"
by simp
then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a"
unfolding Suc by metis
then have th1: "(Fun.swap a z id \<circ> 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 \<circ> 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 comp_assoc)
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 \<open>swapidseq n id\<close>
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"
then have "even n" by presburger
}
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"
by presburger
}
ultimately show "even n"
using H(2)[unfolded swapidseq_cases[of n id]] by auto
qed

subsection \<open>Therefore we have a welldefined notion of parity\<close>

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
apply blast+
done

subsection \<open>And it has the expected composition properties\<close>

lemma evenperm_id[simp]: "evenperm id = True"
by (rule evenperm_unique[where n = 0]) simp_all

lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap)

lemma evenperm_comp:
assumes p: "permutation p"
and q:"permutation q"
shows "evenperm (p \<circ> 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

subsection \<open>A more abstract characterization of permutations\<close>

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
then show ?thesis unfolding bij_iff
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
then show ?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 \<open>a \<noteq> b\<close> have th: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
from finite_subset[OF th fS] show ?case  .
qed
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)
then show ?case by simp
next
case (insert a F p)
let ?r = "Fun.swap a (p a) id \<circ> p"
let ?q = "Fun.swap a (p a) id \<circ> ?r"
have raa: "?r a = a"
from bij_swap_compose_bij[OF insert(4)] have br: "bij ?r"  .
from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
from insert(3)[OF br th] have rp: "permutation ?r" .
have "permutation ?q"
by (simp add: permutation_compose permutation_swap_id rp)
then show ?case
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 "?b \<and> ?f"
then have "?f" "?b" by blast+
from permutation_lemma[OF this] show ?lhs
by blast
qed

lemma permutation_inverse_works:
assumes p: "permutation p"
shows "inv p \<circ> p = id"
and "p \<circ> inv p = id"
using permutation_bijective [OF p]
unfolding bij_def inj_iff surj_iff by auto

lemma permutation_inverse_compose:
assumes p: "permutation p"
and q: "permutation q"
shows "inv (p \<circ> q) = inv q \<circ> inv p"
proof -
note ps = permutation_inverse_works[OF p]
note qs = permutation_inverse_works[OF q]
have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p"
also have "\<dots> = id"
finally have th0: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" .
have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q"
also have "\<dots> = id"
finally have th1: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
from inv_unique_comp[OF th0 th1] show ?thesis .
qed

subsection \<open>Relation to "permutes"\<close>

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

subsection \<open>Hence a sort of induction principle composing by swaps\<close>

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 \<Longrightarrow> P (Fun.swap a b id \<circ> p)) \<Longrightarrow>
(\<And>p. p permutes S \<Longrightarrow> P p)"
proof (induct S rule: finite_induct)
case empty
then show ?case by auto
next
case (insert x F p)
let ?r = "Fun.swap x (p x) id \<circ> p"
let ?q = "Fun.swap x (p x) id \<circ> ?r"
have qp: "?q = p"
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

subsection \<open>Sign of a permutation as a real number\<close>

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

lemma sign_nz: "sign p \<noteq> 0"

lemma sign_id: "sign id = 1"

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

lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q"

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

lemma sign_idempotent: "sign p * sign p = 1"

subsection \<open>Permuting a list\<close>

text \<open>This function permutes a list by applying a permutation to the indices.\<close>

definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"

lemma permute_list_map:
assumes "f permutes {..<length xs}"
shows   "permute_list f (map g xs) = map g (permute_list f xs)"
using permutes_in_image[OF assms] by (auto simp: permute_list_def)

lemma permute_list_nth:
assumes "f permutes {..<length xs}" "i < length xs"
shows   "permute_list f xs ! i = xs ! f i"
using permutes_in_image[OF assms(1)] assms(2)

lemma permute_list_Nil [simp]: "permute_list f [] = []"

lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"

lemma permute_list_compose:
assumes "g permutes {..<length xs}"
shows   "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)

lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs"

lemma permute_list_id [simp]: "permute_list id xs = xs"

lemma mset_permute_list [simp]:
assumes "f permutes {..<length (xs :: 'a list)}"
shows   "mset (permute_list f xs) = mset xs"
proof (rule multiset_eqI)
fix y :: 'a
from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
using permutes_in_image[OF assms] by auto
have "count (mset (permute_list f xs)) y =
card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})"
by (simp add: permute_list_def count_image_mset atLeast0LessThan)
also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}"
by auto
also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}"
by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
also have "\<dots> = count (mset xs) y" by (simp add: count_mset length_filter_conv_card)
finally show "count (mset (permute_list f xs)) y = count (mset xs) y" by simp
qed

lemma set_permute_list [simp]:
assumes "f permutes {..<length xs}"
shows   "set (permute_list f xs) = set xs"
by (rule mset_eq_setD[OF mset_permute_list]) fact

lemma distinct_permute_list [simp]:
assumes "f permutes {..<length xs}"
shows   "distinct (permute_list f xs) = distinct xs"

lemma permute_list_zip:
assumes "f permutes A" "A = {..<length xs}"
assumes [simp]: "length xs = length ys"
shows   "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
proof -
from permutes_in_image[OF assms(1)] assms(2)
have [simp]: "f i < length ys \<longleftrightarrow> i < length ys" for i by simp
have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]"
also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
by (intro nth_equalityI) simp_all
also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
finally show ?thesis .
qed

lemma map_of_permute:
assumes "\<sigma> permutes fst ` set xs"
shows   "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)" (is "_ = map_of (map ?f _)")
proof
fix x
from assms have "inj \<sigma>" "surj \<sigma>" by (simp_all add: permutes_inj permutes_surj)
thus "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x"
by (induction xs) (auto simp: inv_f_f surj_f_inv_f)
qed

text \<open>
The following few lemmas were contributed by Lukas Bulwahn.
\<close>

lemma count_image_mset_eq_card_vimage:
assumes "finite A"
shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
using assms
proof (induct A)
case empty
show ?case by simp
next
case (insert x F)
show ?case
proof cases
assume "f x = b"
from this have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
using insert.hyps by auto
also have "\<dots> = card (insert x {a \<in> F. f a = f x})"
using insert.hyps(1,2) by simp
also have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
using \<open>f x = b\<close> by (auto intro: arg_cong[where f="card"])
finally show ?thesis using insert by auto
next
assume A: "f x \<noteq> b"
hence "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}" by auto
with insert A show ?thesis by simp
qed
qed

(* Prove image_mset_eq_implies_permutes *)
lemma image_mset_eq_implies_permutes:
fixes f :: "'a \<Rightarrow> 'b"
assumes "finite A"
assumes mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)"
obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
proof -
from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto
have "f ` A = f' ` A"
proof -
have "f ` A = f ` (set_mset (mset_set A))" using \<open>finite A\<close> by simp
also have "\<dots> = f' ` (set_mset (mset_set A))"
by (metis mset_eq multiset.set_map)
also have "\<dots> = f' ` A" using \<open>finite A\<close> by simp
finally show ?thesis .
qed
have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}"
proof
fix b
from mset_eq have
"count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b" by simp
from this  have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
using \<open>finite A\<close>
from this show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
by (intro finite_same_card_bij) simp_all
qed
hence "\<exists>p. \<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}"
by (rule bchoice)
then guess p .. note p = this
define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
have "p' permutes A"
proof (rule bij_imp_permutes)
have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)"
unfolding disjoint_family_on_def by auto
moreover have "bij_betw (\<lambda>a. p (f a) a) {a \<in> A. f a = b} {a \<in> A. f' a = b}" if b: "b \<in> f ` A" for b
using p b by (subst bij_betw_cong[where g="p b"]) auto
ultimately have "bij_betw (\<lambda>a. p (f a) a) (\<Union>b\<in>f ` A. {a \<in> A. f a = b}) (\<Union>b\<in>f ` A. {a \<in> A. f' a = b})"
by (rule bij_betw_UNION_disjoint)
moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A" by auto
moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A" using \<open>f ` A = f' ` A\<close> by auto
ultimately show "bij_betw p' A A"
unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto
next
fix x
assume "x \<notin> A"
from this show "p' x = x"
unfolding p'_def by simp
qed
moreover from p have "\<forall>x\<in>A. f x = f' (p' x)"
unfolding p'_def using bij_betwE by fastforce
ultimately show ?thesis by (rule that)
qed

lemma mset_set_upto_eq_mset_upto:
"mset_set {..<n} = mset [0..<n]"

(* and derive the existing property: *)
lemma mset_eq_permutation:
assumes mset_eq: "mset (xs::'a list) = mset ys"
obtains p where "p permutes {..<length ys}" "permute_list p ys = xs"
proof -
from mset_eq have length_eq: "length xs = length ys"
using mset_eq_length by blast
have "mset_set {..<length ys} = mset [0..<length ys]"
using mset_set_upto_eq_mset_upto by blast
from mset_eq length_eq this have
"image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) = image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})"
by (metis map_nth mset_map)
from image_mset_eq_implies_permutes[OF _ this]
obtain p where "p permutes {..<length ys}"
and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)" by auto
moreover from this length_eq have "permute_list p ys = xs"
by (auto intro!: nth_equalityI simp add: permute_list_nth)
ultimately show thesis using that by blast
qed

lemma permutes_natset_le:
fixes S :: "'a::wellorder set"
assumes p: "p permutes S"
and le: "\<forall>i \<in> S. p i \<le> 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. m < n \<Longrightarrow> 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 simp
}
ultimately have "p n = n"
by blast
}
ultimately show "p n = n"
by blast
qed
}
then show ?thesis
qed

lemma permutes_natset_ge:
fixes S :: "'a::wellorder set"
assumes p: "p permutes S"
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_eqI)
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 \<circ> p | p. p permutes S} = {p . p permutes S}"
apply (rule set_eqI)
apply auto
apply (rule permutes_compose)
using q
apply auto
apply (rule_tac x = "inv q \<circ> x" in exI)
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
done

lemma image_compose_permutations_right:
assumes q: "q permutes S"
shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}"
apply (rule set_eqI)
apply auto
apply (rule permutes_compose)
using q
apply auto
apply (rule_tac x = "x \<circ> inv q" in exI)
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc)
done

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

lemma sum_permutations_inverse:
"sum f {p. p permutes S} = sum (\<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"
fix q r
assume q: "q permutes S"
and r: "r permutes S"
and qr: "inv q = inv r"
then have "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 = sum (f \<circ> inv) ?S"
from sum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 .
qed

lemma setum_permutations_compose_left:
assumes q: "q permutes S"
shows "sum f {p. p permutes S} = sum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
have th0: "?rhs = sum (f \<circ> (op \<circ> q)) ?S"
have th1: "inj_on (op \<circ> q) ?S"
fix p r
assume "p permutes S"
and r: "r permutes S"
and rp: "q \<circ> p = q \<circ> r"
then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r"
with permutes_inj[OF q, unfolded inj_iff] show "p = r"
by simp
qed
have th3: "(op \<circ> q) ` ?S = ?S"
using image_compose_permutations_left[OF q] by auto
from sum.reindex[OF th1, of f] show ?thesis unfolding th0 th1 th3 .
qed

lemma sum_permutations_compose_right:
assumes q: "q permutes S"
shows "sum f {p. p permutes S} = sum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
have th0: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
have th1: "inj_on (\<lambda>p. p \<circ> q) ?S"
fix p r
assume "p permutes S"
and r: "r permutes S"
and rp: "p \<circ> q = r \<circ> q"
then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)"
with permutes_surj[OF q, unfolded surj_iff] show "p = r"
by simp
qed
have th3: "(\<lambda>p. p \<circ> q) ` ?S = ?S"
using image_compose_permutations_right[OF q] by auto
from sum.reindex[OF th1, of f]
show ?thesis unfolding th0 th1 th3 .
qed

subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close>

lemma sum_over_permutations_insert:
assumes fS: "finite S"
and aS: "a \<notin> S"
shows "sum f {p. p permutes (insert a S)} =
sum (\<lambda>b. sum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)"
proof -
have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id \<circ> p)) = f \<circ> (\<lambda>(b,p). Fun.swap a b id \<circ> p)"
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 sum.cartesian_product
unfolding th1[symmetric]
unfolding th0
proof (rule sum.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"
assume c: "c \<in> insert a S"
assume p: "p permutes S"
assume q: "q permutes S"
assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> 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 \<circ> p) a  = (Fun.swap a c id \<circ> q) a"
by simp
then have bc: "b = c"
by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def
cong del: if_weak_cong split: if_split_asm)
from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) =
(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp
then have "p = q"
unfolding o_assoc swap_id_idempotent
with bc have "b = c \<and> p = q"
by blast
}
then show "inj_on ?f (insert a S \<times> ?P)"
unfolding inj_on_def by clarify metis
qed
qed

subsection \<open>Constructing permutations from association lists\<close>

definition list_permutes where
"list_permutes xs A \<longleftrightarrow> set (map fst xs) \<subseteq> A \<and> set (map snd xs) = set (map fst xs) \<and>
distinct (map fst xs) \<and> distinct (map snd xs)"

lemma list_permutesI [simp]:
assumes "set (map fst xs) \<subseteq> A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)"
shows   "list_permutes xs A"
proof -
from assms(2,3) have "distinct (map snd xs)"
by (intro card_distinct) (simp_all add: distinct_card del: set_map)
with assms show ?thesis by (simp add: list_permutes_def)
qed

definition permutation_of_list where
"permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)"

lemma permutation_of_list_Cons:
"permutation_of_list ((x,y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"

fun inverse_permutation_of_list where
"inverse_permutation_of_list [] x = x"
| "inverse_permutation_of_list ((y,x')#xs) x =
(if x = x' then y else inverse_permutation_of_list xs x)"

declare inverse_permutation_of_list.simps [simp del]

lemma inj_on_map_of:
assumes "distinct (map snd xs)"
shows   "inj_on (map_of xs) (set (map fst xs))"
proof (rule inj_onI)
fix x y assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)"
assume eq: "map_of xs x = map_of xs y"
from xy obtain x' y'
where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'"
by (cases "map_of xs x"; cases "map_of xs y")
moreover from x'y' have *: "(x,x') \<in> set xs" "(y,y') \<in> set xs"
by (force dest: map_of_SomeD)+
moreover from * eq x'y' have "x' = y'" by simp
ultimately show "x = y" using assms
by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"])
qed

lemma inj_on_the: "None \<notin> A \<Longrightarrow> inj_on the A"
by (auto simp: inj_on_def option.the_def split: option.splits)

lemma inj_on_map_of':
assumes "distinct (map snd xs)"
shows   "inj_on (the \<circ> map_of xs) (set (map fst xs))"
by (intro comp_inj_on inj_on_map_of assms inj_on_the)
(force simp: eq_commute[of None] map_of_eq_None_iff)

lemma image_map_of:
assumes "distinct (map fst xs)"
shows   "map_of xs ` set (map fst xs) = Some ` set (map snd xs)"
using assms by (auto simp: rev_image_eqI)

lemma the_Some_image [simp]: "the ` Some ` A = A"
by (subst image_image) simp

lemma image_map_of':
assumes "distinct (map fst xs)"
shows   "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)"
by (simp only: image_comp [symmetric] image_map_of assms the_Some_image)

lemma permutation_of_list_permutes [simp]:
assumes "list_permutes xs A"
shows   "permutation_of_list xs permutes A" (is "?f permutes _")
proof (rule permutes_subset[OF bij_imp_permutes])
from assms show "set (map fst xs) \<subseteq> A"
from assms have "inj_on (the \<circ> map_of xs) (set (map fst xs))" (is ?P)
by (intro inj_on_map_of') (simp_all add: list_permutes_def)
also have "?P \<longleftrightarrow> inj_on ?f (set (map fst xs))"
by (intro inj_on_cong)
(auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))"
by (rule inj_on_imp_bij_betw)
also from assms have "?f ` set (map fst xs) = (the \<circ> map_of xs) ` set (map fst xs)"
by (intro image_cong refl)
(auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits)
also from assms have "\<dots> = set (map fst xs)"
by (subst image_map_of') (simp_all add: list_permutes_def)
finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" .
qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+

lemma eval_permutation_of_list [simp]:
"permutation_of_list [] x = x"
"x = x' \<Longrightarrow> permutation_of_list ((x',y)#xs) x = y"
"x \<noteq> x' \<Longrightarrow> permutation_of_list ((x',y')#xs) x = permutation_of_list xs x"

lemma eval_inverse_permutation_of_list [simp]:
"inverse_permutation_of_list [] x = x"
"x = x' \<Longrightarrow> inverse_permutation_of_list ((y,x')#xs) x = y"
"x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x"

lemma permutation_of_list_id:
assumes "x \<notin> set (map fst xs)"
shows   "permutation_of_list xs x = x"
using assms by (induction xs) (auto simp: permutation_of_list_Cons)

lemma permutation_of_list_unique':
assumes "distinct (map fst xs)" "(x, y) \<in> set xs"
shows   "permutation_of_list xs x = y"
using assms by (induction xs) (force simp: permutation_of_list_Cons)+

lemma permutation_of_list_unique:
assumes "list_permutes xs A" "(x,y) \<in> set xs"
shows   "permutation_of_list xs x = y"
using assms by (intro permutation_of_list_unique') (simp_all add: list_permutes_def)

lemma inverse_permutation_of_list_id:
assumes "x \<notin> set (map snd xs)"
shows   "inverse_permutation_of_list xs x = x"
using assms by (induction xs) auto

lemma inverse_permutation_of_list_unique':
assumes "distinct (map snd xs)" "(x, y) \<in> set xs"
shows   "inverse_permutation_of_list xs y = x"
using assms by (induction xs) (force simp: inverse_permutation_of_list.simps)+

lemma inverse_permutation_of_list_unique:
assumes "list_permutes xs A" "(x,y) \<in> set xs"
shows   "inverse_permutation_of_list xs y = x"
using assms by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def)

lemma inverse_permutation_of_list_correct:
assumes "list_permutes xs (A :: 'a set)"
shows   "inverse_permutation_of_list xs = inv (permutation_of_list xs)"
proof (rule ext, rule sym, subst permutes_inv_eq)
from assms show "permutation_of_list xs permutes A" by simp
next
fix x
show "permutation_of_list xs (inverse_permutation_of_list xs x) = x"
proof (cases "x \<in> set (map snd xs)")
case True
then obtain y where "(y, x) \<in> set xs" by force
with assms show ?thesis