author  kuncar 
Tue, 13 Aug 2013 15:59:22 +0200  
changeset 53012  cb82606b8215 
child 53927  abe2b313f0e5 
permissions  rwrr 
53012
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(* Title: HOL/Lifting_Set.thy 
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Author: Brian Huffman and Ondrej Kuncar 
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*) 
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header {* Setup for Lifting/Transfer for the set type *} 
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theory Lifting_Set 
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imports Lifting 
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begin 
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subsection {* Relator and predicator properties *} 
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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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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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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 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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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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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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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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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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subsection {* Quotient theorem for the Lifting package *} 
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lemma Quotient_set[quot_map]: 
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assumes "Quotient R Abs Rep T" 
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shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)" 
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using assms unfolding Quotient_alt_def4 
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apply (simp add: set_rel_OO[symmetric] set_rel_conversep) 
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apply (simp add: set_rel_def, fast) 
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done 
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subsection {* Transfer rules for the Transfer package *} 
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subsubsection {* Unconditional transfer rules *} 
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context 
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begin 
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interpretation lifting_syntax . 
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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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lemma Bex_transfer [transfer_rule]: 
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139 
"(set_rel A ===> (A ===> op =) ===> op =) Bex Bex" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

140 
unfolding set_rel_def fun_rel_def by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

141 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

142 
lemma Pow_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
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143 
"(set_rel A ===> set_rel (set_rel A)) Pow Pow" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

144 
apply (rule fun_relI, rename_tac X Y, rule set_relI) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

145 
apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

146 
apply (simp add: set_rel_def, fast) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

147 
apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

148 
apply (simp add: set_rel_def, fast) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

149 
done 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

150 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

151 
lemma set_rel_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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152 
"((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

153 
set_rel set_rel" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

154 
unfolding fun_rel_def set_rel_def by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

155 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

156 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

157 
subsubsection {* Rules requiring biunique, bitotal or righttotal relations *} 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

158 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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159 
lemma member_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

160 
assumes "bi_unique A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

161 
shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

162 
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

163 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

164 
lemma right_total_Collect_transfer[transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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165 
assumes "right_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

166 
shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

167 
using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

168 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

169 
lemma Collect_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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170 
assumes "bi_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

171 
shows "((A ===> op =) ===> set_rel A) Collect Collect" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

172 
using assms unfolding fun_rel_def set_rel_def bi_total_def by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

173 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

174 
lemma inter_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

175 
assumes "bi_unique A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

176 
shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

177 
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

178 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

179 
lemma Diff_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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180 
assumes "bi_unique A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

181 
shows "(set_rel A ===> set_rel A ===> set_rel A) (op ) (op )" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

182 
using assms unfolding fun_rel_def set_rel_def bi_unique_def 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

183 
unfolding Ball_def Bex_def Diff_eq 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

184 
by (safe, simp, metis, simp, metis) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

185 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

186 
lemma subset_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

187 
assumes [transfer_rule]: "bi_unique A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

188 
shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

189 
unfolding subset_eq [abs_def] by transfer_prover 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

190 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

191 
lemma right_total_UNIV_transfer[transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

192 
assumes "right_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

193 
shows "(set_rel A) (Collect (Domainp A)) UNIV" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

194 
using assms unfolding right_total_def set_rel_def Domainp_iff by blast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

195 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

196 
lemma UNIV_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

197 
assumes "bi_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

198 
shows "(set_rel A) UNIV UNIV" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

199 
using assms unfolding set_rel_def bi_total_def by simp 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

200 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

201 
lemma right_total_Compl_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

202 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

203 
shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

204 
unfolding Compl_eq [abs_def] 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

205 
by (subst Collect_conj_eq[symmetric]) transfer_prover 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

206 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

207 
lemma Compl_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

208 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

209 
shows "(set_rel A ===> set_rel A) uminus uminus" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

210 
unfolding Compl_eq [abs_def] by transfer_prover 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

211 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

212 
lemma right_total_Inter_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

213 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

214 
shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

215 
unfolding Inter_eq[abs_def] 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

216 
by (subst Collect_conj_eq[symmetric]) transfer_prover 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

217 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

218 
lemma Inter_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

219 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

220 
shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

221 
unfolding Inter_eq [abs_def] by transfer_prover 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

222 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

223 
lemma filter_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

224 
assumes [transfer_rule]: "bi_unique A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

225 
shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

226 
unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

227 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

228 
lemma bi_unique_set_rel_lemma: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

229 
assumes "bi_unique R" and "set_rel R X Y" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

230 
obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

231 
proof 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

232 
let ?f = "\<lambda>x. THE y. R x y" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

233 
from assms show f: "\<forall>x\<in>X. R x (?f x)" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

234 
apply (clarsimp simp add: set_rel_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

235 
apply (drule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

236 
apply (rule theI2, assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

237 
apply (simp add: bi_unique_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

238 
apply assumption 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

239 
done 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

240 
from assms show "Y = image ?f X" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

241 
apply safe 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

242 
apply (clarsimp simp add: set_rel_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

243 
apply (drule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

244 
apply (rule image_eqI) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

245 
apply (rule the_equality [symmetric], assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

246 
apply (simp add: bi_unique_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

247 
apply assumption 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

248 
apply (clarsimp simp add: set_rel_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

249 
apply (frule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

250 
apply (rule theI2, assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

251 
apply (clarsimp simp add: bi_unique_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

252 
apply (simp add: bi_unique_def, metis) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

253 
done 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

254 
show "inj_on ?f X" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

255 
apply (rule inj_onI) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

256 
apply (drule f [rule_format]) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

257 
apply (drule f [rule_format]) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

258 
apply (simp add: assms(1) [unfolded bi_unique_def]) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

259 
done 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

260 
qed 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

261 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

262 
lemma finite_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

263 
"bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

264 
by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

265 
auto dest: finite_imageD) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

266 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

267 
lemma card_transfer [transfer_rule]: 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

268 
"bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

269 
by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

270 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

271 
end 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

272 

cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
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273 
end 