(* Title: HOL/Library/Permutations.thy
Author: Amine Chaieb, University of Cambridge
*)
header {* Permutations, both general and specifically on finite sets.*}
theory Permutations
imports Parity Fact
begin
subsection {* Transpositions *}
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)"
by (simp add: Fun.swap_def)
subsection {* Basic consequences of the definition *}
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_image: "p permutes S \<Longrightarrow> p ` S = S"
unfolding permutes_def
apply (rule set_eqI)
apply (simp add: image_iff)
apply metis
done
lemma permutes_inj: "p permutes S \<Longrightarrow> inj p"
unfolding permutes_def inj_on_def by blast
lemma permutes_surj: "p permutes s \<Longrightarrow> surj p"
unfolding permutes_def surj_def by metis
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
subsection {* Group properties *}
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
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 \<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 `x \<notin> 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"
using fact_gt_zero_nat by (auto intro: card_ge_0_finite)
then have pF'f: "finite ?pF'"
using H0 `finite F`
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'"
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) `x \<notin> 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 "\<dots> = ?g (c,q) x"
using ths(5) `x \<notin> F` eq
by (auto simp add: swap_def fun_upd_def fun_eq_iff)
also have "\<dots> = c"
using ths(5) `x \<notin> 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 \<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"
by (simp add: o_assoc)
with bc have "(b, p) = (c, q)"
by simp
}
then show ?thesis
unfolding inj_on_def by blast
qed
from `x \<notin> F` H0 have n0: "n \<noteq> 0"
using `finite F` 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 `finite F` `finite ?pF`
apply (simp only: Collect_split 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] fact_gt_zero_nat
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 \<circ> 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 \<circ> p) S"
by (rule reindex)
moreover from `p permutes S` have "p ` S = S"
by (rule permutes_image)
ultimately show ?thesis
by simp
qed
lemma setsum_permute:
assumes "p permutes S"
shows "setsum f S = setsum (f \<circ> p) S"
using assms by (fact setsum.permute)
lemma setsum_permute_natseg:
assumes pS: "p permutes {m .. n}"
shows "setsum f {m .. n} = setsum (f \<circ> p) {m .. n}"
using setsum_permute [OF pS, of f ] pS by blast
lemma setprod_permute:
assumes "p permutes S"
shows "setprod f S = setprod (f \<circ> p) S"
using assms by (fact setprod.permute)
lemma setprod_permute_natseg:
assumes pS: "p permutes {m .. n}"
shows "setprod f {m .. n} = setprod (f \<circ> p) {m .. n}"
using setprod_permute [OF pS, of f ] pS by blast
subsection {* 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 \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
by (simp add: fun_eq_iff Fun.swap_def)
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"
by (simp add: fun_eq_iff Fun.swap_def)
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"
by (simp add: fun_eq_iff Fun.swap_def)
subsection {* 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 \<circ> p)"
declare id[unfolded id_def, simp]
definition "permutation p \<longleftrightarrow> (\<exists>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 \<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"
by (simp add: o_assoc)
also have "\<dots> = id"
by (simp add: q(2))
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"
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 \<Longrightarrow> permutation (inv p)"
using permutation_def swapidseq_inverse by blast
subsection {* The identity map only has even transposition sequences *}
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)"
by (simp add: o_assoc H)
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 `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"
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 {* 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
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 \<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 {* 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
then show ?thesis unfolding bij_iff
apply (auto simp add: fun_eq_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 `a \<noteq> b` have th: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
by (auto simp add: Fun.swap_def)
from finite_subset[OF th fS] show ?case .
qed
qed
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_ompose_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
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"
by (simp add: Fun.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: Fun.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)
then show ?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 "?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"
by (simp add: o_assoc)
also have "\<dots> = id"
by (simp add: ps qs)
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"
by (simp add: o_assoc)
also have "\<dots> = id"
by (simp add: ps qs)
finally have th1: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
from inv_unique_comp[OF th0 th1] show ?thesis .
qed
subsection {* 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
subsection {* 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 \<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"
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
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 \<noteq> 0"
by (simp add: sign_def)
lemma sign_id: "sign id = 1"
by (simp add: sign_def)
lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p"
by (simp add: sign_def evenperm_inv)
lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> 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)
subsection {* More lemmas about permutations *}
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
by (auto simp add: fun_eq_iff)
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"
by (simp add: permutes_def) metis
lemma setsum_permutations_inverse:
"setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p . p permutes S}"
have th0: "inj_on inv ?S"
proof (auto simp add: inj_on_def)
fix q r
assume q: "q permutes S"
and r: "r permutes S"
and qr: "inv q = inv r"
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 \<circ> inv) ?S"
by (simp add: o_def)
from setsum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 .
qed
lemma setum_permutations_compose_left:
assumes q: "q permutes S"
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
have th0: "?rhs = setsum (f \<circ> (op \<circ> q)) ?S"
by (simp add: o_def)
have th1: "inj_on (op \<circ> q) ?S"
proof (auto simp add: inj_on_def)
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"
by (simp add: comp_assoc)
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 setsum.reindex[OF th1, of f] show ?thesis unfolding th0 th1 th3 .
qed
lemma sum_permutations_compose_right:
assumes q: "q permutes S"
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
have th0: "?rhs = setsum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
by (simp add: o_def)
have th1: "inj_on (\<lambda>p. p \<circ> q) ?S"
proof (auto simp add: inj_on_def)
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)"
by (simp add: o_assoc)
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 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 \<notin> S"
shows "setsum f {p. p permutes (insert a S)} =
setsum (\<lambda>b. setsum (\<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)"
by (simp add: fun_eq_iff)
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"
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: split_if_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
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 by clarify metis
qed
qed
end