--- a/src/HOL/Library/Permutations.thy Sat Apr 01 15:35:32 2017 +0200
+++ b/src/HOL/Library/Permutations.thy Sat Apr 01 18:50:26 2017 +0200
@@ -5,39 +5,33 @@
section \<open>Permutations, both general and specifically on finite sets.\<close>
theory Permutations
-imports Binomial Multiset Disjoint_Sets
+ 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 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"
+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 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)
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"
+ assumes "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])
+ using surj_f_inv_f[OF bij_is_surj[OF \<open>bij p\<close>]]
+ by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF \<open>bij p\<close>])
-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
+lemma bij_swap_compose_bij:
+ assumes "bij p"
+ shows "bij (Fun.swap a b id \<circ> p)"
+ by (simp only: bij_swap_comp[OF \<open>bij p\<close>] bij_swap_iff \<open>bij p\<close>)
subsection \<open>Basic consequences of the definition\<close>
@@ -48,9 +42,8 @@
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_not_in: "f permutes S \<Longrightarrow> x \<notin> S \<Longrightarrow> f x = x"
+ by (auto simp: permutes_def)
lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S"
unfolding permutes_def
@@ -63,46 +56,46 @@
unfolding permutes_def inj_def by blast
lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A"
- unfolding permutes_def inj_on_def by auto
+ by (auto simp: permutes_def inj_on_def)
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)
+ 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)
+ 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"
+ assumes permutes: "p permutes S"
shows "p \<circ> inv p = id"
and "inv p \<circ> p = id"
- using permutes_inj[OF pS] permutes_surj[OF pS]
+ using permutes_inj[OF permutes] permutes_surj[OF permutes]
unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
lemma permutes_inverses:
fixes p :: "'a \<Rightarrow> 'a"
- assumes pS: "p permutes S"
+ assumes permutes: "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
+ using permutes_inv_o[OF permutes, 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
+ by (auto simp add: fun_eq_iff permutes_def)
lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
- unfolding fun_eq_iff permutes_def by simp metis
+ by (simp add: fun_eq_iff permutes_def) metis (*somewhat slow*)
lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
- unfolding permutes_def by simp
+ by (simp add: permutes_def)
lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y"
unfolding permutes_def inv_def
@@ -124,42 +117,44 @@
(* Next three lemmas contributed by Lukas Bulwahn *)
lemma permutes_bij_inv_into:
- fixes A :: "'a set" and B :: "'b set"
+ fixes A :: "'a set"
+ and B :: "'b set"
assumes "p permutes A"
- assumes "bij_betw f A B"
+ and "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
+ from assms have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
+ by (auto simp add: permutes_imp_bij bij_betw_inv_into)
+ then have "bij_betw (f \<circ> p \<circ> inv_into A f) B B"
+ by (simp add: bij_betw_trans)
+ then 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 \<circ> p \<circ> 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
+ then 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)
+ 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)
+ have "f x = f' (p x)" if "x \<in># mset_set A" for x
+ using assms(2)[of x] that by (cases "finite A") auto
+ with assms have "image_mset f (mset_set A) = image_mset (f' \<circ> p) (mset_set A)"
+ by (auto intro!: image_mset_cong)
also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
by (simp add: image_mset.compositionality)
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
+ from assms permutes_image_mset have "image_mset p (mset_set A) = mset_set A"
+ by blast
+ then show ?thesis by simp
qed
finally show ?thesis ..
qed
@@ -168,36 +163,41 @@
subsection \<open>Group properties\<close>
lemma permutes_id: "id permutes S"
- unfolding permutes_def by simp
+ by (simp add: permutes_def)
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"
+ assumes "p permutes S"
shows "inv p permutes S"
- using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis
+ using assms unfolding permutes_def permutes_inv_eq[OF assms] by metis
lemma permutes_inv_inv:
- assumes pS: "p permutes S"
+ assumes "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]]
+ unfolding fun_eq_iff permutes_inv_eq[OF assms] permutes_inv_eq[OF permutes_inv[OF assms]]
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'"
+ 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"
+ show "inv p x = p' x" for 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)
+ case True
+ from assms have "p' x = p' (p (inv p x))"
+ by (simp add: permutes_inverses)
+ also from permutes_inv[OF perm] True have "\<dots> = inv p x"
+ by (subst inv) (simp_all add: permutes_in_image)
+ finally show ?thesis ..
+ next
+ case False
+ with permutes_inv[OF perm] show ?thesis
+ by (simp_all add: outside permutes_not_in)
+ qed
qed
lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
@@ -207,58 +207,54 @@
subsection \<open>The number of permutations on a finite set\<close>
lemma permutes_insert_lemma:
- assumes pS: "p permutes (insert a S)"
+ assumes "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_compose[OF assms])
apply (rule permutes_swap_id, simp)
- using permutes_in_image[OF pS, of a]
+ using permutes_in_image[OF assms, 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}}"
+ (\<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"
+ 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)" for p
+ proof -
+ have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S"
+ if p: "p permutes insert a S"
+ proof -
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"
+ have *: "p = Fun.swap a ?b id \<circ> ?q"
+ by (simp add: fun_eq_iff o_assoc)
+ have **: "?b \<in> insert a S"
+ unfolding permutes_in_image[OF p] by simp
+ from permutes_insert_lemma[OF p] * ** show ?thesis
+ by blast
+ qed
+ moreover have "p permutes insert a S"
+ if bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" for b q
+ proof -
+ from permutes_subset[OF bq(3), of "insert a S"] have q: "q permutes insert a S"
by auto
- have aS: "a \<in> insert a S"
+ have a: "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"
+ from bq(1) permutes_compose[OF q permutes_swap_id[OF a bq(2)]] show ?thesis
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
+ ultimately show ?thesis by blast
+ qed
+ then show ?thesis by auto
qed
lemma card_permutations:
- assumes Sn: "card S = n"
- and fS: "finite S"
+ assumes "card S = n"
+ and "finite S"
shows "card {p. p permutes S} = fact n"
- using fS Sn
+ using assms(2,1)
proof (induct arbitrary: n)
case empty
then show ?case by simp
@@ -266,21 +262,20 @@
case (insert x F)
{
fix n
- assume H0: "card (insert x F) = n"
+ assume card_insert: "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
+ have xfgpF': "?xF = ?g ` ?pF'"
+ by (rule permutes_insert[of x F])
+ from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have Fs: "card F = n - 1"
+ by auto
+ from \<open>finite F\<close> insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
+ 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>
+ with \<open>finite F\<close> card_insert have pF'f: "finite ?pF'"
apply (simp only: Collect_case_prod Collect_mem_eq)
apply (rule finite_cartesian_product)
apply simp_all
@@ -290,64 +285,54 @@
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"
+ assume bp: "(b, p) \<in> ?pF'"
+ assume cq: "(c, q) \<in> ?pF'"
+ assume eq: "?g (b, p) = ?g (c, q)"
+ from bp cq have pF: "p permutes F" and qF: "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" .
+ from pF \<open>x \<notin> F\<close> eq have "b = ?g (b, p) x"
+ by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
+ also from qF \<open>x \<notin> F\<close> eq have "\<dots> = ?g (c, q) x"
+ by (auto simp: swap_def fun_upd_def fun_eq_iff)
+ also from qF \<open>x \<notin> F\<close> have "\<dots> = c"
+ by (auto simp: permutes_def Fun.swap_def fun_upd_def fun_eq_iff)
+ finally have "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)"
+ 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)"
+ with \<open>b = c\<close> 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
+ from \<open>x \<notin> F\<close> \<open>finite F\<close> card_insert have "n \<noteq> 0"
+ by auto
then have "\<exists>m. n = Suc m"
by presburger
- then obtain m where n[simp]: "n = Suc m"
+ then obtain m where n: "n = Suc m"
by blast
- from pFs H0 have xFc: "card ?xF = fact n"
+ from pFs card_insert have *: "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
+ by (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) (simp add: n)
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
+ by (simp add: xfgpF' n)
+ from * have "card ?xF = fact n"
+ unfolding xFf by blast
}
- then show ?case
- using insert by simp
+ with insert show ?case by simp
qed
lemma finite_permutations:
- assumes fS: "finite S"
+ assumes "finite S"
shows "finite {p. p permutes S}"
- using card_permutations[OF refl fS]
- by (auto intro: card_ge_0_finite)
+ using card_permutations[OF refl assms] by (auto intro: card_ge_0_finite)
subsection \<open>Permutations of index set for iterated operations\<close>
@@ -387,9 +372,9 @@
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)"
+ 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]
@@ -410,12 +395,16 @@
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
+proof (cases "a = b")
+ case True
+ then show ?thesis by simp
+next
+ case False
+ then show ?thesis
+ unfolding permutation_def
+ using swapidseq_swap[of a b] by blast
+qed
+
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)
@@ -423,13 +412,13 @@
then show ?case by simp
next
case (comp_Suc n p a b m q)
- have th: "Suc n + m = Suc (n + m)"
+ have eq: "Suc n + m = Suc (n + m)"
by arith
show ?case
- unfolding th comp_assoc
+ apply (simp only: eq comp_assoc)
apply (rule swapidseq.comp_Suc)
using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
- apply blast+
+ apply blast+
done
qed
@@ -437,10 +426,8 @@
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
+ by (induct n p rule: swapidseq.induct)
+ (use swapidseq_swap[of a b] in \<open>auto simp add: comp_assoc intro: swapidseq.comp_Suc\<close>)
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)
@@ -453,27 +440,27 @@
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)"
+ from swapidseq_swap[of a b] H(3) have *: "swapidseq 1 (Fun.swap a b id)"
by simp
- from swapidseq_comp_add[OF q(1) th0] have th1: "swapidseq (Suc n) ?q"
+ from swapidseq_comp_add[OF q(1) *] have **: "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" .
+ finally have ***: "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
+ with ** *** show ?case
by blast
qed
lemma swapidseq_inverse:
- assumes H: "swapidseq n p"
+ assumes "swapidseq n p"
shows "swapidseq n (inv p)"
- using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto
+ using swapidseq_inverse_exists[OF assms] inv_unique_comp[of p] by auto
lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)"
using permutation_def swapidseq_inverse by blast
@@ -494,61 +481,60 @@
(\<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"
+ assume neq: "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 (simp_all only: swap_commute)
apply (case_tac "a = c \<and> b = d")
- apply (clarsimp simp only: swap_commute swap_id_idempotent)
+ 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 (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="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
+ with neq 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
+ 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)"
+ 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 (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"
+ assumes "swapidseq n p"
+ and "a \<noteq> b"
+ and "(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
+ using assms
proof (induct n arbitrary: p a b)
case 0
then show ?case
@@ -559,49 +545,44 @@
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"
+ consider "Fun.swap a b id \<circ> Fun.swap c d id = id"
+ | x y z where "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"
+ using swap_general[OF Suc.prems(2) cdqm(4)] by metis
+ then show ?case
+ proof cases
+ case 1
+ then show ?thesis
+ by (simp only: cdqm o_assoc) (simp add: cdqm)
+ next
+ case prems: 2
+ then 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 prems have *: "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" for h
+ by (simp add: Fun.swap_def)
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)
+ by (simp add: o_assoc prems)
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"
+ then have "(Fun.swap a z id \<circ> q) a = a"
+ by (simp only: *)
+ from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az this]
+ have **: "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
+ from \<open>n \<noteq> 0\<close> have ***: "Suc n - 1 = Suc (n - 1)"
+ by auto
+ show ?thesis
+ apply (simp only: cdqm(2) prems o_assoc ***)
apply (simp only: Suc_not_Zero simp_thms comp_assoc)
apply (rule comp_Suc)
- using th2 H
- apply blast+
+ using ** prems
+ apply blast+
done
- }
- ultimately show ?case
- using swap_general[OF Suc.prems(2) cdqm(4)] by metis
+ qed
qed
lemma swapidseq_identity_even:
@@ -609,26 +590,24 @@
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"
+ case H: (1 n)
+ consider "n = 0"
+ | a b :: 'a and q m where "n = Suc m" "id = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
+ using H(2)[unfolded swapidseq_cases[of n id]] by auto
+ then show ?case
+ proof cases
+ case 1
+ then show ?thesis by presburger
+ next
+ case h: 2
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"
+ from H(1)[rule_format, OF mn m(2)] h(1) m(1) show ?thesis
by presburger
- }
- ultimately show "even n"
- using H(2)[unfolded swapidseq_cases[of n id]] by auto
+ qed
qed
@@ -641,11 +620,9 @@
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"
+ 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
+ from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] show ?thesis
by arith
qed
@@ -655,9 +632,9 @@
shows "evenperm p = b"
unfolding n[symmetric] evenperm_def
apply (rule swapidseq_even_even[where p = p])
- apply (rule someI[where x = n])
+ apply (rule someI[where x = n])
using p
- apply blast+
+ apply blast+
done
@@ -670,29 +647,26 @@
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)"
+ assumes "permutation p" "permutation q"
+ shows "evenperm (p \<circ> q) \<longleftrightarrow> evenperm p = evenperm q"
proof -
- from p q obtain n m where n: "swapidseq n p" and m: "swapidseq m q"
+ from assms 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)"
+ have "even (n + m) \<longleftrightarrow> (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
+ and evenperm_unique[OF swapidseq_comp_add[OF n m] this] show ?thesis
by blast
qed
lemma evenperm_inv:
- assumes p: "permutation p"
+ assumes "permutation p"
shows "evenperm (inv p) = evenperm p"
proof -
- from p obtain n where n: "swapidseq n p"
+ from assms 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 .
+ show ?thesis
+ by (rule evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]])
qed
@@ -701,67 +675,71 @@
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
unfolding bij_def inj_def surj_def
apply auto
- apply metis
+ apply metis
apply metis
done
lemma permutation_bijective:
- assumes p: "permutation p"
+ assumes "permutation p"
shows "bij p"
proof -
- from p obtain n where n: "swapidseq n p"
+ from assms 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"
+ 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
+ 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"
+ assumes "permutation p"
shows "finite {x. p x \<noteq> x}"
proof -
- from p obtain n where n: "swapidseq n p"
+ from assms obtain n where "swapidseq n p"
unfolding permutation_def by blast
- from n show ?thesis
+ then 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"
+ from comp_Suc.hyps(2) have *: "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"
- by (auto simp add: Fun.swap_def)
- from finite_subset[OF th fS] show ?case .
+ from \<open>a \<noteq> b\<close> have **: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
+ by (auto simp: Fun.swap_def)
+ show ?case
+ by (rule finite_subset[OF ** *])
qed
qed
lemma permutation_lemma:
- assumes fS: "finite S"
- and p: "bij p"
- and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
+ assumes "finite S"
+ and "bij p"
+ and "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
shows "permutation p"
- using fS p pS
+ using assms
proof (induct S arbitrary: p rule: finite_induct)
- case (empty p)
- then show ?case by simp
+ case empty
+ 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"
+ have *: "?r a = a"
by (simp add: Fun.swap_def)
- 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"
+ from insert * have **: "\<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)
+ have "bij ?r"
+ by (rule bij_swap_compose_bij[OF insert(4)])
+ have "permutation ?r"
+ by (rule insert(3)[OF \<open>bij ?r\<close> **])
+ then have "permutation ?q"
+ by (simp add: permutation_compose permutation_swap_id)
then show ?case
by (simp add: o_assoc)
qed
@@ -769,8 +747,8 @@
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"
+ assume ?lhs
+ with permutation_bijective permutation_finite_support show "?b \<and> ?f"
by auto
next
assume "?b \<and> ?f"
@@ -780,11 +758,10 @@
qed
lemma permutation_inverse_works:
- assumes p: "permutation p"
+ assumes "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
+ using permutation_bijective [OF assms] by (auto simp: bij_def inj_iff surj_iff)
lemma permutation_inverse_compose:
assumes p: "permutation p"
@@ -797,33 +774,34 @@
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" .
+ finally have *: "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 .
+ finally have **: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
+ show ?thesis
+ by (rule inv_unique_comp[OF * **])
qed
-subsection \<open>Relation to "permutes"\<close>
+subsection \<open>Relation to \<open>permutes\<close>\<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 (rule exI[where x="{x. p x \<noteq> x}"])
+ apply simp
apply clarsimp
apply (rule_tac B="S" in finite_subset)
- apply auto
+ 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>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
@@ -842,12 +820,11 @@
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)]
+ 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 .
+ from insert.prems(2)[OF xF pxF Pr Pr rp] qp show ?case
+ by (simp only:)
qed
@@ -878,17 +855,17 @@
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]"
+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)"
+ 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"
+ shows "permute_list f xs ! i = xs ! f i"
using permutes_in_image[OF assms(1)] assms(2)
by (simp add: permute_list_def)
@@ -900,7 +877,7 @@
lemma permute_list_compose:
assumes "g permutes {..<length xs}"
- shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f 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"
@@ -910,44 +887,46 @@
by (simp add: id_def)
lemma mset_permute_list [simp]:
- assumes "f permutes {..<length (xs :: 'a list)}"
- shows "mset (permute_list f xs) = mset xs"
+ fixes xs :: "'a list"
+ assumes "f permutes {..<length xs}"
+ 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})"
+ 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
+ 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"
+ 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"
+ shows "distinct (permute_list f xs) = distinct xs"
by (simp add: distinct_count_atmost_1 assms)
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)"
+ 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
+ from permutes_in_image[OF assms(1)] assms(2) have *: "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]"
by (simp_all add: permute_list_def zip_map_map)
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
+ by (intro nth_equalityI) (simp_all add: *)
also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
by (simp_all add: permute_list_def zip_map_map)
finally show ?thesis .
@@ -955,20 +934,19 @@
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 _)")
+ 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)
+ from assms have "inj \<sigma>" "surj \<sigma>"
+ by (simp_all add: permutes_inj permutes_surj)
+ then show "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x" for x
+ by (induct xs) (auto simp: inv_f_f surj_f_inv_f)
qed
subsection \<open>More lemmas about permutations\<close>
-text \<open>
- The following few lemmas were contributed by Lukas Bulwahn.
-\<close>
+text \<open>The following few lemmas were contributed by Lukas Bulwahn.\<close>
lemma count_image_mset_eq_card_vimage:
assumes "finite A"
@@ -980,19 +958,23 @@
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
+ proof (cases "f x = b")
+ case True
+ with insert.hyps
+ have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
+ by auto
+ also from insert.hyps(1,2) have "\<dots> = card (insert x {a \<in> F. f a = f x})"
+ by simp
+ also from \<open>f x = b\<close> have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
+ 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
+ case False
+ then have "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}"
+ by auto
+ with insert False show ?thesis
+ by simp
qed
qed
@@ -1000,123 +982,116 @@
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)"
+ and 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))"
+ from \<open>finite A\<close> have "f ` A = f ` (set_mset (mset_set A))"
+ 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
+ also from \<open>finite A\<close> have "\<dots> = f' ` A"
+ 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 mset_eq have "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b"
+ by simp
+ with \<open>finite A\<close> have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
by (simp add: count_image_mset_eq_card_vimage)
- from this show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
+ then 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}"
+ then have "\<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
+ then obtain p where p: "\<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}" ..
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 (auto simp: disjoint_family_on_def)
+ moreover
+ have "bij_betw (\<lambda>a. p (f a) a) {a \<in> A. f a = b} {a \<in> A. f' a = b}" if "b \<in> f ` A" for b
+ using p that 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
+ moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A"
+ by auto
+ moreover from \<open>f ` A = f' ` A\<close> have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A"
+ 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
+ show "\<And>x. x \<notin> A \<Longrightarrow> p' x = x"
+ by (simp add: p'_def)
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)
+ ultimately show ?thesis ..
qed
-lemma mset_set_upto_eq_mset_upto:
- "mset_set {..<n} = mset [0..<n]"
- by (induct n) (auto simp add: add.commute lessThan_Suc)
+lemma mset_set_upto_eq_mset_upto: "mset_set {..<n} = mset [0..<n]"
+ by (induct n) (auto simp: add.commute lessThan_Suc)
(* and derive the existing property: *)
lemma mset_eq_permutation:
- assumes mset_eq: "mset (xs::'a list) = mset ys"
+ fixes xs ys :: "'a list"
+ assumes mset_eq: "mset xs = 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
+ by (rule mset_eq_length)
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 (rule mset_set_upto_eq_mset_upto)
+ with mset_eq length_eq 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
+ obtain p where p: "p permutes {..<length ys}" and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)"
+ by auto
+ with length_eq have "permute_list p ys = xs"
+ by (auto intro!: nth_equalityI simp: permute_list_nth)
+ with p show thesis ..
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"
+ assumes "p permutes S"
+ and "\<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_def by blast
- with h have False
- by simp
- }
- ultimately have "p n = n"
- by blast
- }
- ultimately show "p n = n"
- by blast
+ have "p n = n" for n
+ using assms
+ proof (induct n arbitrary: S rule: less_induct)
+ case (less n)
+ show ?case
+ proof (cases "n \<in> S")
+ case False
+ with less(2) show ?thesis
+ unfolding permutes_def by metis
+ next
+ case True
+ with less(3) have "p n < n \<or> p n = n"
+ by auto
+ then show ?thesis
+ proof
+ assume "p n < n"
+ with less have "p (p n) = p n"
+ by metis
+ with permutes_inj[OF less(2)] have "p n = n"
+ unfolding inj_def by blast
+ with \<open>p n < n\<close> have False
+ by simp
+ then show ?thesis ..
+ qed
qed
- }
- then show ?thesis
- by (auto simp add: fun_eq_iff)
+ qed
+ then show ?thesis by (auto simp: fun_eq_iff)
qed
lemma permutes_natset_ge:
@@ -1125,25 +1100,23 @@
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"
+ have "i \<ge> inv p i" if "i \<in> S" for i
+ proof -
+ from that 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"
+ with permutes_inverses[OF p] show ?thesis
by simp
- }
- then have th: "\<forall>i\<in>S. inv p i \<le> i"
+ qed
+ then have "\<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"
+ from permutes_natset_le[OF permutes_inv[OF p] this] 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
+ apply simp_all
done
qed
@@ -1151,31 +1124,31 @@
apply (rule set_eqI)
apply auto
using permutes_inv_inv permutes_inv
- apply auto
+ 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}"
+ assumes "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 permutes_compose)
+ using assms
+ 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"
+ assumes "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 permutes_compose)
+ using assms
+ 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
@@ -1183,12 +1156,11 @@
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 sum_permutations_inverse:
- "sum f {p. p permutes S} = sum (\<lambda>p. f(inv p)) {p. p permutes S}"
+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"
+ have *: "inj_on inv ?S"
proof (auto simp add: inj_on_def)
fix q r
assume q: "q permutes S"
@@ -1199,11 +1171,12 @@
with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r"
by metis
qed
- have th1: "inv ` ?S = ?S"
+ have **: "inv ` ?S = ?S"
using image_inverse_permutations by blast
- have th2: "?rhs = sum (f \<circ> inv) ?S"
+ have ***: "?rhs = sum (f \<circ> inv) ?S"
by (simp add: o_def)
- from sum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 .
+ from sum.reindex[OF *, of f] show ?thesis
+ by (simp only: ** ***)
qed
lemma setum_permutations_compose_left:
@@ -1212,9 +1185,9 @@
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
- have th0: "?rhs = sum (f \<circ> (op \<circ> q)) ?S"
+ have *: "?rhs = sum (f \<circ> (op \<circ> q)) ?S"
by (simp add: o_def)
- have th1: "inj_on (op \<circ> q) ?S"
+ have **: "inj_on (op \<circ> q) ?S"
proof (auto simp add: inj_on_def)
fix p r
assume "p permutes S"
@@ -1225,9 +1198,10 @@
with permutes_inj[OF q, unfolded inj_iff] show "p = r"
by simp
qed
- have th3: "(op \<circ> q) ` ?S = ?S"
+ have "(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 .
+ with * sum.reindex[OF **, of f] show ?thesis
+ by (simp only:)
qed
lemma sum_permutations_compose_right:
@@ -1236,9 +1210,9 @@
(is "?lhs = ?rhs")
proof -
let ?S = "{p. p permutes S}"
- have th0: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
+ have *: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
by (simp add: o_def)
- have th1: "inj_on (\<lambda>p. p \<circ> q) ?S"
+ have **: "inj_on (\<lambda>p. p \<circ> q) ?S"
proof (auto simp add: inj_on_def)
fix p r
assume "p permutes S"
@@ -1249,10 +1223,10 @@
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 .
+ from image_compose_permutations_right[OF q] have "(\<lambda>p. p \<circ> q) ` ?S = ?S"
+ by auto
+ with * sum.reindex[OF **, of f] show ?thesis
+ by (simp only:)
qed
@@ -1264,17 +1238,12 @@
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 *: "\<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"
+ have **: "\<And>P Q. {(a, b). a \<in> P \<and> b \<in> Q} = P \<times> Q"
by blast
show ?thesis
- unfolding permutes_insert
- unfolding sum.cartesian_product
- unfolding th1[symmetric]
- unfolding th0
+ unfolding * ** sum.cartesian_product permutes_insert
proof (rule sum.reindex)
let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
let ?P = "{p. p permutes S}"
@@ -1295,8 +1264,7 @@
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)
+ unfolding o_assoc swap_id_idempotent by simp
with bc have "b = c \<and> p = q"
by blast
}
@@ -1308,48 +1276,53 @@
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)"
+definition list_permutes :: "('a \<times> 'a) list \<Rightarrow> 'a set \<Rightarrow> bool"
+ 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"
+ 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)
+ 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)"
+definition permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
+ 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')"
+ "permutation_of_list ((x, y) # xs) x' = (if x = x' then y else permutation_of_list xs x')"
by (simp add: permutation_of_list_def)
-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)"
+fun inverse_permutation_of_list :: "('a \<times> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
+ 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))"
+ 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)"
+ 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")
- (simp_all add: map_of_eq_None_iff)
- moreover from x'y' have *: "(x,x') \<in> set xs" "(y,y') \<in> set xs"
+ 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") (simp_all add: map_of_eq_None_iff)
+ 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')"])
+ 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"
@@ -1357,13 +1330,13 @@
lemma inj_on_map_of':
assumes "distinct (map snd xs)"
- shows "inj_on (the \<circ> map_of xs) (set (map fst 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)
+ (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)"
+ 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"
@@ -1371,12 +1344,13 @@
lemma image_map_of':
assumes "distinct (map fst xs)"
- shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd 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 _")
+ 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"
by (simp add: list_permutes_def)
@@ -1384,12 +1358,12 @@
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)
+ (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)
+ (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))" .
@@ -1407,52 +1381,47 @@
"x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x"
by (simp_all add: inverse_permutation_of_list.simps)
-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_id: "x \<notin> set (map fst xs) \<Longrightarrow> permutation_of_list xs x = x"
+ by (induct 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)+
+ "distinct (map fst xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
+ by (induct 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)
+ "list_permutes xs A \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> permutation_of_list xs x = y"
+ 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
+ "x \<notin> set (map snd xs) \<Longrightarrow> inverse_permutation_of_list xs x = x"
+ by (induct 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)+
+ "distinct (map snd xs) \<Longrightarrow> (x, y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
+ by (induct 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)
+ "list_permutes xs A \<Longrightarrow> (x,y) \<in> set xs \<Longrightarrow> inverse_permutation_of_list xs y = x"
+ 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)"
+ fixes A :: "'a set"
+ assumes "list_permutes xs A"
+ 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"
+ from assms show "permutation_of_list xs permutes A"
+ by simp
+ show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" for x
proof (cases "x \<in> set (map snd xs)")
case True
- then obtain y where "(y, x) \<in> set xs" by force
+ then obtain y where "(y, x) \<in> set xs" by auto
with assms show ?thesis
by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique)
- qed (insert assms, auto simp: list_permutes_def
- inverse_permutation_of_list_id permutation_of_list_id)
+ next
+ case False
+ with assms show ?thesis
+ by (auto simp: list_permutes_def inverse_permutation_of_list_id permutation_of_list_id)
+ qed
qed
end
-