# Theory Permutations

theory Permutations
imports Binomial
```(*  Title:      HOL/Library/Permutations.thy
Author:     Amine Chaieb, University of Cambridge
*)

section ‹Permutations, both general and specifically on finite sets.›

theory Permutations
imports Binomial
begin

subsection ‹Transpositions›

lemma swap_id_idempotent [simp]:
"Fun.swap a b id ∘ 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)"

subsection ‹Basic consequences of the definition›

definition permutes  (infixr "permutes" 41)
where "(p permutes S) ⟷ (∀x. x ∉ S ⟶ p x = x) ∧ (∀y. ∃!x. p x = y)"

lemma permutes_in_image: "p permutes S ⟹ p x ∈ S ⟷ x ∈ S"
unfolding permutes_def by metis

lemma permutes_image: "p permutes S ⟹ p ` S = S"
unfolding permutes_def
apply (rule set_eqI)
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_bij: "p permutes s ⟹ bij p"
unfolding bij_def by (metis permutes_inj permutes_surj)

lemma permutes_imp_bij: "p permutes S ⟹ 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 ⟹ (⋀x. x ∉ S ⟹ p x = x) ⟹ 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 ∘ inv p = id"
and "inv p ∘ 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 ⇒ '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 ⟹ S ⊆ T ⟹ p permutes T"
unfolding permutes_def by blast

lemma permutes_empty[simp]: "p permutes {} ⟷ p = id"
unfolding fun_eq_iff permutes_def by simp metis

lemma permutes_sing[simp]: "p permutes {a} ⟷ p = id"
unfolding fun_eq_iff permutes_def by simp metis

lemma permutes_univ: "p permutes UNIV ⟷ (∀y. ∃!x. p x = y)"
unfolding permutes_def by simp

lemma permutes_inv_eq: "p permutes S ⟹ inv p y = x ⟷ 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 ∈ S ⟹ b ∈ S ⟹ 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 ⟹ (∀x ∈ S - T. p x = x) ⟹ p permutes T"
by (simp add: Ball_def permutes_def) metis

subsection ‹Group properties›

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

lemma permutes_compose: "p permutes S ⟹ q permutes S ⟹ q ∘ 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

subsection ‹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 ∘ 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)} =
(λ(b,p). Fun.swap a b id ∘ p) ` {(b,p). b ∈ insert a S ∧ p ∈ {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 ∘ p"
have th0: "p = Fun.swap a ?b id ∘ ?q"
unfolding fun_eq_iff o_assoc by simp
have th1: "?b ∈ insert a S"
unfolding permutes_in_image[OF pS] by simp
from permutes_insert_lemma[OF pS] th0 th1
have "∃b q. p = Fun.swap a b id ∘ q ∧ b ∈ insert a S ∧ q permutes S" by blast
}
moreover
{
fix b q
assume bq: "p = Fun.swap a b id ∘ q" "b ∈ 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 ∈ 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 ⟷
(∃b q. p = Fun.swap a b id ∘ q ∧ b ∈ insert a S ∧ 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 ∈ insert x F ∧ p ∈ ?pF}"
let ?g = "(λ(b, p). Fun.swap x b id ∘ p)"
from permutes_insert[of x F]
have xfgpF': "?xF = ?g ` ?pF'" .
have Fs: "card F = n - 1"
using ‹x ∉ F› H0 ‹finite F› by auto
from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
using ‹finite F› by auto
then have "finite ?pF"
by (auto intro: card_ge_0_finite)
then have pF'f: "finite ?pF'"
using H0 ‹finite F›
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) ∈ ?pF'"
assume cq: "(c,q) ∈ ?pF'"
assume eq: "?g (b,p) = ?g (c,q)"
from bp cq have ths: "b ∈ insert x F" "c ∈ insert x F" "x ∈ insert x F"
"p permutes F" "q permutes F"
by auto
from ths(4) ‹x ∉ F› 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 "… = ?g (c,q) x"
using ths(5) ‹x ∉ F› eq
by (auto simp add: swap_def fun_upd_def fun_eq_iff)
also have "… = c"
using ths(5) ‹x ∉ F›
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 ∘ p = Fun.swap x b id ∘ q"
by simp
then have "Fun.swap x b id ∘ (Fun.swap x b id ∘ p) =
Fun.swap x b id ∘ (Fun.swap x b id ∘ 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 ‹x ∉ F› H0 have n0: "n ≠ 0"
using ‹finite F› by auto
then have "∃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 ‹finite F› ‹finite ?pF›
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 ‹Permutations of index set for iterated operations›

lemma (in comm_monoid_set) permute:
assumes "p permutes S"
shows "F g S = F (g ∘ p) S"
proof -
from ‹p permutes S› 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 ∘ p) S"
by (rule reindex)
moreover from ‹p permutes S› have "p ` S = S"
by (rule permutes_image)
ultimately show ?thesis
by simp
qed

subsection ‹Various combinations of transpositions with 2, 1 and 0 common elements›

lemma swap_id_common:" a ≠ c ⟹ b ≠ c ⟹
Fun.swap a b id ∘ Fun.swap a c id = Fun.swap b c id ∘ Fun.swap a b id"

lemma swap_id_common': "a ≠ b ⟹ a ≠ c ⟹
Fun.swap a c id ∘ Fun.swap b c id = Fun.swap b c id ∘ Fun.swap a b id"

lemma swap_id_independent: "a ≠ c ⟹ a ≠ d ⟹ b ≠ c ⟹ b ≠ d ⟹
Fun.swap a b id ∘ Fun.swap c d id = Fun.swap c d id ∘ Fun.swap a b id"

subsection ‹Permutations as transposition sequences›

inductive swapidseq :: "nat ⇒ ('a ⇒ 'a) ⇒ bool"
where
id[simp]: "swapidseq 0 id"
| comp_Suc: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (Fun.swap a b id ∘ p)"

declare id[unfolded id_def, simp]

definition "permutation p ⟷ (∃n. swapidseq n p)"

subsection ‹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]
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 ⟹ swapidseq m q ⟹ swapidseq (n + m) (p ∘ 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 ⟹ permutation q ⟹ permutation (p ∘ q)"
unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis

lemma swapidseq_endswap: "swapidseq n p ⟹ a ≠ b ⟹ swapidseq (Suc n) (p ∘ 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 ⟹ ∃q. swapidseq n q ∧ p ∘ q = id ∧ q ∘ 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 ∘ q = id" "q ∘ p = id"
by blast
let ?q = "q ∘ 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 ∘ p ∘ ?q = Fun.swap a b id ∘ (p ∘ q) ∘ Fun.swap a b id"
also have "… = id"
finally have th2: "Fun.swap a b id ∘ p ∘ ?q = id" .
have "?q ∘ (Fun.swap a b id ∘ p) = q ∘ (Fun.swap a b id ∘ Fun.swap a b id) ∘ p"
by (simp only: o_assoc)
then have "?q ∘ (Fun.swap a b id ∘ 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 ⟹ permutation (inv p)"
using permutation_def swapidseq_inverse by blast

subsection ‹The identity map only has even transposition sequences›

lemma symmetry_lemma:
assumes "⋀a b c d. P a b c d ⟹ P a b d c"
and "⋀a b c d. a ≠ b ⟹ c ≠ d ⟹
a = c ∧ b = d ∨ a = c ∧ b ≠ d ∨ a ≠ c ∧ b = d ∨ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ⟹
P a b c d"
shows "⋀a b c d. a ≠ b ⟶ c ≠ d ⟶  P a b c d"
using assms by metis

lemma swap_general: "a ≠ b ⟹ c ≠ d ⟹
Fun.swap a b id ∘ Fun.swap c d id = id ∨
(∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧
Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ Fun.swap a z id)"
proof -
assume H: "a ≠ b" "c ≠ d"
have "a ≠ b ⟶ c ≠ d ⟶
(Fun.swap a b id ∘ Fun.swap c d id = id ∨
(∃x y z. x ≠ a ∧ y ≠ a ∧ z ≠ a ∧ x ≠ y ∧
Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ 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 ∧ b = d")
apply (clarsimp simp only: swap_commute swap_id_idempotent)
apply (case_tac "a = c ∧ b ≠ 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 ≠ c ∧ 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 ⟷ p = id"
using swapidseq.cases[of 0 p "p = id"]
by auto

lemma swapidseq_cases: "swapidseq n p ⟷
n = 0 ∧ p = id ∨ (∃a b q m. n = Suc m ∧ p = Fun.swap a b id ∘ q ∧ swapidseq m q ∧ a ≠ 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 ≠ b"
and pa: "(Fun.swap a b id ∘ p) a = a"
shows "n ≠ 0 ∧ swapidseq (n - 1) (Fun.swap a b id ∘ 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 ∘ q" "swapidseq m q" "c ≠ d" "n = m"
by auto
{
assume H: "Fun.swap a b id ∘ 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 ≠ a" "y ≠ a" "z ≠ a" "x ≠ y"
"Fun.swap a b id ∘ Fun.swap c d id = Fun.swap x y id ∘ Fun.swap a z id"
from H have az: "a ≠ z"
by simp

{
fix h
have "(Fun.swap x y id ∘ h) a = a ⟷ h a = a"
using H by (simp add: Fun.swap_def)
}
note th3 = this
from cdqm(2) have "Fun.swap a b id ∘ p = Fun.swap a b id ∘ (Fun.swap c d id ∘ q)"
by simp
then have "Fun.swap a b id ∘ p = Fun.swap x y id ∘ (Fun.swap a z id ∘ q)"
then have "(Fun.swap a b id ∘ p) a = (Fun.swap x y id ∘ (Fun.swap a z id ∘ q)) a"
by simp
then have "(Fun.swap x y id ∘ (Fun.swap a z id ∘ q)) a = a"
unfolding Suc by metis
then have th1: "(Fun.swap a z id ∘ 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 ∘ q)" "n ≠ 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 ⇒ 'a)"
shows "even n"
using ‹swapidseq n id›
proof (induct n rule: nat_less_induct)
fix n
assume H: "∀m<n. swapidseq m (id::'a ⇒ 'a) ⟶ even m" "swapidseq n (id :: 'a ⇒ '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 ⇒ 'a) = Fun.swap a b id ∘ q" "swapidseq m q" "a ≠ b"
from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
have m: "m ≠ 0" "swapidseq (m - 1) (id :: 'a ⇒ '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 ‹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 ⟷ even n"
proof -
from swapidseq_inverse_exists[OF n]
obtain q where q: "swapidseq n q" "p ∘ q = id" "q ∘ 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 ‹And it has the expected composition properties›

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 ∘ 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 ⟷ 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 ‹A more abstract characterization of permutations›

lemma bij_iff: "bij f ⟷ (∀x. ∃!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 ∘ q = id" "q ∘ 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 ≠ 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 ≠ x})"
from comp_Suc.hyps(2) have fS: "finite ?S"
by simp
from ‹a ≠ b› have th: "{x. (Fun.swap a b id ∘ p) x ≠ x} ⊆ ?S"
from finite_subset[OF th fS] show ?case  .
qed
qed

lemma bij_inv_eq_iff: "bij p ⟹ x = inv p y ⟷ 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 ∘ 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_ompose_bij: "bij p ⟹ bij (Fun.swap a b id ∘ 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: "∀x. x∉ S ⟶ 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 ∘ p"
let ?q = "Fun.swap a (p a) id ∘ ?r"
have raa: "?r a = a"
from bij_swap_ompose_bij[OF insert(4)]
have br: "bij ?r"  .

from insert raa have th: "∀x. x ∉ F ⟶ ?r x = x"
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)
then show ?case
qed

lemma permutation: "permutation p ⟷ bij p ∧ finite {x. p x ≠ x}"
(is "?lhs ⟷ ?b ∧ ?f")
proof
assume p: ?lhs
from p permutation_bijective permutation_finite_support show "?b ∧ ?f"
by auto
next
assume "?b ∧ ?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 ∘ p = id"
and "p ∘ 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 ∘ q) = inv q ∘ inv p"
proof -
note ps = permutation_inverse_works[OF p]
note qs = permutation_inverse_works[OF q]
have "p ∘ q ∘ (inv q ∘ inv p) = p ∘ (q ∘ inv q) ∘ inv p"
also have "… = id"
finally have th0: "p ∘ q ∘ (inv q ∘ inv p) = id" .
have "inv q ∘ inv p ∘ (p ∘ q) = inv q ∘ (inv p ∘ p) ∘ q"
also have "… = id"
finally have th1: "inv q ∘ inv p ∘ (p ∘ q) = id" .
from inv_unique_comp[OF th0 th1] show ?thesis .
qed

subsection ‹Relation to "permutes"›

lemma permutation_permutes: "permutation p ⟷ (∃S. finite S ∧ p permutes S)"
unfolding permutation permutes_def bij_iff[symmetric]
apply (rule iffI, clarify)
apply (rule exI[where x="{x. p x ≠ x}"])
apply simp
apply clarsimp
apply (rule_tac B="S" in finite_subset)
apply auto
done

subsection ‹Hence a sort of induction principle composing by swaps›

lemma permutes_induct: "finite S ⟹ P id ⟹
(⋀ a b p. a ∈ S ⟹ b ∈ S ⟹ P p ⟹ P p ⟹ permutation p ⟹ P (Fun.swap a b id ∘ p)) ⟹
(⋀p. p permutes S ⟹ 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 ∘ p"
let ?q = "Fun.swap x (p x) id ∘ ?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 ∈ insert x F"
by simp
have xF: "x ∈ 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 ‹Sign of a permutation as a real number›

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

lemma sign_nz: "sign p ≠ 0"

lemma sign_id: "sign id = 1"

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

lemma sign_compose: "permutation p ⟹ permutation q ⟹ sign (p ∘ 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"

lemma permutes_natset_le:
fixes S :: "'a::wellorder set"
assumes p: "p permutes S"
and le: "∀i ∈ S. p i ≤ i"
shows "p = id"
proof -
{
fix n
have "p n = n"
using p le
proof (induct n arbitrary: S rule: less_induct)
fix n S
assume H:
"⋀m S. m < n ⟹ p permutes S ⟹ ∀i∈S. p i ≤ i ⟹ p m = m"
"p permutes S" "∀i ∈S. p i ≤ i"
{
assume "n ∉ S"
with H(2) have "p n = n"
unfolding permutes_def by metis
}
moreover
{
assume ns: "n ∈ S"
from H(3)  ns have "p n < n ∨ 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: "∀i ∈ S. p i ≥ i"
shows "p = id"
proof -
{
fix i
assume i: "i ∈ S"
from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i ∈ S"
by simp
with le have "p (inv p i) ≥ inv p i"
by blast
with permutes_inverses[OF p] have "i ≥ inv p i"
by simp
}
then have th: "∀i∈S. inv p i ≤ 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 ∘ 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 ∘ 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 ∘ 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 ∘ 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} ⟹ i ∈ {1..n} ⟹ 1 ≤ p i ∧ p i ≤ n"

lemma setsum_permutations_inverse:
"setsum f {p. p permutes S} = setsum (λp. 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 = setsum (f ∘ inv) ?S"
from setsum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 .
qed

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

lemma sum_permutations_compose_right:
assumes q: "q permutes S"
shows "setsum f {p. p permutes S} = setsum (λp. f(p ∘ q)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
have th0: "?rhs = setsum (f ∘ (λp. p ∘ q)) ?S"
have th1: "inj_on (λp. p ∘ q) ?S"
fix p r
assume "p permutes S"
and r: "r permutes S"
and rp: "p ∘ q = r ∘ q"
then have "p ∘ (q ∘ inv q) = r ∘ (q ∘ inv q)"
with permutes_surj[OF q, unfolded surj_iff] show "p = r"
by simp
qed
have th3: "(λp. p ∘ 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

subsection ‹Sum over a set of permutations (could generalize to iteration)›

lemma setsum_over_permutations_insert:
assumes fS: "finite S"
and aS: "a ∉ S"
shows "setsum f {p. p permutes (insert a S)} =
setsum (λb. setsum (λq. f (Fun.swap a b id ∘ q)) {p. p permutes S}) (insert a S)"
proof -
have th0: "⋀f a b. (λ(b,p). f (Fun.swap a b id ∘ p)) = f ∘ (λ(b,p). Fun.swap a b id ∘ p)"
have th1: "⋀P Q. P × Q = {(a,b). a ∈ P ∧ b ∈ Q}"
by blast
have th2: "⋀P Q. P ⟹ (P ⟹ Q) ⟹ P ∧ Q"
by blast
show ?thesis
unfolding permutes_insert
unfolding setsum.cartesian_product
unfolding th1[symmetric]
unfolding th0
proof (rule setsum.reindex)
let ?f = "(λ(b, y). Fun.swap a b id ∘ y)"
let ?P = "{p. p permutes S}"
{
fix b c p q
assume b: "b ∈ insert a S"
assume c: "c ∈ insert a S"
assume p: "p permutes S"
assume q: "q permutes S"
assume eq: "Fun.swap a b id ∘ p = Fun.swap a c id ∘ 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 ∘ p) a  = (Fun.swap a c id ∘ 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: split_if_asm)
from eq[unfolded bc] have "(λp. Fun.swap a c id ∘ p) (Fun.swap a c id ∘ p) =
(λp. Fun.swap a c id ∘ p) (Fun.swap a c id ∘ q)" by simp
then have "p = q"
unfolding o_assoc swap_id_idempotent
with bc have "b = c ∧ p = q"
by blast
}
then show "inj_on ?f (insert a S × ?P)"
unfolding inj_on_def by clarify metis
qed
qed

end

```