author | huffman |
Sun, 22 Apr 2012 20:16:30 +0200 | |
changeset 47680 | 49aa3686e566 |
parent 47660 | 7a5c681c0265 |
child 47777 | f29e7dcd7c40 |
permissions | -rw-r--r-- |
47455 | 1 |
(* Title: HOL/Library/Quotient_Set.thy |
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Author: Cezary Kaliszyk and Christian Urban |
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*) |
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|
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header {* Quotient infrastructure for the set type *} |
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|
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theory Quotient_Set |
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imports Main Quotient_Syntax |
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begin |
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subsection {* Relator for set type *} |
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|
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definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" |
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where "set_rel R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))" |
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|
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lemma set_relI: |
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assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y" |
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assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" |
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shows "set_rel R A B" |
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using assms unfolding set_rel_def by simp |
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|
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lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)" |
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unfolding set_rel_def by auto |
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|
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lemma set_rel_OO: "set_rel (R OO S) = set_rel R OO set_rel S" |
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apply (intro ext, rename_tac X Z) |
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apply (rule iffI) |
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apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI) |
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apply (simp add: set_rel_def, fast) |
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apply (simp add: set_rel_def, fast) |
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apply (simp add: set_rel_def, fast) |
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done |
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|
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lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)" |
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unfolding set_rel_def fun_eq_iff by auto |
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|
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lemma reflp_set_rel: "reflp R \<Longrightarrow> reflp (set_rel R)" |
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unfolding reflp_def set_rel_def by fast |
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|
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lemma symp_set_rel: "symp R \<Longrightarrow> symp (set_rel R)" |
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unfolding symp_def set_rel_def by fast |
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|
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lemma transp_set_rel: "transp R \<Longrightarrow> transp (set_rel R)" |
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unfolding transp_def set_rel_def by fast |
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|
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lemma equivp_set_rel: "equivp R \<Longrightarrow> equivp (set_rel R)" |
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by (blast intro: equivpI reflp_set_rel symp_set_rel transp_set_rel |
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elim: equivpE) |
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|
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lemma right_total_set_rel [transfer_rule]: |
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"right_total A \<Longrightarrow> right_total (set_rel A)" |
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unfolding right_total_def set_rel_def |
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by (rule allI, rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast) |
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|
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lemma right_unique_set_rel [transfer_rule]: |
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"right_unique A \<Longrightarrow> right_unique (set_rel A)" |
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unfolding right_unique_def set_rel_def by fast |
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|
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lemma bi_total_set_rel [transfer_rule]: |
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"bi_total A \<Longrightarrow> bi_total (set_rel A)" |
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unfolding bi_total_def set_rel_def |
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apply safe |
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apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast) |
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apply (rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast) |
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done |
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|
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lemma bi_unique_set_rel [transfer_rule]: |
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"bi_unique A \<Longrightarrow> bi_unique (set_rel A)" |
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unfolding bi_unique_def set_rel_def by fast |
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|
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subsection {* Transfer rules for transfer package *} |
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|
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subsubsection {* Unconditional transfer rules *} |
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|
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lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}" |
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unfolding set_rel_def by simp |
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lemma insert_transfer [transfer_rule]: |
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"(A ===> set_rel A ===> set_rel A) insert insert" |
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unfolding fun_rel_def set_rel_def by auto |
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lemma union_transfer [transfer_rule]: |
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"(set_rel A ===> set_rel A ===> set_rel A) union union" |
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unfolding fun_rel_def set_rel_def by auto |
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lemma Union_transfer [transfer_rule]: |
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"(set_rel (set_rel A) ===> set_rel A) Union Union" |
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unfolding fun_rel_def set_rel_def by simp fast |
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lemma image_transfer [transfer_rule]: |
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"((A ===> B) ===> set_rel A ===> set_rel B) image image" |
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unfolding fun_rel_def set_rel_def by simp fast |
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lemma UNION_transfer [transfer_rule]: |
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"(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION" |
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unfolding SUP_def [abs_def] by transfer_prover |
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lemma Ball_transfer [transfer_rule]: |
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"(set_rel A ===> (A ===> op =) ===> op =) Ball Ball" |
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unfolding set_rel_def fun_rel_def by fast |
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|
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lemma Bex_transfer [transfer_rule]: |
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"(set_rel A ===> (A ===> op =) ===> op =) Bex Bex" |
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unfolding set_rel_def fun_rel_def by fast |
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|
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lemma Pow_transfer [transfer_rule]: |
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"(set_rel A ===> set_rel (set_rel A)) Pow Pow" |
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apply (rule fun_relI, rename_tac X Y, rule set_relI) |
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apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp) |
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apply (simp add: set_rel_def, fast) |
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apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp) |
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apply (simp add: set_rel_def, fast) |
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done |
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subsubsection {* Rules requiring bi-unique or bi-total relations *} |
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lemma member_transfer [transfer_rule]: |
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assumes "bi_unique A" |
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shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)" |
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120 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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121 |
|
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122 |
lemma Collect_transfer [transfer_rule]: |
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123 |
assumes "bi_total A" |
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|
124 |
shows "((A ===> op =) ===> set_rel A) Collect Collect" |
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125 |
using assms unfolding fun_rel_def set_rel_def bi_total_def by fast |
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126 |
|
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127 |
lemma inter_transfer [transfer_rule]: |
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|
128 |
assumes "bi_unique A" |
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|
129 |
shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" |
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130 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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131 |
|
47680 | 132 |
lemma Diff_transfer [transfer_rule]: |
133 |
assumes "bi_unique A" |
|
134 |
shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)" |
|
135 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def |
|
136 |
unfolding Ball_def Bex_def Diff_eq |
|
137 |
by (safe, simp, metis, simp, metis) |
|
138 |
||
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139 |
lemma subset_transfer [transfer_rule]: |
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140 |
assumes [transfer_rule]: "bi_unique A" |
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|
141 |
shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)" |
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|
142 |
unfolding subset_eq [abs_def] by transfer_prover |
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|
143 |
|
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|
144 |
lemma UNIV_transfer [transfer_rule]: |
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145 |
assumes "bi_total A" |
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|
146 |
shows "(set_rel A) UNIV UNIV" |
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147 |
using assms unfolding set_rel_def bi_total_def by simp |
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|
148 |
|
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|
149 |
lemma Compl_transfer [transfer_rule]: |
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150 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
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|
151 |
shows "(set_rel A ===> set_rel A) uminus uminus" |
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changeset
|
152 |
unfolding Compl_eq [abs_def] by transfer_prover |
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|
153 |
|
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changeset
|
154 |
lemma Inter_transfer [transfer_rule]: |
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|
155 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
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|
156 |
shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" |
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|
157 |
unfolding Inter_eq [abs_def] by transfer_prover |
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|
158 |
|
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changeset
|
159 |
lemma finite_transfer [transfer_rule]: |
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|
160 |
assumes "bi_unique A" |
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|
161 |
shows "(set_rel A ===> op =) finite finite" |
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|
162 |
apply (rule fun_relI, rename_tac X Y) |
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|
163 |
apply (rule iffI) |
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changeset
|
164 |
apply (subgoal_tac "Y \<subseteq> (\<lambda>x. THE y. A x y) ` X") |
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|
165 |
apply (erule finite_subset, erule finite_imageI) |
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|
166 |
apply (rule subsetI, rename_tac y) |
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|
167 |
apply (clarsimp simp add: set_rel_def) |
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|
168 |
apply (drule (1) bspec, clarify) |
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|
169 |
apply (rule image_eqI) |
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|
170 |
apply (rule the_equality [symmetric]) |
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|
171 |
apply assumption |
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|
172 |
apply (simp add: assms [unfolded bi_unique_def]) |
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|
173 |
apply assumption |
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changeset
|
174 |
apply (subgoal_tac "X \<subseteq> (\<lambda>y. THE x. A x y) ` Y") |
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|
175 |
apply (erule finite_subset, erule finite_imageI) |
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changeset
|
176 |
apply (rule subsetI, rename_tac x) |
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changeset
|
177 |
apply (clarsimp simp add: set_rel_def) |
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changeset
|
178 |
apply (drule (1) bspec, clarify) |
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changeset
|
179 |
apply (rule image_eqI) |
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changeset
|
180 |
apply (rule the_equality [symmetric]) |
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changeset
|
181 |
apply assumption |
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changeset
|
182 |
apply (simp add: assms [unfolded bi_unique_def]) |
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changeset
|
183 |
apply assumption |
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|
184 |
done |
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changeset
|
185 |
|
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|
186 |
subsection {* Setup for lifting package *} |
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|
187 |
|
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|
188 |
lemma Quotient_set: |
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|
189 |
assumes "Quotient R Abs Rep T" |
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|
190 |
shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)" |
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changeset
|
191 |
using assms unfolding Quotient_alt_def4 |
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changeset
|
192 |
apply (simp add: set_rel_OO set_rel_conversep) |
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changeset
|
193 |
apply (simp add: set_rel_def, fast) |
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changeset
|
194 |
done |
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changeset
|
195 |
|
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|
196 |
declare [[map set = (set_rel, Quotient_set)]] |
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changeset
|
197 |
|
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changeset
|
198 |
lemma set_invariant_commute [invariant_commute]: |
6b9d20a095ae
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|
199 |
"set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)" |
6b9d20a095ae
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changeset
|
200 |
unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast |
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changeset
|
201 |
|
6b9d20a095ae
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changeset
|
202 |
subsection {* Contravariant set map (vimage) and set relator *} |
47626
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changeset
|
203 |
|
47647
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changeset
|
204 |
definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys" |
47626
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changeset
|
205 |
|
47647
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changeset
|
206 |
lemma vset_rel_eq [id_simps]: |
ec29cc09599d
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changeset
|
207 |
"vset_rel op = = op =" |
ec29cc09599d
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diff
changeset
|
208 |
by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff vset_rel_def) |
47626
f7b1034cb9ce
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changeset
|
209 |
|
47647
ec29cc09599d
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changeset
|
210 |
lemma vset_rel_equivp: |
47626
f7b1034cb9ce
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changeset
|
211 |
assumes e: "equivp R" |
47647
ec29cc09599d
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diff
changeset
|
212 |
shows "vset_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)" |
ec29cc09599d
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changeset
|
213 |
unfolding vset_rel_def |
47626
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changeset
|
214 |
using equivp_reflp[OF e] |
f7b1034cb9ce
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changeset
|
215 |
by auto (metis, metis equivp_symp[OF e]) |
f7b1034cb9ce
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changeset
|
216 |
|
44413
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diff
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|
217 |
lemma set_quotient [quot_thm]: |
47308 | 218 |
assumes "Quotient3 R Abs Rep" |
47647
ec29cc09599d
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changeset
|
219 |
shows "Quotient3 (vset_rel R) (vimage Rep) (vimage Abs)" |
47308 | 220 |
proof (rule Quotient3I) |
221 |
from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep) |
|
44413
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|
222 |
then show "\<And>xs. Rep -` (Abs -` xs) = xs" |
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|
223 |
unfolding vimage_def by auto |
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|
224 |
next |
47647
ec29cc09599d
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|
225 |
show "\<And>xs. vset_rel R (Abs -` xs) (Abs -` xs)" |
ec29cc09599d
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|
226 |
unfolding vset_rel_def vimage_def |
47308 | 227 |
by auto (metis Quotient3_rel_abs[OF assms])+ |
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|
228 |
next |
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|
229 |
fix r s |
47647
ec29cc09599d
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changeset
|
230 |
show "vset_rel R r s = (vset_rel R r r \<and> vset_rel R s s \<and> Rep -` r = Rep -` s)" |
ec29cc09599d
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diff
changeset
|
231 |
unfolding vset_rel_def vimage_def set_eq_iff |
47308 | 232 |
by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+ |
44413
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|
233 |
qed |
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changeset
|
234 |
|
47647
ec29cc09599d
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changeset
|
235 |
declare [[mapQ3 set = (vset_rel, set_quotient)]] |
47094 | 236 |
|
44413
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|
237 |
lemma empty_set_rsp[quot_respect]: |
47647
ec29cc09599d
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changeset
|
238 |
"vset_rel R {} {}" |
ec29cc09599d
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changeset
|
239 |
unfolding vset_rel_def by simp |
44413
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changeset
|
240 |
|
80d460bc6fa8
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parents:
diff
changeset
|
241 |
lemma collect_rsp[quot_respect]: |
47308 | 242 |
assumes "Quotient3 R Abs Rep" |
47647
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
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parents:
47626
diff
changeset
|
243 |
shows "((R ===> op =) ===> vset_rel R) Collect Collect" |
ec29cc09599d
renamed contravariant relator set_rel to vset_rel, to make room for new covariant relator
huffman
parents:
47626
diff
changeset
|
244 |
by (intro fun_relI) (simp add: fun_rel_def vset_rel_def) |
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lemma collect_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "((Abs ---> id) ---> op -` Rep) Collect = Collect" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF assms]) |
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lemma union_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<union> op \<union>" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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lemma union_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<union> = op \<union>" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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lemma diff_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op - op -" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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lemma diff_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff) |
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lemma inter_rsp[quot_respect]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<inter> op \<inter>" |
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by (intro fun_relI) (auto simp add: vset_rel_def) |
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lemma inter_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<inter> = op \<inter>" |
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unfolding fun_eq_iff |
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by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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lemma mem_prs[quot_preserve]: |
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assumes "Quotient3 R Abs Rep" |
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shows "(Rep ---> op -` Abs ---> id) op \<in> = op \<in>" |
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by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms]) |
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lemma mem_rsp[quot_respect]: |
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shows "(R ===> vset_rel R ===> op =) op \<in> op \<in>" |
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by (intro fun_relI) (simp add: vset_rel_def) |
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|
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end |