author | eberlm <eberlm@in.tum.de> |
Mon, 19 Sep 2016 17:37:22 +0200 | |
changeset 63921 | 0a5184877cb7 |
parent 63539 | 70d4d9e5707b |
child 64267 | b9a1486e79be |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Library/Permutations.thy |
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Author: Amine Chaieb, University of Cambridge |
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*) |
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Permutations, both general and specifically on finite sets.
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section \<open>Permutations, both general and specifically on finite sets.\<close> |
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Permutations, both general and specifically on finite sets.
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Permutations, both general and specifically on finite sets.
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theory Permutations |
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imports Binomial Multiset Disjoint_Sets |
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begin |
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subsection \<open>Transpositions\<close> |
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lemma swap_id_idempotent [simp]: |
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"Fun.swap a b id \<circ> Fun.swap a b id = id" |
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by (rule ext, auto simp add: Fun.swap_def) |
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lemma inv_swap_id: |
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"inv (Fun.swap a b id) = Fun.swap a b id" |
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by (rule inv_unique_comp) simp_all |
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lemma swap_id_eq: |
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"Fun.swap a b id x = (if x = a then b else if x = b then a else x)" |
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by (simp add: Fun.swap_def) |
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subsection \<open>Basic consequences of the definition\<close> |
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definition permutes (infixr "permutes" 41) |
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where "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)" |
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lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S" |
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unfolding permutes_def by metis |
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lemma permutes_not_in: |
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assumes "f permutes S" "x \<notin> S" shows "f x = x" |
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using assms by (auto simp: permutes_def) |
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lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S" |
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unfolding permutes_def |
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apply (rule set_eqI) |
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apply (simp add: image_iff) |
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Permutations, both general and specifically on finite sets.
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apply metis |
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Permutations, both general and specifically on finite sets.
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done |
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lemma permutes_inj: "p permutes S \<Longrightarrow> inj p" |
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unfolding permutes_def inj_on_def by blast |
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lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A" |
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unfolding permutes_def inj_on_def by auto |
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lemma permutes_surj: "p permutes s \<Longrightarrow> surj p" |
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unfolding permutes_def surj_def by metis |
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lemma permutes_bij: "p permutes s \<Longrightarrow> bij p" |
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unfolding bij_def by (metis permutes_inj permutes_surj) |
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lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S" |
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by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI) |
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lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S" |
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unfolding permutes_def bij_betw_def inj_on_def |
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by auto (metis image_iff)+ |
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lemma permutes_inv_o: |
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assumes pS: "p permutes S" |
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shows "p \<circ> inv p = id" |
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and "inv p \<circ> p = id" |
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using permutes_inj[OF pS] permutes_surj[OF pS] |
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unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+ |
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lemma permutes_inverses: |
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fixes p :: "'a \<Rightarrow> 'a" |
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assumes pS: "p permutes S" |
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shows "p (inv p x) = x" |
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and "inv p (p x) = x" |
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using permutes_inv_o[OF pS, unfolded fun_eq_iff o_def] by auto |
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lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T" |
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unfolding permutes_def by blast |
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80 |
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lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id" |
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unfolding fun_eq_iff permutes_def by simp metis |
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lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id" |
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unfolding fun_eq_iff permutes_def by simp metis |
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lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)" |
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unfolding permutes_def by simp |
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lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y" |
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unfolding permutes_def inv_def |
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apply auto |
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apply (erule allE[where x=y]) |
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apply (erule allE[where x=y]) |
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apply (rule someI_ex) |
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apply blast |
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apply (rule some1_equality) |
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apply blast |
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apply blast |
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done |
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lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S" |
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unfolding permutes_def Fun.swap_def fun_upd_def by auto metis |
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lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T" |
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by (simp add: Ball_def permutes_def) metis |
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||
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(* Next three lemmas contributed by Lukas Bulwahn *) |
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lemma permutes_bij_inv_into: |
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fixes A :: "'a set" and B :: "'b set" |
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assumes "p permutes A" |
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assumes "bij_betw f A B" |
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shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B" |
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114 |
proof (rule bij_imp_permutes) |
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115 |
have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A" |
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116 |
using assms by (auto simp add: permutes_imp_bij bij_betw_inv_into) |
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from this have "bij_betw (f o p o inv_into A f) B B" by (simp add: bij_betw_trans) |
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from this show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B" |
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119 |
by (subst bij_betw_cong[where g="f o p o inv_into A f"]) auto |
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120 |
next |
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121 |
fix x |
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122 |
assume "x \<notin> B" |
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123 |
from this show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto |
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124 |
qed |
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125 |
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126 |
lemma permutes_image_mset: |
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127 |
assumes "p permutes A" |
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128 |
shows "image_mset p (mset_set A) = mset_set A" |
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129 |
using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image) |
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130 |
|
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lemma permutes_implies_image_mset_eq: |
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132 |
assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)" |
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eberlm <eberlm@in.tum.de>
parents:
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133 |
shows "image_mset f' (mset_set A) = image_mset f (mset_set A)" |
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eberlm <eberlm@in.tum.de>
parents:
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134 |
proof - |
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eberlm <eberlm@in.tum.de>
parents:
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135 |
have "f x = f' (p x)" if x: "x \<in># mset_set A" for x |
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parents:
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136 |
using assms(2)[of x] x by (cases "finite A") auto |
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parents:
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137 |
from this have "image_mset f (mset_set A) = image_mset (f' o p) (mset_set A)" |
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eberlm <eberlm@in.tum.de>
parents:
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138 |
using assms by (auto intro!: image_mset_cong) |
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parents:
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139 |
also have "\<dots> = image_mset f' (image_mset p (mset_set A))" |
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parents:
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140 |
by (simp add: image_mset.compositionality) |
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parents:
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141 |
also have "\<dots> = image_mset f' (mset_set A)" |
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eberlm <eberlm@in.tum.de>
parents:
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142 |
proof - |
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parents:
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143 |
from assms have "image_mset p (mset_set A) = mset_set A" |
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parents:
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144 |
using permutes_image_mset by blast |
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eberlm <eberlm@in.tum.de>
parents:
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145 |
from this show ?thesis by simp |
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parents:
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146 |
qed |
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parents:
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147 |
finally show ?thesis .. |
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parents:
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148 |
qed |
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eberlm <eberlm@in.tum.de>
parents:
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149 |
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parents:
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|
60500 | 151 |
subsection \<open>Group properties\<close> |
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chaieb
parents:
diff
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|
152 |
|
54681 | 153 |
lemma permutes_id: "id permutes S" |
154 |
unfolding permutes_def by simp |
|
29840
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chaieb
parents:
diff
changeset
|
155 |
|
54681 | 156 |
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S" |
29840
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chaieb
parents:
diff
changeset
|
157 |
unfolding permutes_def o_def by metis |
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chaieb
parents:
diff
changeset
|
158 |
|
54681 | 159 |
lemma permutes_inv: |
160 |
assumes pS: "p permutes S" |
|
161 |
shows "inv p permutes S" |
|
30488 | 162 |
using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis |
29840
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chaieb
parents:
diff
changeset
|
163 |
|
54681 | 164 |
lemma permutes_inv_inv: |
165 |
assumes pS: "p permutes S" |
|
166 |
shows "inv (inv p) = p" |
|
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renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
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diff
changeset
|
167 |
unfolding fun_eq_iff permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]] |
29840
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chaieb
parents:
diff
changeset
|
168 |
by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
169 |
|
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|
170 |
lemma permutes_invI: |
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|
171 |
assumes perm: "p permutes S" |
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changeset
|
172 |
and inv: "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x" |
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changeset
|
173 |
and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x" |
af0e964aad7b
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changeset
|
174 |
shows "inv p = p'" |
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|
175 |
proof |
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changeset
|
176 |
fix x show "inv p x = p' x" |
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|
177 |
proof (cases "x \<in> S") |
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|
178 |
assume [simp]: "x \<in> S" |
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|
179 |
from assms have "p' x = p' (p (inv p x))" by (simp add: permutes_inverses) |
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|
180 |
also from permutes_inv[OF perm] |
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|
181 |
have "\<dots> = inv p x" by (subst inv) (simp_all add: permutes_in_image) |
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|
182 |
finally show "inv p x = p' x" .. |
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|
183 |
qed (insert permutes_inv[OF perm], simp_all add: outside permutes_not_in) |
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|
184 |
qed |
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diff
changeset
|
185 |
|
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|
186 |
lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A" |
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|
187 |
by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv]) |
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|
188 |
|
54681 | 189 |
|
60500 | 190 |
subsection \<open>The number of permutations on a finite set\<close> |
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
191 |
|
30488 | 192 |
lemma permutes_insert_lemma: |
29840
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chaieb
parents:
diff
changeset
|
193 |
assumes pS: "p permutes (insert a S)" |
54681 | 194 |
shows "Fun.swap a (p a) id \<circ> p permutes S" |
29840
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chaieb
parents:
diff
changeset
|
195 |
apply (rule permutes_superset[where S = "insert a S"]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
196 |
apply (rule permutes_compose[OF pS]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
197 |
apply (rule permutes_swap_id, simp) |
54681 | 198 |
using permutes_in_image[OF pS, of a] |
199 |
apply simp |
|
56545 | 200 |
apply (auto simp add: Ball_def Fun.swap_def) |
29840
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chaieb
parents:
diff
changeset
|
201 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
202 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
203 |
lemma permutes_insert: "{p. p permutes (insert a S)} = |
54681 | 204 |
(\<lambda>(b,p). Fun.swap a b id \<circ> p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}" |
205 |
proof - |
|
206 |
{ |
|
207 |
fix p |
|
208 |
{ |
|
209 |
assume pS: "p permutes insert a S" |
|
29840
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chaieb
parents:
diff
changeset
|
210 |
let ?b = "p a" |
54681 | 211 |
let ?q = "Fun.swap a (p a) id \<circ> p" |
212 |
have th0: "p = Fun.swap a ?b id \<circ> ?q" |
|
213 |
unfolding fun_eq_iff o_assoc by simp |
|
214 |
have th1: "?b \<in> insert a S" |
|
215 |
unfolding permutes_in_image[OF pS] by simp |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
216 |
from permutes_insert_lemma[OF pS] th0 th1 |
54681 | 217 |
have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S" by blast |
218 |
} |
|
29840
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chaieb
parents:
diff
changeset
|
219 |
moreover |
54681 | 220 |
{ |
221 |
fix b q |
|
222 |
assume bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S" |
|
30488 | 223 |
from permutes_subset[OF bq(3), of "insert a S"] |
54681 | 224 |
have qS: "q permutes insert a S" |
225 |
by auto |
|
226 |
have aS: "a \<in> insert a S" |
|
227 |
by simp |
|
29840
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chaieb
parents:
diff
changeset
|
228 |
from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]] |
54681 | 229 |
have "p permutes insert a S" |
230 |
by simp |
|
231 |
} |
|
232 |
ultimately have "p permutes insert a S \<longleftrightarrow> |
|
233 |
(\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)" |
|
234 |
by blast |
|
235 |
} |
|
236 |
then show ?thesis |
|
237 |
by auto |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
238 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
239 |
|
54681 | 240 |
lemma card_permutations: |
241 |
assumes Sn: "card S = n" |
|
242 |
and fS: "finite S" |
|
33715 | 243 |
shows "card {p. p permutes S} = fact n" |
54681 | 244 |
using fS Sn |
245 |
proof (induct arbitrary: n) |
|
246 |
case empty |
|
247 |
then show ?case by simp |
|
33715 | 248 |
next |
249 |
case (insert x F) |
|
54681 | 250 |
{ |
251 |
fix n |
|
252 |
assume H0: "card (insert x F) = n" |
|
33715 | 253 |
let ?xF = "{p. p permutes insert x F}" |
254 |
let ?pF = "{p. p permutes F}" |
|
255 |
let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}" |
|
256 |
let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)" |
|
257 |
from permutes_insert[of x F] |
|
258 |
have xfgpF': "?xF = ?g ` ?pF'" . |
|
54681 | 259 |
have Fs: "card F = n - 1" |
60500 | 260 |
using \<open>x \<notin> F\<close> H0 \<open>finite F\<close> by auto |
54681 | 261 |
from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)" |
60500 | 262 |
using \<open>finite F\<close> by auto |
54681 | 263 |
then have "finite ?pF" |
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
264 |
by (auto intro: card_ge_0_finite) |
54681 | 265 |
then have pF'f: "finite ?pF'" |
60500 | 266 |
using H0 \<open>finite F\<close> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
60601
diff
changeset
|
267 |
apply (simp only: Collect_case_prod Collect_mem_eq) |
33715 | 268 |
apply (rule finite_cartesian_product) |
269 |
apply simp_all |
|
270 |
done |
|
29840
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Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
271 |
|
33715 | 272 |
have ginj: "inj_on ?g ?pF'" |
54681 | 273 |
proof - |
33715 | 274 |
{ |
54681 | 275 |
fix b p c q |
276 |
assume bp: "(b,p) \<in> ?pF'" |
|
277 |
assume cq: "(c,q) \<in> ?pF'" |
|
278 |
assume eq: "?g (b,p) = ?g (c,q)" |
|
279 |
from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" |
|
280 |
"p permutes F" "q permutes F" |
|
281 |
by auto |
|
60500 | 282 |
from ths(4) \<open>x \<notin> F\<close> eq have "b = ?g (b,p) x" |
54681 | 283 |
unfolding permutes_def |
56545 | 284 |
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff) |
54681 | 285 |
also have "\<dots> = ?g (c,q) x" |
60500 | 286 |
using ths(5) \<open>x \<notin> F\<close> eq |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
287 |
by (auto simp add: swap_def fun_upd_def fun_eq_iff) |
54681 | 288 |
also have "\<dots> = c" |
60500 | 289 |
using ths(5) \<open>x \<notin> F\<close> |
54681 | 290 |
unfolding permutes_def |
56545 | 291 |
by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff) |
33715 | 292 |
finally have bc: "b = c" . |
54681 | 293 |
then have "Fun.swap x b id = Fun.swap x c id" |
294 |
by simp |
|
295 |
with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q" |
|
296 |
by simp |
|
297 |
then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) = |
|
298 |
Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)" |
|
299 |
by simp |
|
300 |
then have "p = q" |
|
301 |
by (simp add: o_assoc) |
|
302 |
with bc have "(b, p) = (c, q)" |
|
303 |
by simp |
|
33715 | 304 |
} |
54681 | 305 |
then show ?thesis |
306 |
unfolding inj_on_def by blast |
|
33715 | 307 |
qed |
60500 | 308 |
from \<open>x \<notin> F\<close> H0 have n0: "n \<noteq> 0" |
309 |
using \<open>finite F\<close> by auto |
|
54681 | 310 |
then have "\<exists>m. n = Suc m" |
311 |
by presburger |
|
312 |
then obtain m where n[simp]: "n = Suc m" |
|
313 |
by blast |
|
33715 | 314 |
from pFs H0 have xFc: "card ?xF = fact n" |
54681 | 315 |
unfolding xfgpF' card_image[OF ginj] |
60500 | 316 |
using \<open>finite F\<close> \<open>finite ?pF\<close> |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
60601
diff
changeset
|
317 |
apply (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product) |
54681 | 318 |
apply simp |
319 |
done |
|
320 |
from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" |
|
321 |
unfolding xfgpF' by simp |
|
33715 | 322 |
have "card ?xF = fact n" |
323 |
using xFf xFc unfolding xFf by blast |
|
324 |
} |
|
54681 | 325 |
then show ?case |
326 |
using insert by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
327 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
328 |
|
54681 | 329 |
lemma finite_permutations: |
330 |
assumes fS: "finite S" |
|
331 |
shows "finite {p. p permutes S}" |
|
59730
b7c394c7a619
The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents:
59669
diff
changeset
|
332 |
using card_permutations[OF refl fS] |
33715 | 333 |
by (auto intro: card_ge_0_finite) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
334 |
|
54681 | 335 |
|
60500 | 336 |
subsection \<open>Permutations of index set for iterated operations\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
337 |
|
51489 | 338 |
lemma (in comm_monoid_set) permute: |
339 |
assumes "p permutes S" |
|
54681 | 340 |
shows "F g S = F (g \<circ> p) S" |
51489 | 341 |
proof - |
60500 | 342 |
from \<open>p permutes S\<close> have "inj p" |
54681 | 343 |
by (rule permutes_inj) |
344 |
then have "inj_on p S" |
|
345 |
by (auto intro: subset_inj_on) |
|
346 |
then have "F g (p ` S) = F (g \<circ> p) S" |
|
347 |
by (rule reindex) |
|
60500 | 348 |
moreover from \<open>p permutes S\<close> have "p ` S = S" |
54681 | 349 |
by (rule permutes_image) |
350 |
ultimately show ?thesis |
|
351 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
352 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
353 |
|
54681 | 354 |
|
60500 | 355 |
subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close> |
54681 | 356 |
|
357 |
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow> |
|
358 |
Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id" |
|
56545 | 359 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
360 |
|
54681 | 361 |
lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> |
362 |
Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id" |
|
56545 | 363 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
364 |
|
54681 | 365 |
lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow> |
366 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id" |
|
56545 | 367 |
by (simp add: fun_eq_iff Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
368 |
|
54681 | 369 |
|
60500 | 370 |
subsection \<open>Permutations as transposition sequences\<close> |
54681 | 371 |
|
372 |
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" |
|
373 |
where |
|
374 |
id[simp]: "swapidseq 0 id" |
|
375 |
| comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)" |
|
376 |
||
377 |
declare id[unfolded id_def, simp] |
|
378 |
||
379 |
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
380 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
381 |
|
60500 | 382 |
subsection \<open>Some closure properties of the set of permutations, with lengths\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
383 |
|
54681 | 384 |
lemma permutation_id[simp]: "permutation id" |
385 |
unfolding permutation_def by (rule exI[where x=0]) simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
386 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
387 |
declare permutation_id[unfolded id_def, simp] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
388 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
389 |
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
390 |
apply clarsimp |
54681 | 391 |
using comp_Suc[of 0 id a b] |
392 |
apply simp |
|
393 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
394 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
395 |
lemma permutation_swap_id: "permutation (Fun.swap a b id)" |
54681 | 396 |
apply (cases "a = b") |
397 |
apply simp_all |
|
398 |
unfolding permutation_def |
|
399 |
using swapidseq_swap[of a b] |
|
400 |
apply blast |
|
401 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
402 |
|
54681 | 403 |
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)" |
404 |
proof (induct n p arbitrary: m q rule: swapidseq.induct) |
|
405 |
case (id m q) |
|
406 |
then show ?case by simp |
|
407 |
next |
|
408 |
case (comp_Suc n p a b m q) |
|
409 |
have th: "Suc n + m = Suc (n + m)" |
|
410 |
by arith |
|
411 |
show ?case |
|
412 |
unfolding th comp_assoc |
|
413 |
apply (rule swapidseq.comp_Suc) |
|
414 |
using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3) |
|
415 |
apply blast+ |
|
416 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
417 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
418 |
|
54681 | 419 |
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
420 |
unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
421 |
|
54681 | 422 |
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
423 |
apply (induct n p rule: swapidseq.induct) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
424 |
using swapidseq_swap[of a b] |
54681 | 425 |
apply (auto simp add: comp_assoc intro: swapidseq.comp_Suc) |
426 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
427 |
|
54681 | 428 |
lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id" |
429 |
proof (induct n p rule: swapidseq.induct) |
|
430 |
case id |
|
431 |
then show ?case |
|
432 |
by (rule exI[where x=id]) simp |
|
30488 | 433 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
434 |
case (comp_Suc n p a b) |
54681 | 435 |
from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
436 |
by blast |
|
437 |
let ?q = "q \<circ> Fun.swap a b id" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
438 |
note H = comp_Suc.hyps |
54681 | 439 |
from swapidseq_swap[of a b] H(3) have th0: "swapidseq 1 (Fun.swap a b id)" |
440 |
by simp |
|
441 |
from swapidseq_comp_add[OF q(1) th0] have th1: "swapidseq (Suc n) ?q" |
|
442 |
by simp |
|
443 |
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" |
|
444 |
by (simp add: o_assoc) |
|
445 |
also have "\<dots> = id" |
|
446 |
by (simp add: q(2)) |
|
447 |
finally have th2: "Fun.swap a b id \<circ> p \<circ> ?q = id" . |
|
448 |
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" |
|
449 |
by (simp only: o_assoc) |
|
450 |
then have "?q \<circ> (Fun.swap a b id \<circ> p) = id" |
|
451 |
by (simp add: q(3)) |
|
452 |
with th1 th2 show ?case |
|
453 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
454 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
455 |
|
54681 | 456 |
lemma swapidseq_inverse: |
457 |
assumes H: "swapidseq n p" |
|
458 |
shows "swapidseq n (inv p)" |
|
459 |
using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto |
|
460 |
||
461 |
lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)" |
|
462 |
using permutation_def swapidseq_inverse by blast |
|
463 |
||
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
464 |
|
60500 | 465 |
subsection \<open>The identity map only has even transposition sequences\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
466 |
|
54681 | 467 |
lemma symmetry_lemma: |
468 |
assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c" |
|
469 |
and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> |
|
470 |
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> |
|
471 |
P a b c d" |
|
472 |
shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> P a b c d" |
|
473 |
using assms by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
474 |
|
54681 | 475 |
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> |
476 |
Fun.swap a b id \<circ> Fun.swap c d id = id \<or> |
|
477 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> |
|
478 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)" |
|
479 |
proof - |
|
480 |
assume H: "a \<noteq> b" "c \<noteq> d" |
|
481 |
have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow> |
|
482 |
(Fun.swap a b id \<circ> Fun.swap c d id = id \<or> |
|
483 |
(\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> |
|
484 |
Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))" |
|
485 |
apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d]) |
|
56545 | 486 |
apply (simp_all only: swap_commute) |
54681 | 487 |
apply (case_tac "a = c \<and> b = d") |
56608 | 488 |
apply (clarsimp simp only: swap_commute swap_id_idempotent) |
54681 | 489 |
apply (case_tac "a = c \<and> b \<noteq> d") |
490 |
apply (rule disjI2) |
|
491 |
apply (rule_tac x="b" in exI) |
|
492 |
apply (rule_tac x="d" in exI) |
|
493 |
apply (rule_tac x="b" in exI) |
|
56545 | 494 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 495 |
apply (case_tac "a \<noteq> c \<and> b = d") |
496 |
apply (rule disjI2) |
|
497 |
apply (rule_tac x="c" in exI) |
|
498 |
apply (rule_tac x="d" in exI) |
|
499 |
apply (rule_tac x="c" in exI) |
|
56545 | 500 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 501 |
apply (rule disjI2) |
502 |
apply (rule_tac x="c" in exI) |
|
503 |
apply (rule_tac x="d" in exI) |
|
504 |
apply (rule_tac x="b" in exI) |
|
56545 | 505 |
apply (clarsimp simp add: fun_eq_iff Fun.swap_def) |
54681 | 506 |
done |
507 |
with H show ?thesis by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
508 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
509 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
510 |
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
511 |
using swapidseq.cases[of 0 p "p = id"] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
512 |
by auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
513 |
|
54681 | 514 |
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> |
515 |
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)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
516 |
apply (rule iffI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
517 |
apply (erule swapidseq.cases[of n p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
518 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
519 |
apply (rule disjI2) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
520 |
apply (rule_tac x= "a" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
521 |
apply (rule_tac x= "b" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
522 |
apply (rule_tac x= "pa" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
523 |
apply (rule_tac x= "na" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
524 |
apply simp |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
525 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
526 |
apply (rule comp_Suc, simp_all) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
527 |
done |
54681 | 528 |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
529 |
lemma fixing_swapidseq_decrease: |
54681 | 530 |
assumes spn: "swapidseq n p" |
531 |
and ab: "a \<noteq> b" |
|
532 |
and pa: "(Fun.swap a b id \<circ> p) a = a" |
|
533 |
shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
534 |
using spn ab pa |
54681 | 535 |
proof (induct n arbitrary: p a b) |
536 |
case 0 |
|
537 |
then show ?case |
|
56545 | 538 |
by (auto simp add: Fun.swap_def fun_upd_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
539 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
540 |
case (Suc n p a b) |
54681 | 541 |
from Suc.prems(1) swapidseq_cases[of "Suc n" p] |
542 |
obtain c d q m where |
|
543 |
cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
544 |
by auto |
54681 | 545 |
{ |
546 |
assume H: "Fun.swap a b id \<circ> Fun.swap c d id = id" |
|
547 |
have ?case by (simp only: cdqm o_assoc H) (simp add: cdqm) |
|
548 |
} |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
549 |
moreover |
54681 | 550 |
{ |
551 |
fix x y z |
|
552 |
assume H: "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y" |
|
553 |
"Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id" |
|
554 |
from H have az: "a \<noteq> z" |
|
555 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
556 |
|
54681 | 557 |
{ |
558 |
fix h |
|
559 |
have "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a" |
|
56545 | 560 |
using H by (simp add: Fun.swap_def) |
54681 | 561 |
} |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
562 |
note th3 = this |
54681 | 563 |
from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)" |
564 |
by simp |
|
565 |
then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)" |
|
566 |
by (simp add: o_assoc H) |
|
567 |
then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a" |
|
568 |
by simp |
|
569 |
then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a" |
|
570 |
unfolding Suc by metis |
|
571 |
then have th1: "(Fun.swap a z id \<circ> q) a = a" |
|
572 |
unfolding th3 . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
573 |
from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1] |
54681 | 574 |
have th2: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0" |
575 |
by blast+ |
|
576 |
have th: "Suc n - 1 = Suc (n - 1)" |
|
577 |
using th2(2) by auto |
|
578 |
have ?case |
|
579 |
unfolding cdqm(2) H o_assoc th |
|
49739 | 580 |
apply (simp only: Suc_not_Zero simp_thms comp_assoc) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
581 |
apply (rule comp_Suc) |
54681 | 582 |
using th2 H |
583 |
apply blast+ |
|
584 |
done |
|
585 |
} |
|
586 |
ultimately show ?case |
|
587 |
using swap_general[OF Suc.prems(2) cdqm(4)] by metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
588 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
589 |
|
30488 | 590 |
lemma swapidseq_identity_even: |
54681 | 591 |
assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" |
592 |
shows "even n" |
|
60500 | 593 |
using \<open>swapidseq n id\<close> |
54681 | 594 |
proof (induct n rule: nat_less_induct) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
595 |
fix n |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
596 |
assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)" |
54681 | 597 |
{ |
598 |
assume "n = 0" |
|
599 |
then have "even n" by presburger |
|
600 |
} |
|
30488 | 601 |
moreover |
54681 | 602 |
{ |
603 |
fix a b :: 'a and q m |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
604 |
assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
605 |
from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]] |
54681 | 606 |
have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" |
607 |
by auto |
|
608 |
from h m have mn: "m - 1 < n" |
|
609 |
by arith |
|
610 |
from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" |
|
611 |
by presburger |
|
612 |
} |
|
613 |
ultimately show "even n" |
|
614 |
using H(2)[unfolded swapidseq_cases[of n id]] by auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
615 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
616 |
|
54681 | 617 |
|
60500 | 618 |
subsection \<open>Therefore we have a welldefined notion of parity\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
619 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
620 |
definition "evenperm p = even (SOME n. swapidseq n p)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
621 |
|
54681 | 622 |
lemma swapidseq_even_even: |
623 |
assumes m: "swapidseq m p" |
|
624 |
and n: "swapidseq n p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
625 |
shows "even m \<longleftrightarrow> even n" |
54681 | 626 |
proof - |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
627 |
from swapidseq_inverse_exists[OF n] |
54681 | 628 |
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
629 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
630 |
from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]] |
54681 | 631 |
show ?thesis |
632 |
by arith |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
633 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
634 |
|
54681 | 635 |
lemma evenperm_unique: |
636 |
assumes p: "swapidseq n p" |
|
637 |
and n:"even n = b" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
638 |
shows "evenperm p = b" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
639 |
unfolding n[symmetric] evenperm_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
640 |
apply (rule swapidseq_even_even[where p = p]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
641 |
apply (rule someI[where x = n]) |
54681 | 642 |
using p |
643 |
apply blast+ |
|
644 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
645 |
|
54681 | 646 |
|
60500 | 647 |
subsection \<open>And it has the expected composition properties\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
648 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
649 |
lemma evenperm_id[simp]: "evenperm id = True" |
54681 | 650 |
by (rule evenperm_unique[where n = 0]) simp_all |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
651 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
652 |
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)" |
54681 | 653 |
by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
654 |
|
30488 | 655 |
lemma evenperm_comp: |
54681 | 656 |
assumes p: "permutation p" |
657 |
and q:"permutation q" |
|
658 |
shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)" |
|
659 |
proof - |
|
660 |
from p q obtain n m where n: "swapidseq n p" and m: "swapidseq m q" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
661 |
unfolding permutation_def by blast |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
662 |
note nm = swapidseq_comp_add[OF n m] |
54681 | 663 |
have th: "even (n + m) = (even n \<longleftrightarrow> even m)" |
664 |
by arith |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
665 |
from evenperm_unique[OF n refl] evenperm_unique[OF m refl] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
666 |
evenperm_unique[OF nm th] |
54681 | 667 |
show ?thesis |
668 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
669 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
670 |
|
54681 | 671 |
lemma evenperm_inv: |
672 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
673 |
shows "evenperm (inv p) = evenperm p" |
54681 | 674 |
proof - |
675 |
from p obtain n where n: "swapidseq n p" |
|
676 |
unfolding permutation_def by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
677 |
from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
678 |
show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
679 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
680 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
681 |
|
60500 | 682 |
subsection \<open>A more abstract characterization of permutations\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
683 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
684 |
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
685 |
unfolding bij_def inj_on_def surj_def |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
686 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
687 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
688 |
apply metis |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
689 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
690 |
|
30488 | 691 |
lemma permutation_bijective: |
692 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
693 |
shows "bij p" |
54681 | 694 |
proof - |
695 |
from p obtain n where n: "swapidseq n p" |
|
696 |
unfolding permutation_def by blast |
|
697 |
from swapidseq_inverse_exists[OF n] |
|
698 |
obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" |
|
699 |
by blast |
|
700 |
then show ?thesis unfolding bij_iff |
|
701 |
apply (auto simp add: fun_eq_iff) |
|
702 |
apply metis |
|
703 |
done |
|
30488 | 704 |
qed |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
705 |
|
54681 | 706 |
lemma permutation_finite_support: |
707 |
assumes p: "permutation p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
708 |
shows "finite {x. p x \<noteq> x}" |
54681 | 709 |
proof - |
710 |
from p obtain n where n: "swapidseq n p" |
|
711 |
unfolding permutation_def by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
712 |
from n show ?thesis |
54681 | 713 |
proof (induct n p rule: swapidseq.induct) |
714 |
case id |
|
715 |
then show ?case by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
716 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
717 |
case (comp_Suc n p a b) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
718 |
let ?S = "insert a (insert b {x. p x \<noteq> x})" |
54681 | 719 |
from comp_Suc.hyps(2) have fS: "finite ?S" |
720 |
by simp |
|
60500 | 721 |
from \<open>a \<noteq> b\<close> have th: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S" |
56545 | 722 |
by (auto simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
723 |
from finite_subset[OF th fS] show ?case . |
54681 | 724 |
qed |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
725 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
726 |
|
54681 | 727 |
lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" |
728 |
using surj_f_inv_f[of p] by (auto simp add: bij_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
729 |
|
30488 | 730 |
lemma bij_swap_comp: |
54681 | 731 |
assumes bp: "bij p" |
732 |
shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
733 |
using surj_f_inv_f[OF bij_is_surj[OF bp]] |
56545 | 734 |
by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp]) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
735 |
|
54681 | 736 |
lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)" |
737 |
proof - |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
738 |
assume H: "bij p" |
30488 | 739 |
show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
740 |
unfolding bij_swap_comp[OF H] bij_swap_iff |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
741 |
using H . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
742 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
743 |
|
30488 | 744 |
lemma permutation_lemma: |
54681 | 745 |
assumes fS: "finite S" |
746 |
and p: "bij p" |
|
747 |
and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
748 |
shows "permutation p" |
54681 | 749 |
using fS p pS |
750 |
proof (induct S arbitrary: p rule: finite_induct) |
|
751 |
case (empty p) |
|
752 |
then show ?case by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
753 |
next |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
754 |
case (insert a F p) |
54681 | 755 |
let ?r = "Fun.swap a (p a) id \<circ> p" |
756 |
let ?q = "Fun.swap a (p a) id \<circ> ?r" |
|
757 |
have raa: "?r a = a" |
|
56545 | 758 |
by (simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
759 |
from bij_swap_ompose_bij[OF insert(4)] |
30488 | 760 |
have br: "bij ?r" . |
761 |
||
762 |
from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x" |
|
56545 | 763 |
apply (clarsimp simp add: Fun.swap_def) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
764 |
apply (erule_tac x="x" in allE) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
765 |
apply auto |
54681 | 766 |
unfolding bij_iff |
767 |
apply metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
768 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
769 |
from insert(3)[OF br th] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
770 |
have rp: "permutation ?r" . |
54681 | 771 |
have "permutation ?q" |
772 |
by (simp add: permutation_compose permutation_swap_id rp) |
|
773 |
then show ?case |
|
774 |
by (simp add: o_assoc) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
775 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
776 |
|
30488 | 777 |
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
778 |
(is "?lhs \<longleftrightarrow> ?b \<and> ?f") |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
779 |
proof |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
780 |
assume p: ?lhs |
54681 | 781 |
from p permutation_bijective permutation_finite_support show "?b \<and> ?f" |
782 |
by auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
783 |
next |
54681 | 784 |
assume "?b \<and> ?f" |
785 |
then have "?f" "?b" by blast+ |
|
786 |
from permutation_lemma[OF this] show ?lhs |
|
787 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
788 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
789 |
|
54681 | 790 |
lemma permutation_inverse_works: |
791 |
assumes p: "permutation p" |
|
792 |
shows "inv p \<circ> p = id" |
|
793 |
and "p \<circ> inv p = id" |
|
44227
78e033e8ba05
get Library/Permutations.thy compiled and working again
huffman
parents:
41959
diff
changeset
|
794 |
using permutation_bijective [OF p] |
78e033e8ba05
get Library/Permutations.thy compiled and working again
huffman
parents:
41959
diff
changeset
|
795 |
unfolding bij_def inj_iff surj_iff by auto |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
796 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
797 |
lemma permutation_inverse_compose: |
54681 | 798 |
assumes p: "permutation p" |
799 |
and q: "permutation q" |
|
800 |
shows "inv (p \<circ> q) = inv q \<circ> inv p" |
|
801 |
proof - |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
802 |
note ps = permutation_inverse_works[OF p] |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
803 |
note qs = permutation_inverse_works[OF q] |
54681 | 804 |
have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p" |
805 |
by (simp add: o_assoc) |
|
806 |
also have "\<dots> = id" |
|
807 |
by (simp add: ps qs) |
|
808 |
finally have th0: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" . |
|
809 |
have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q" |
|
810 |
by (simp add: o_assoc) |
|
811 |
also have "\<dots> = id" |
|
812 |
by (simp add: ps qs) |
|
813 |
finally have th1: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
814 |
from inv_unique_comp[OF th0 th1] show ?thesis . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
815 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
816 |
|
54681 | 817 |
|
60500 | 818 |
subsection \<open>Relation to "permutes"\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
819 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
820 |
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)" |
54681 | 821 |
unfolding permutation permutes_def bij_iff[symmetric] |
822 |
apply (rule iffI, clarify) |
|
823 |
apply (rule exI[where x="{x. p x \<noteq> x}"]) |
|
824 |
apply simp |
|
825 |
apply clarsimp |
|
826 |
apply (rule_tac B="S" in finite_subset) |
|
827 |
apply auto |
|
828 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
829 |
|
54681 | 830 |
|
60500 | 831 |
subsection \<open>Hence a sort of induction principle composing by swaps\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
832 |
|
54681 | 833 |
lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow> |
834 |
(\<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> |
|
835 |
(\<And>p. p permutes S \<Longrightarrow> P p)" |
|
836 |
proof (induct S rule: finite_induct) |
|
837 |
case empty |
|
838 |
then show ?case by auto |
|
30488 | 839 |
next |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
840 |
case (insert x F p) |
54681 | 841 |
let ?r = "Fun.swap x (p x) id \<circ> p" |
842 |
let ?q = "Fun.swap x (p x) id \<circ> ?r" |
|
843 |
have qp: "?q = p" |
|
844 |
by (simp add: o_assoc) |
|
845 |
from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" |
|
846 |
by blast |
|
30488 | 847 |
from permutes_in_image[OF insert.prems(3), of x] |
54681 | 848 |
have pxF: "p x \<in> insert x F" |
849 |
by simp |
|
850 |
have xF: "x \<in> insert x F" |
|
851 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
852 |
have rp: "permutation ?r" |
30488 | 853 |
unfolding permutation_permutes using insert.hyps(1) |
54681 | 854 |
permutes_insert_lemma[OF insert.prems(3)] |
855 |
by blast |
|
30488 | 856 |
from insert.prems(2)[OF xF pxF Pr Pr rp] |
54681 | 857 |
show ?case |
858 |
unfolding qp . |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
859 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
860 |
|
54681 | 861 |
|
60500 | 862 |
subsection \<open>Sign of a permutation as a real number\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
863 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
864 |
definition "sign p = (if evenperm p then (1::int) else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
865 |
|
54681 | 866 |
lemma sign_nz: "sign p \<noteq> 0" |
867 |
by (simp add: sign_def) |
|
868 |
||
869 |
lemma sign_id: "sign id = 1" |
|
870 |
by (simp add: sign_def) |
|
871 |
||
872 |
lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
873 |
by (simp add: sign_def evenperm_inv) |
54681 | 874 |
|
875 |
lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q" |
|
876 |
by (simp add: sign_def evenperm_comp) |
|
877 |
||
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
878 |
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
879 |
by (simp add: sign_def evenperm_swap) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
880 |
|
54681 | 881 |
lemma sign_idempotent: "sign p * sign p = 1" |
882 |
by (simp add: sign_def) |
|
883 |
||
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
884 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
885 |
subsection \<open>Permuting a list\<close> |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
886 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
887 |
text \<open>This function permutes a list by applying a permutation to the indices.\<close> |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
888 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
889 |
definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
890 |
"permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
891 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
892 |
lemma permute_list_map: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
893 |
assumes "f permutes {..<length xs}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
894 |
shows "permute_list f (map g xs) = map g (permute_list f xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
895 |
using permutes_in_image[OF assms] by (auto simp: permute_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
896 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
897 |
lemma permute_list_nth: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
898 |
assumes "f permutes {..<length xs}" "i < length xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
899 |
shows "permute_list f xs ! i = xs ! f i" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
900 |
using permutes_in_image[OF assms(1)] assms(2) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
901 |
by (simp add: permute_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
902 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
903 |
lemma permute_list_Nil [simp]: "permute_list f [] = []" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
904 |
by (simp add: permute_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
905 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
906 |
lemma length_permute_list [simp]: "length (permute_list f xs) = length xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
907 |
by (simp add: permute_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
908 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
909 |
lemma permute_list_compose: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
910 |
assumes "g permutes {..<length xs}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
911 |
shows "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
912 |
using assms[THEN permutes_in_image] by (auto simp add: permute_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
913 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
914 |
lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
915 |
by (simp add: permute_list_def map_nth) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
916 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
917 |
lemma permute_list_id [simp]: "permute_list id xs = xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
918 |
by (simp add: id_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
919 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
920 |
lemma mset_permute_list [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
921 |
assumes "f permutes {..<length (xs :: 'a list)}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
922 |
shows "mset (permute_list f xs) = mset xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
923 |
proof (rule multiset_eqI) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
924 |
fix y :: 'a |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
925 |
from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
926 |
using permutes_in_image[OF assms] by auto |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
927 |
have "count (mset (permute_list f xs)) y = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
928 |
card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
929 |
by (simp add: permute_list_def mset_map count_image_mset atLeast0LessThan) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
930 |
also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
931 |
by auto |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
932 |
also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
933 |
by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
934 |
also have "\<dots> = count (mset xs) y" by (simp add: count_mset length_filter_conv_card) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
935 |
finally show "count (mset (permute_list f xs)) y = count (mset xs) y" by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
936 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
937 |
|
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
938 |
lemma set_permute_list [simp]: |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
939 |
assumes "f permutes {..<length xs}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
940 |
shows "set (permute_list f xs) = set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
941 |
by (rule mset_eq_setD[OF mset_permute_list]) fact |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
942 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
943 |
lemma distinct_permute_list [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
944 |
assumes "f permutes {..<length xs}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
945 |
shows "distinct (permute_list f xs) = distinct xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
946 |
by (simp add: distinct_count_atmost_1 assms) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
947 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
948 |
lemma permute_list_zip: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
949 |
assumes "f permutes A" "A = {..<length xs}" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
950 |
assumes [simp]: "length xs = length ys" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
951 |
shows "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
952 |
proof - |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
953 |
from permutes_in_image[OF assms(1)] assms(2) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
954 |
have [simp]: "f i < length ys \<longleftrightarrow> i < length ys" for i by simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
955 |
have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
956 |
by (simp_all add: permute_list_def zip_map_map) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
957 |
also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
958 |
by (intro nth_equalityI) simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
959 |
also have "\<dots> = zip (permute_list f xs) (permute_list f ys)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
960 |
by (simp_all add: permute_list_def zip_map_map) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
961 |
finally show ?thesis . |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
962 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
963 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
964 |
lemma map_of_permute: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
965 |
assumes "\<sigma> permutes fst ` set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
966 |
shows "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)" (is "_ = map_of (map ?f _)") |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
967 |
proof |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
968 |
fix x |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
969 |
from assms have "inj \<sigma>" "surj \<sigma>" by (simp_all add: permutes_inj permutes_surj) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
970 |
thus "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
971 |
by (induction xs) (auto simp: inv_f_f surj_f_inv_f) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
972 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
973 |
|
54681 | 974 |
|
60500 | 975 |
subsection \<open>More lemmas about permutations\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
976 |
|
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
977 |
text \<open> |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
978 |
The following few lemmas were contributed by Lukas Bulwahn. |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
979 |
\<close> |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
980 |
|
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
981 |
lemma count_image_mset_eq_card_vimage: |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
982 |
assumes "finite A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
983 |
shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
984 |
using assms |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
985 |
proof (induct A) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
986 |
case empty |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
987 |
show ?case by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
988 |
next |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
989 |
case (insert x F) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
990 |
show ?case |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
991 |
proof cases |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
992 |
assume "f x = b" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
993 |
from this have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
994 |
using insert.hyps by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
995 |
also have "\<dots> = card (insert x {a \<in> F. f a = f x})" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
996 |
using insert.hyps(1,2) by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
997 |
also have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
998 |
using \<open>f x = b\<close> by (auto intro: arg_cong[where f="card"]) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
999 |
finally show ?thesis using insert by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1000 |
next |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1001 |
assume A: "f x \<noteq> b" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1002 |
hence "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}" by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1003 |
with insert A show ?thesis by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1004 |
qed |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1005 |
qed |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1006 |
|
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1007 |
(* Prove image_mset_eq_implies_permutes *) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1008 |
lemma image_mset_eq_implies_permutes: |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1009 |
fixes f :: "'a \<Rightarrow> 'b" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1010 |
assumes "finite A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1011 |
assumes mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1012 |
obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1013 |
proof - |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1014 |
from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1015 |
have "f ` A = f' ` A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1016 |
proof - |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1017 |
have "f ` A = f ` (set_mset (mset_set A))" using \<open>finite A\<close> by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1018 |
also have "\<dots> = f' ` (set_mset (mset_set A))" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1019 |
by (metis mset_eq multiset.set_map) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1020 |
also have "\<dots> = f' ` A" using \<open>finite A\<close> by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1021 |
finally show ?thesis . |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1022 |
qed |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1023 |
have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1024 |
proof |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1025 |
fix b |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1026 |
from mset_eq have |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1027 |
"count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b" by simp |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1028 |
from this have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1029 |
using \<open>finite A\<close> |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1030 |
by (simp add: count_image_mset_eq_card_vimage) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1031 |
from this show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1032 |
by (intro finite_same_card_bij) simp_all |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1033 |
qed |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1034 |
hence "\<exists>p. \<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1035 |
by (rule bchoice) |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1036 |
then guess p .. note p = this |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1037 |
define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1038 |
have "p' permutes A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1039 |
proof (rule bij_imp_permutes) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1040 |
have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1041 |
unfolding disjoint_family_on_def by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1042 |
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 |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1043 |
using p b by (subst bij_betw_cong[where g="p b"]) auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1044 |
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})" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1045 |
by (rule bij_betw_UNION_disjoint) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1046 |
moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A" by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1047 |
moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A" using \<open>f ` A = f' ` A\<close> by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1048 |
ultimately show "bij_betw p' A A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1049 |
unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1050 |
next |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1051 |
fix x |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1052 |
assume "x \<notin> A" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1053 |
from this show "p' x = x" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1054 |
unfolding p'_def by simp |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1055 |
qed |
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1056 |
moreover from p have "\<forall>x\<in>A. f x = f' (p' x)" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1057 |
unfolding p'_def using bij_betwE by fastforce |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1058 |
ultimately show ?thesis by (rule that) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1059 |
qed |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1060 |
|
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1061 |
lemma mset_set_upto_eq_mset_upto: |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1062 |
"mset_set {..<n} = mset [0..<n]" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1063 |
by (induct n) (auto simp add: add.commute lessThan_Suc) |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1064 |
|
63921
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1065 |
(* and derive the existing property: *) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1066 |
lemma mset_eq_permutation: |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1067 |
assumes mset_eq: "mset (xs::'a list) = mset ys" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1068 |
obtains p where "p permutes {..<length ys}" "permute_list p ys = xs" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1069 |
proof - |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1070 |
from mset_eq have length_eq: "length xs = length ys" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1071 |
using mset_eq_length by blast |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1072 |
have "mset_set {..<length ys} = mset [0..<length ys]" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1073 |
using mset_set_upto_eq_mset_upto by blast |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1074 |
from mset_eq length_eq this have |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1075 |
"image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) = image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1076 |
by (metis map_nth mset_map) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1077 |
from image_mset_eq_implies_permutes[OF _ this] |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1078 |
obtain p where "p permutes {..<length ys}" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1079 |
and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)" by auto |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1080 |
moreover from this length_eq have "permute_list p ys = xs" |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1081 |
by (auto intro!: nth_equalityI simp add: permute_list_nth) |
0a5184877cb7
Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents:
63539
diff
changeset
|
1082 |
ultimately show thesis using that by blast |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1083 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1084 |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1085 |
lemma permutes_natset_le: |
54681 | 1086 |
fixes S :: "'a::wellorder set" |
1087 |
assumes p: "p permutes S" |
|
1088 |
and le: "\<forall>i \<in> S. p i \<le> i" |
|
1089 |
shows "p = id" |
|
1090 |
proof - |
|
1091 |
{ |
|
1092 |
fix n |
|
30488 | 1093 |
have "p n = n" |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1094 |
using p le |
54681 | 1095 |
proof (induct n arbitrary: S rule: less_induct) |
1096 |
fix n S |
|
1097 |
assume H: |
|
1098 |
"\<And>m S. m < n \<Longrightarrow> p permutes S \<Longrightarrow> \<forall>i\<in>S. p i \<le> i \<Longrightarrow> p m = m" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
1099 |
"p permutes S" "\<forall>i \<in>S. p i \<le> i" |
54681 | 1100 |
{ |
1101 |
assume "n \<notin> S" |
|
1102 |
with H(2) have "p n = n" |
|
1103 |
unfolding permutes_def by metis |
|
1104 |
} |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1105 |
moreover |
54681 | 1106 |
{ |
1107 |
assume ns: "n \<in> S" |
|
1108 |
from H(3) ns have "p n < n \<or> p n = n" |
|
1109 |
by auto |
|
1110 |
moreover { |
|
1111 |
assume h: "p n < n" |
|
1112 |
from H h have "p (p n) = p n" |
|
1113 |
by metis |
|
1114 |
with permutes_inj[OF H(2)] have "p n = n" |
|
1115 |
unfolding inj_on_def by blast |
|
1116 |
with h have False |
|
1117 |
by simp |
|
1118 |
} |
|
1119 |
ultimately have "p n = n" |
|
1120 |
by blast |
|
1121 |
} |
|
1122 |
ultimately show "p n = n" |
|
1123 |
by blast |
|
1124 |
qed |
|
1125 |
} |
|
1126 |
then show ?thesis |
|
1127 |
by (auto simp add: fun_eq_iff) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1128 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1129 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1130 |
lemma permutes_natset_ge: |
54681 | 1131 |
fixes S :: "'a::wellorder set" |
1132 |
assumes p: "p permutes S" |
|
1133 |
and le: "\<forall>i \<in> S. p i \<ge> i" |
|
1134 |
shows "p = id" |
|
1135 |
proof - |
|
1136 |
{ |
|
1137 |
fix i |
|
1138 |
assume i: "i \<in> S" |
|
1139 |
from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" |
|
1140 |
by simp |
|
1141 |
with le have "p (inv p i) \<ge> inv p i" |
|
1142 |
by blast |
|
1143 |
with permutes_inverses[OF p] have "i \<ge> inv p i" |
|
1144 |
by simp |
|
1145 |
} |
|
1146 |
then have th: "\<forall>i\<in>S. inv p i \<le> i" |
|
1147 |
by blast |
|
30488 | 1148 |
from permutes_natset_le[OF permutes_inv[OF p] th] |
54681 | 1149 |
have "inv p = inv id" |
1150 |
by simp |
|
30488 | 1151 |
then show ?thesis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1152 |
apply (subst permutes_inv_inv[OF p, symmetric]) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1153 |
apply (rule inv_unique_comp) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1154 |
apply simp_all |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1155 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1156 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1157 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1158 |
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}" |
54681 | 1159 |
apply (rule set_eqI) |
1160 |
apply auto |
|
1161 |
using permutes_inv_inv permutes_inv |
|
1162 |
apply auto |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1163 |
apply (rule_tac x="inv x" in exI) |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1164 |
apply auto |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1165 |
done |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1166 |
|
30488 | 1167 |
lemma image_compose_permutations_left: |
54681 | 1168 |
assumes q: "q permutes S" |
1169 |
shows "{q \<circ> p | p. p permutes S} = {p . p permutes S}" |
|
1170 |
apply (rule set_eqI) |
|
1171 |
apply auto |
|
1172 |
apply (rule permutes_compose) |
|
1173 |
using q |
|
1174 |
apply auto |
|
1175 |
apply (rule_tac x = "inv q \<circ> x" in exI) |
|
1176 |
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o) |
|
1177 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1178 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1179 |
lemma image_compose_permutations_right: |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1180 |
assumes q: "q permutes S" |
54681 | 1181 |
shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}" |
1182 |
apply (rule set_eqI) |
|
1183 |
apply auto |
|
1184 |
apply (rule permutes_compose) |
|
1185 |
using q |
|
1186 |
apply auto |
|
1187 |
apply (rule_tac x = "x \<circ> inv q" in exI) |
|
1188 |
apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc) |
|
1189 |
done |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1190 |
|
54681 | 1191 |
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n" |
1192 |
by (simp add: permutes_def) metis |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1193 |
|
54681 | 1194 |
lemma setsum_permutations_inverse: |
1195 |
"setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" |
|
1196 |
(is "?lhs = ?rhs") |
|
1197 |
proof - |
|
30036 | 1198 |
let ?S = "{p . p permutes S}" |
54681 | 1199 |
have th0: "inj_on inv ?S" |
1200 |
proof (auto simp add: inj_on_def) |
|
1201 |
fix q r |
|
1202 |
assume q: "q permutes S" |
|
1203 |
and r: "r permutes S" |
|
1204 |
and qr: "inv q = inv r" |
|
1205 |
then have "inv (inv q) = inv (inv r)" |
|
1206 |
by simp |
|
1207 |
with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r" |
|
1208 |
by metis |
|
1209 |
qed |
|
1210 |
have th1: "inv ` ?S = ?S" |
|
1211 |
using image_inverse_permutations by blast |
|
1212 |
have th2: "?rhs = setsum (f \<circ> inv) ?S" |
|
1213 |
by (simp add: o_def) |
|
57418 | 1214 |
from setsum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 . |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1215 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1216 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1217 |
lemma setum_permutations_compose_left: |
30036 | 1218 |
assumes q: "q permutes S" |
54681 | 1219 |
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}" |
1220 |
(is "?lhs = ?rhs") |
|
1221 |
proof - |
|
30036 | 1222 |
let ?S = "{p. p permutes S}" |
54681 | 1223 |
have th0: "?rhs = setsum (f \<circ> (op \<circ> q)) ?S" |
1224 |
by (simp add: o_def) |
|
1225 |
have th1: "inj_on (op \<circ> q) ?S" |
|
1226 |
proof (auto simp add: inj_on_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1227 |
fix p r |
54681 | 1228 |
assume "p permutes S" |
1229 |
and r: "r permutes S" |
|
1230 |
and rp: "q \<circ> p = q \<circ> r" |
|
1231 |
then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r" |
|
1232 |
by (simp add: comp_assoc) |
|
1233 |
with permutes_inj[OF q, unfolded inj_iff] show "p = r" |
|
1234 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1235 |
qed |
54681 | 1236 |
have th3: "(op \<circ> q) ` ?S = ?S" |
1237 |
using image_compose_permutations_left[OF q] by auto |
|
57418 | 1238 |
from setsum.reindex[OF th1, of f] show ?thesis unfolding th0 th1 th3 . |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1239 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1240 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1241 |
lemma sum_permutations_compose_right: |
30036 | 1242 |
assumes q: "q permutes S" |
54681 | 1243 |
shows "setsum f {p. p permutes S} = setsum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}" |
1244 |
(is "?lhs = ?rhs") |
|
1245 |
proof - |
|
30036 | 1246 |
let ?S = "{p. p permutes S}" |
54681 | 1247 |
have th0: "?rhs = setsum (f \<circ> (\<lambda>p. p \<circ> q)) ?S" |
1248 |
by (simp add: o_def) |
|
1249 |
have th1: "inj_on (\<lambda>p. p \<circ> q) ?S" |
|
1250 |
proof (auto simp add: inj_on_def) |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1251 |
fix p r |
54681 | 1252 |
assume "p permutes S" |
1253 |
and r: "r permutes S" |
|
1254 |
and rp: "p \<circ> q = r \<circ> q" |
|
1255 |
then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)" |
|
1256 |
by (simp add: o_assoc) |
|
1257 |
with permutes_surj[OF q, unfolded surj_iff] show "p = r" |
|
1258 |
by simp |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1259 |
qed |
54681 | 1260 |
have th3: "(\<lambda>p. p \<circ> q) ` ?S = ?S" |
1261 |
using image_compose_permutations_right[OF q] by auto |
|
57418 | 1262 |
from setsum.reindex[OF th1, of f] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1263 |
show ?thesis unfolding th0 th1 th3 . |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1264 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1265 |
|
54681 | 1266 |
|
60500 | 1267 |
subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close> |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1268 |
|
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1269 |
lemma setsum_over_permutations_insert: |
54681 | 1270 |
assumes fS: "finite S" |
1271 |
and aS: "a \<notin> S" |
|
1272 |
shows "setsum f {p. p permutes (insert a S)} = |
|
1273 |
setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id \<circ> q)) {p. p permutes S}) (insert a S)" |
|
1274 |
proof - |
|
1275 |
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)" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
1276 |
by (simp add: fun_eq_iff) |
54681 | 1277 |
have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" |
1278 |
by blast |
|
1279 |
have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" |
|
1280 |
by blast |
|
30488 | 1281 |
show ?thesis |
1282 |
unfolding permutes_insert |
|
57418 | 1283 |
unfolding setsum.cartesian_product |
57129
7edb7550663e
introduce more powerful reindexing rules for big operators
hoelzl
parents:
56608
diff
changeset
|
1284 |
unfolding th1[symmetric] |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1285 |
unfolding th0 |
57418 | 1286 |
proof (rule setsum.reindex) |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1287 |
let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)" |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1288 |
let ?P = "{p. p permutes S}" |
54681 | 1289 |
{ |
1290 |
fix b c p q |
|
1291 |
assume b: "b \<in> insert a S" |
|
1292 |
assume c: "c \<in> insert a S" |
|
1293 |
assume p: "p permutes S" |
|
1294 |
assume q: "q permutes S" |
|
1295 |
assume eq: "Fun.swap a b id \<circ> p = Fun.swap a c id \<circ> q" |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1296 |
from p q aS have pa: "p a = a" and qa: "q a = a" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
1297 |
unfolding permutes_def by metis+ |
54681 | 1298 |
from eq have "(Fun.swap a b id \<circ> p) a = (Fun.swap a c id \<circ> q) a" |
1299 |
by simp |
|
1300 |
then have bc: "b = c" |
|
56545 | 1301 |
by (simp add: permutes_def pa qa o_def fun_upd_def Fun.swap_def id_def |
62390 | 1302 |
cong del: if_weak_cong split: if_split_asm) |
54681 | 1303 |
from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> p) = |
1304 |
(\<lambda>p. Fun.swap a c id \<circ> p) (Fun.swap a c id \<circ> q)" by simp |
|
1305 |
then have "p = q" |
|
1306 |
unfolding o_assoc swap_id_idempotent |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
32456
diff
changeset
|
1307 |
by (simp add: o_def) |
54681 | 1308 |
with bc have "b = c \<and> p = q" |
1309 |
by blast |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1310 |
} |
30488 | 1311 |
then show "inj_on ?f (insert a S \<times> ?P)" |
54681 | 1312 |
unfolding inj_on_def by clarify metis |
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1313 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1314 |
qed |
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
parents:
diff
changeset
|
1315 |
|
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1316 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1317 |
subsection \<open>Constructing permutations from association lists\<close> |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1318 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1319 |
definition list_permutes where |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1320 |
"list_permutes xs A \<longleftrightarrow> set (map fst xs) \<subseteq> A \<and> set (map snd xs) = set (map fst xs) \<and> |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1321 |
distinct (map fst xs) \<and> distinct (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1322 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1323 |
lemma list_permutesI [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1324 |
assumes "set (map fst xs) \<subseteq> A" "set (map snd xs) = set (map fst xs)" "distinct (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1325 |
shows "list_permutes xs A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1326 |
proof - |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1327 |
from assms(2,3) have "distinct (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1328 |
by (intro card_distinct) (simp_all add: distinct_card del: set_map) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1329 |
with assms show ?thesis by (simp add: list_permutes_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1330 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1331 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1332 |
definition permutation_of_list where |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1333 |
"permutation_of_list xs x = (case map_of xs x of None \<Rightarrow> x | Some y \<Rightarrow> y)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1334 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1335 |
lemma permutation_of_list_Cons: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1336 |
"permutation_of_list ((x,y) # xs) x' = (if x = x' then y else permutation_of_list xs x')" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1337 |
by (simp add: permutation_of_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1338 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1339 |
fun inverse_permutation_of_list where |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1340 |
"inverse_permutation_of_list [] x = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1341 |
| "inverse_permutation_of_list ((y,x')#xs) x = |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1342 |
(if x = x' then y else inverse_permutation_of_list xs x)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1343 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1344 |
declare inverse_permutation_of_list.simps [simp del] |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1345 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1346 |
lemma inj_on_map_of: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1347 |
assumes "distinct (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1348 |
shows "inj_on (map_of xs) (set (map fst xs))" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1349 |
proof (rule inj_onI) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1350 |
fix x y assume xy: "x \<in> set (map fst xs)" "y \<in> set (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1351 |
assume eq: "map_of xs x = map_of xs y" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1352 |
from xy obtain x' y' |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1353 |
where x'y': "map_of xs x = Some x'" "map_of xs y = Some y'" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1354 |
by (cases "map_of xs x"; cases "map_of xs y") |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1355 |
(simp_all add: map_of_eq_None_iff) |
63539 | 1356 |
moreover from x'y' have *: "(x,x') \<in> set xs" "(y,y') \<in> set xs" |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1357 |
by (force dest: map_of_SomeD)+ |
63539 | 1358 |
moreover from * eq x'y' have "x' = y'" by simp |
63099
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1359 |
ultimately show "x = y" using assms |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1360 |
by (force simp: distinct_map dest: inj_onD[of _ _ "(x,x')" "(y,y')"]) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1361 |
qed |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1362 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1363 |
lemma inj_on_the: "None \<notin> A \<Longrightarrow> inj_on the A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1364 |
by (auto simp: inj_on_def option.the_def split: option.splits) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1365 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1366 |
lemma inj_on_map_of': |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1367 |
assumes "distinct (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1368 |
shows "inj_on (the \<circ> map_of xs) (set (map fst xs))" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1369 |
by (intro comp_inj_on inj_on_map_of assms inj_on_the) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1370 |
(force simp: eq_commute[of None] map_of_eq_None_iff) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1371 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1372 |
lemma image_map_of: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1373 |
assumes "distinct (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1374 |
shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1375 |
using assms by (auto simp: rev_image_eqI) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1376 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1377 |
lemma the_Some_image [simp]: "the ` Some ` A = A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1378 |
by (subst image_image) simp |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1379 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1380 |
lemma image_map_of': |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1381 |
assumes "distinct (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1382 |
shows "(the \<circ> map_of xs) ` set (map fst xs) = set (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1383 |
by (simp only: image_comp [symmetric] image_map_of assms the_Some_image) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1384 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1385 |
lemma permutation_of_list_permutes [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1386 |
assumes "list_permutes xs A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1387 |
shows "permutation_of_list xs permutes A" (is "?f permutes _") |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1388 |
proof (rule permutes_subset[OF bij_imp_permutes]) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1389 |
from assms show "set (map fst xs) \<subseteq> A" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1390 |
by (simp add: list_permutes_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1391 |
from assms have "inj_on (the \<circ> map_of xs) (set (map fst xs))" (is ?P) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1392 |
by (intro inj_on_map_of') (simp_all add: list_permutes_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1393 |
also have "?P \<longleftrightarrow> inj_on ?f (set (map fst xs))" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1394 |
by (intro inj_on_cong) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1395 |
(auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1396 |
finally have "bij_betw ?f (set (map fst xs)) (?f ` set (map fst xs))" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1397 |
by (rule inj_on_imp_bij_betw) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1398 |
also from assms have "?f ` set (map fst xs) = (the \<circ> map_of xs) ` set (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1399 |
by (intro image_cong refl) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1400 |
(auto simp: permutation_of_list_def map_of_eq_None_iff split: option.splits) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1401 |
also from assms have "\<dots> = set (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1402 |
by (subst image_map_of') (simp_all add: list_permutes_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1403 |
finally show "bij_betw ?f (set (map fst xs)) (set (map fst xs))" . |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1404 |
qed (force simp: permutation_of_list_def dest!: map_of_SomeD split: option.splits)+ |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1405 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1406 |
lemma eval_permutation_of_list [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1407 |
"permutation_of_list [] x = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1408 |
"x = x' \<Longrightarrow> permutation_of_list ((x',y)#xs) x = y" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1409 |
"x \<noteq> x' \<Longrightarrow> permutation_of_list ((x',y')#xs) x = permutation_of_list xs x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1410 |
by (simp_all add: permutation_of_list_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1411 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1412 |
lemma eval_inverse_permutation_of_list [simp]: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1413 |
"inverse_permutation_of_list [] x = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1414 |
"x = x' \<Longrightarrow> inverse_permutation_of_list ((y,x')#xs) x = y" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1415 |
"x \<noteq> x' \<Longrightarrow> inverse_permutation_of_list ((y',x')#xs) x = inverse_permutation_of_list xs x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1416 |
by (simp_all add: inverse_permutation_of_list.simps) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1417 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1418 |
lemma permutation_of_list_id: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1419 |
assumes "x \<notin> set (map fst xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1420 |
shows "permutation_of_list xs x = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1421 |
using assms by (induction xs) (auto simp: permutation_of_list_Cons) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1422 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1423 |
lemma permutation_of_list_unique': |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1424 |
assumes "distinct (map fst xs)" "(x, y) \<in> set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1425 |
shows "permutation_of_list xs x = y" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1426 |
using assms by (induction xs) (force simp: permutation_of_list_Cons)+ |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1427 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1428 |
lemma permutation_of_list_unique: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1429 |
assumes "list_permutes xs A" "(x,y) \<in> set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1430 |
shows "permutation_of_list xs x = y" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1431 |
using assms by (intro permutation_of_list_unique') (simp_all add: list_permutes_def) |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1432 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1433 |
lemma inverse_permutation_of_list_id: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1434 |
assumes "x \<notin> set (map snd xs)" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1435 |
shows "inverse_permutation_of_list xs x = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1436 |
using assms by (induction xs) auto |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1437 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1438 |
lemma inverse_permutation_of_list_unique': |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1439 |
assumes "distinct (map snd xs)" "(x, y) \<in> set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1440 |
shows "inverse_permutation_of_list xs y = x" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1441 |
using assms by (induction xs) (force simp: inverse_permutation_of_list.simps)+ |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1442 |
|
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1443 |
lemma inverse_permutation_of_list_unique: |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
changeset
|
1444 |
assumes "list_permutes xs A" "(x,y) \<in> set xs" |
af0e964aad7b
Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents:
62390
diff
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|
1445 |
shows "inverse_permutation_of_list xs y = x" |
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|
1446 |
using assms by (intro inverse_permutation_of_list_unique') (simp_all add: list_permutes_def) |
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|
1447 |
|
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|
1448 |
lemma inverse_permutation_of_list_correct: |
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|
1449 |
assumes "list_permutes xs (A :: 'a set)" |
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|
1450 |
shows "inverse_permutation_of_list xs = inv (permutation_of_list xs)" |
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|
1451 |
proof (rule ext, rule sym, subst permutes_inv_eq) |
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|
1452 |
from assms show "permutation_of_list xs permutes A" by simp |
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|
1453 |
next |
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|
1454 |
fix x |
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|
1455 |
show "permutation_of_list xs (inverse_permutation_of_list xs x) = x" |
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|
1456 |
proof (cases "x \<in> set (map snd xs)") |
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|
1457 |
case True |
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|
1458 |
then obtain y where "(y, x) \<in> set xs" by force |
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|
1459 |
with assms show ?thesis |
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|
1460 |
by (simp add: inverse_permutation_of_list_unique permutation_of_list_unique) |
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|
1461 |
qed (insert assms, auto simp: list_permutes_def |
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|
1462 |
inverse_permutation_of_list_id permutation_of_list_id) |
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|
1463 |
qed |
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|
1464 |
|
29840
cfab6a76aa13
Permutations, both general and specifically on finite sets.
chaieb
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|
1465 |
end |
51489 | 1466 |