author | kuncar |
Wed, 15 May 2013 12:10:39 +0200 | |
changeset 51994 | 82cc2aeb7d13 |
parent 51956 | a4d81cdebf8b |
child 52359 | 0eafa146b399 |
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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Quotient Package: some infrastructure for lifting inside sets
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|
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Quotient Package: some infrastructure for lifting inside sets
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header {* Quotient infrastructure for the set type *} |
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|
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Quotient Package: some infrastructure for lifting inside sets
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theory Quotient_Set |
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imports Main Quotient_Syntax |
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Quotient Package: some infrastructure for lifting inside sets
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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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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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lemma set_rel_mono[relator_mono]: |
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assumes "A \<le> B" |
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shows "set_rel A \<le> set_rel B" |
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using assms unfolding set_rel_def by blast |
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lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)" |
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apply (rule sym) |
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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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lemma Domainp_set[relator_domain]: |
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assumes "Domainp T = R" |
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shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)" |
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using assms unfolding set_rel_def Domainp_iff[abs_def] |
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apply (intro ext) |
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apply (rule iffI) |
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apply blast |
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apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast) |
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done |
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lemma reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)" |
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unfolding reflp_def set_rel_def by fast |
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lemma left_total_set_rel[reflexivity_rule]: |
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"left_total A \<Longrightarrow> left_total (set_rel A)" |
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unfolding left_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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done |
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lemma left_unique_set_rel[reflexivity_rule]: |
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"left_unique A \<Longrightarrow> left_unique (set_rel A)" |
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unfolding left_unique_def set_rel_def |
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by fast |
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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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|
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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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|
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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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|
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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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|
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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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||
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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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134 |
lemma Pow_transfer [transfer_rule]: |
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135 |
"(set_rel A ===> set_rel (set_rel A)) Pow Pow" |
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|
136 |
apply (rule fun_relI, rename_tac X Y, rule set_relI) |
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|
137 |
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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|
138 |
apply (simp add: set_rel_def, fast) |
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|
139 |
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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|
140 |
apply (simp add: set_rel_def, fast) |
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|
141 |
done |
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|
142 |
|
47922 | 143 |
lemma set_rel_transfer [transfer_rule]: |
144 |
"((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =) |
|
145 |
set_rel set_rel" |
|
146 |
unfolding fun_rel_def set_rel_def by fast |
|
147 |
||
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|
148 |
|
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|
149 |
subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *} |
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150 |
|
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|
151 |
lemma member_transfer [transfer_rule]: |
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|
152 |
assumes "bi_unique A" |
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|
153 |
shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)" |
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|
154 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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changeset
|
155 |
|
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156 |
lemma right_total_Collect_transfer[transfer_rule]: |
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|
157 |
assumes "right_total A" |
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|
158 |
shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect" |
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changeset
|
159 |
using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast |
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changeset
|
160 |
|
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|
161 |
lemma Collect_transfer [transfer_rule]: |
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|
162 |
assumes "bi_total A" |
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changeset
|
163 |
shows "((A ===> op =) ===> set_rel A) Collect Collect" |
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changeset
|
164 |
using assms unfolding fun_rel_def set_rel_def bi_total_def by fast |
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changeset
|
165 |
|
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changeset
|
166 |
lemma inter_transfer [transfer_rule]: |
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|
167 |
assumes "bi_unique A" |
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changeset
|
168 |
shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" |
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changeset
|
169 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast |
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changeset
|
170 |
|
47680 | 171 |
lemma Diff_transfer [transfer_rule]: |
172 |
assumes "bi_unique A" |
|
173 |
shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)" |
|
174 |
using assms unfolding fun_rel_def set_rel_def bi_unique_def |
|
175 |
unfolding Ball_def Bex_def Diff_eq |
|
176 |
by (safe, simp, metis, simp, metis) |
|
177 |
||
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changeset
|
178 |
lemma subset_transfer [transfer_rule]: |
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changeset
|
179 |
assumes [transfer_rule]: "bi_unique A" |
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parents:
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changeset
|
180 |
shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)" |
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parents:
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changeset
|
181 |
unfolding subset_eq [abs_def] by transfer_prover |
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changeset
|
182 |
|
51956
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changeset
|
183 |
lemma right_total_UNIV_transfer[transfer_rule]: |
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changeset
|
184 |
assumes "right_total A" |
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changeset
|
185 |
shows "(set_rel A) (Collect (Domainp A)) UNIV" |
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changeset
|
186 |
using assms unfolding right_total_def set_rel_def Domainp_iff by blast |
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better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
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changeset
|
187 |
|
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changeset
|
188 |
lemma UNIV_transfer [transfer_rule]: |
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parents:
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changeset
|
189 |
assumes "bi_total A" |
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parents:
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changeset
|
190 |
shows "(set_rel A) UNIV UNIV" |
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parents:
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diff
changeset
|
191 |
using assms unfolding set_rel_def bi_total_def by simp |
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added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
192 |
|
51956
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parents:
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diff
changeset
|
193 |
lemma right_total_Compl_transfer [transfer_rule]: |
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parents:
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diff
changeset
|
194 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
195 |
shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
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parents:
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diff
changeset
|
196 |
unfolding Compl_eq [abs_def] |
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parents:
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diff
changeset
|
197 |
by (subst Collect_conj_eq[symmetric]) transfer_prover |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
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parents:
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diff
changeset
|
198 |
|
47648
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added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
199 |
lemma Compl_transfer [transfer_rule]: |
6b9d20a095ae
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parents:
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diff
changeset
|
200 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
201 |
shows "(set_rel A ===> set_rel A) uminus uminus" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
202 |
unfolding Compl_eq [abs_def] by transfer_prover |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
203 |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
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parents:
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diff
changeset
|
204 |
lemma right_total_Inter_transfer [transfer_rule]: |
a4d81cdebf8b
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kuncar
parents:
51377
diff
changeset
|
205 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
206 |
shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
207 |
unfolding Inter_eq[abs_def] |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
208 |
by (subst Collect_conj_eq[symmetric]) transfer_prover |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
209 |
|
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
210 |
lemma Inter_transfer [transfer_rule]: |
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
211 |
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
6b9d20a095ae
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huffman
parents:
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diff
changeset
|
212 |
shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
213 |
unfolding Inter_eq [abs_def] by transfer_prover |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
214 |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
215 |
lemma filter_transfer [transfer_rule]: |
a4d81cdebf8b
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kuncar
parents:
51377
diff
changeset
|
216 |
assumes [transfer_rule]: "bi_unique A" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
217 |
shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter" |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
218 |
unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51377
diff
changeset
|
219 |
|
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
220 |
lemma finite_transfer [transfer_rule]: |
6b9d20a095ae
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parents:
47647
diff
changeset
|
221 |
assumes "bi_unique A" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
222 |
shows "(set_rel A ===> op =) finite finite" |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
223 |
apply (rule fun_relI, rename_tac X Y) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
224 |
apply (rule iffI) |
6b9d20a095ae
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parents:
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diff
changeset
|
225 |
apply (subgoal_tac "Y \<subseteq> (\<lambda>x. THE y. A x y) ` X") |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
226 |
apply (erule finite_subset, erule finite_imageI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
227 |
apply (rule subsetI, rename_tac y) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
228 |
apply (clarsimp simp add: set_rel_def) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
229 |
apply (drule (1) bspec, clarify) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
230 |
apply (rule image_eqI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
231 |
apply (rule the_equality [symmetric]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
232 |
apply assumption |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
233 |
apply (simp add: assms [unfolded bi_unique_def]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
234 |
apply assumption |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
235 |
apply (subgoal_tac "X \<subseteq> (\<lambda>y. THE x. A x y) ` Y") |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
236 |
apply (erule finite_subset, erule finite_imageI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
237 |
apply (rule subsetI, rename_tac x) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
238 |
apply (clarsimp simp add: set_rel_def) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
239 |
apply (drule (1) bspec, clarify) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
240 |
apply (rule image_eqI) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
241 |
apply (rule the_equality [symmetric]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
242 |
apply assumption |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
243 |
apply (simp add: assms [unfolded bi_unique_def]) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
244 |
apply assumption |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
245 |
done |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
246 |
|
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
247 |
subsection {* Setup for lifting package *} |
6b9d20a095ae
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parents:
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diff
changeset
|
248 |
|
47777
f29e7dcd7c40
use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents:
47680
diff
changeset
|
249 |
lemma Quotient_set[quot_map]: |
47648
6b9d20a095ae
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parents:
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diff
changeset
|
250 |
assumes "Quotient R Abs Rep T" |
6b9d20a095ae
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parents:
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diff
changeset
|
251 |
shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)" |
6b9d20a095ae
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parents:
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changeset
|
252 |
using assms unfolding Quotient_alt_def4 |
51377 | 253 |
apply (simp add: set_rel_OO[symmetric] set_rel_conversep) |
47648
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
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parents:
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diff
changeset
|
254 |
apply (simp add: set_rel_def, fast) |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
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diff
changeset
|
255 |
done |
6b9d20a095ae
added covariant relator set_rel, with transfer rules for set operations
huffman
parents:
47647
diff
changeset
|
256 |
|
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lemma set_invariant_commute [invariant_commute]: |
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"set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)" |
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unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast |
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|
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subsection {* Contravariant set map (vimage) and set relator *} |
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|
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definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys" |
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|
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lemma vset_rel_eq [id_simps]: |
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"vset_rel op = = op =" |
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by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff vset_rel_def) |
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268 |
|
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lemma vset_rel_equivp: |
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assumes e: "equivp R" |
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271 |
shows "vset_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)" |
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272 |
unfolding vset_rel_def |
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273 |
using equivp_reflp[OF e] |
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by auto (metis, metis equivp_symp[OF e]) |
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275 |
|
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lemma set_quotient [quot_thm]: |
47308 | 277 |
assumes "Quotient3 R Abs Rep" |
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278 |
shows "Quotient3 (vset_rel R) (vimage Rep) (vimage Abs)" |
47308 | 279 |
proof (rule Quotient3I) |
280 |
from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep) |
|
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then show "\<And>xs. Rep -` (Abs -` xs) = xs" |
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unfolding vimage_def by auto |
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283 |
next |
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284 |
show "\<And>xs. vset_rel R (Abs -` xs) (Abs -` xs)" |
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285 |
unfolding vset_rel_def vimage_def |
47308 | 286 |
by auto (metis Quotient3_rel_abs[OF assms])+ |
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next |
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fix r s |
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289 |
show "vset_rel R r s = (vset_rel R r r \<and> vset_rel R s s \<and> Rep -` r = Rep -` s)" |
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290 |
unfolding vset_rel_def vimage_def set_eq_iff |
47308 | 291 |
by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+ |
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292 |
qed |
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293 |
|
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294 |
declare [[mapQ3 set = (vset_rel, set_quotient)]] |
47094 | 295 |
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lemma empty_set_rsp[quot_respect]: |
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297 |
"vset_rel R {} {}" |
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298 |
unfolding vset_rel_def by simp |
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299 |
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300 |
lemma collect_rsp[quot_respect]: |
47308 | 301 |
assumes "Quotient3 R Abs Rep" |
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302 |
shows "((R ===> op =) ===> vset_rel R) Collect Collect" |
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303 |
by (intro fun_relI) (simp add: fun_rel_def vset_rel_def) |
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lemma collect_prs[quot_preserve]: |
47308 | 306 |
assumes "Quotient3 R Abs Rep" |
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307 |
shows "((Abs ---> id) ---> op -` Rep) Collect = Collect" |
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308 |
unfolding fun_eq_iff |
47308 | 309 |
by (simp add: Quotient3_abs_rep[OF assms]) |
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311 |
lemma union_rsp[quot_respect]: |
47308 | 312 |
assumes "Quotient3 R Abs Rep" |
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313 |
shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<union> op \<union>" |
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314 |
by (intro fun_relI) (simp add: vset_rel_def) |
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315 |
|
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316 |
lemma union_prs[quot_preserve]: |
47308 | 317 |
assumes "Quotient3 R Abs Rep" |
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318 |
shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<union> = op \<union>" |
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319 |
unfolding fun_eq_iff |
47308 | 320 |
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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321 |
|
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322 |
lemma diff_rsp[quot_respect]: |
47308 | 323 |
assumes "Quotient3 R Abs Rep" |
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|
324 |
shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op - op -" |
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325 |
by (intro fun_relI) (simp add: vset_rel_def) |
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326 |
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327 |
lemma diff_prs[quot_preserve]: |
47308 | 328 |
assumes "Quotient3 R Abs Rep" |
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329 |
shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -" |
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330 |
unfolding fun_eq_iff |
47308 | 331 |
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff) |
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332 |
|
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333 |
lemma inter_rsp[quot_respect]: |
47308 | 334 |
assumes "Quotient3 R Abs Rep" |
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|
335 |
shows "(vset_rel R ===> vset_rel R ===> vset_rel R) op \<inter> op \<inter>" |
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336 |
by (intro fun_relI) (auto simp add: vset_rel_def) |
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337 |
|
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338 |
lemma inter_prs[quot_preserve]: |
47308 | 339 |
assumes "Quotient3 R Abs Rep" |
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340 |
shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<inter> = op \<inter>" |
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|
341 |
unfolding fun_eq_iff |
47308 | 342 |
by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]]) |
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343 |
|
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344 |
lemma mem_prs[quot_preserve]: |
47308 | 345 |
assumes "Quotient3 R Abs Rep" |
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346 |
shows "(Rep ---> op -` Abs ---> id) op \<in> = op \<in>" |
47308 | 347 |
by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms]) |
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348 |
|
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|
349 |
lemma mem_rsp[quot_respect]: |
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|
350 |
shows "(R ===> vset_rel R ===> op =) op \<in> op \<in>" |
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|
351 |
by (intro fun_relI) (simp add: vset_rel_def) |
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352 |
|
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353 |
end |