author  kuncar 
Fri, 27 Sep 2013 14:43:26 +0200  
changeset 53952  b2781a3ce958 
parent 53945  4191acef9d0e 
child 54257  5c7a3b6b05a9 
permissions  rwrr 
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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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53927  22 
lemma set_relD1: "\<lbrakk> set_rel R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y" 
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and set_relD2: "\<lbrakk> set_rel R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y" 

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by(simp_all add: set_rel_def) 

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lemma set_rel_conversep [simp]: "set_rel A\<inverse>\<inverse> = (set_rel A)\<inverse>\<inverse>" 
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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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using left_total_set_rel[of "A\<inverse>\<inverse>"] by simp 
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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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by(simp add: bi_total_conv_left_right left_total_set_rel right_total_set_rel) 
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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]) 
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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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"(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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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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144 
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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145 
apply (simp add: set_rel_def, fast) 
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146 
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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147 
apply (simp add: set_rel_def, fast) 
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148 
done 
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149 

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150 
lemma set_rel_transfer [transfer_rule]: 
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151 
"((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =) 
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152 
set_rel set_rel" 
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153 
unfolding fun_rel_def set_rel_def by fast 
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154 

53927  155 
lemma SUPR_parametric [transfer_rule]: 
156 
"(set_rel R ===> (R ===> op =) ===> op =) SUPR SUPR" 

157 
proof(rule fun_relI)+ 

158 
fix A B f and g :: "'b \<Rightarrow> 'c" 

159 
assume AB: "set_rel R A B" 

160 
and fg: "(R ===> op =) f g" 

161 
show "SUPR A f = SUPR B g" 

162 
by(rule SUPR_eq)(auto 4 4 dest: set_relD1[OF AB] set_relD2[OF AB] fun_relD[OF fg] intro: rev_bexI) 

163 
qed 

53012
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164 

53952  165 
lemma bind_transfer [transfer_rule]: 
166 
"(set_rel A ===> (A ===> set_rel B) ===> set_rel B) Set.bind Set.bind" 

167 
unfolding bind_UNION[abs_def] by transfer_prover 

168 

53012
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169 
subsubsection {* Rules requiring biunique, bitotal or righttotal relations *} 
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170 

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171 
lemma member_transfer [transfer_rule]: 
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172 
assumes "bi_unique A" 
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173 
shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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174 
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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175 

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176 
lemma right_total_Collect_transfer[transfer_rule]: 
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177 
assumes "right_total A" 
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178 
shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect" 
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179 
using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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180 

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181 
lemma Collect_transfer [transfer_rule]: 
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182 
assumes "bi_total A" 
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183 
shows "((A ===> op =) ===> set_rel A) Collect Collect" 
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184 
using assms unfolding fun_rel_def set_rel_def bi_total_def by fast 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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185 

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186 
lemma inter_transfer [transfer_rule]: 
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187 
assumes "bi_unique A" 
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188 
shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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189 
using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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190 

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191 
lemma Diff_transfer [transfer_rule]: 
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192 
assumes "bi_unique A" 
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193 
shows "(set_rel A ===> set_rel A ===> set_rel A) (op ) (op )" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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194 
using assms unfolding fun_rel_def set_rel_def bi_unique_def 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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195 
unfolding Ball_def Bex_def Diff_eq 
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196 
by (safe, simp, metis, simp, metis) 
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197 

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198 
lemma subset_transfer [transfer_rule]: 
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199 
assumes [transfer_rule]: "bi_unique A" 
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200 
shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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201 
unfolding subset_eq [abs_def] by transfer_prover 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
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202 

cb82606b8215
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203 
lemma right_total_UNIV_transfer[transfer_rule]: 
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204 
assumes "right_total A" 
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205 
shows "(set_rel A) (Collect (Domainp A)) UNIV" 
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206 
using assms unfolding right_total_def set_rel_def Domainp_iff by blast 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

207 

cb82606b8215
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208 
lemma UNIV_transfer [transfer_rule]: 
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209 
assumes "bi_total A" 
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210 
shows "(set_rel A) UNIV UNIV" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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211 
using assms unfolding set_rel_def bi_total_def by simp 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

212 

cb82606b8215
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213 
lemma right_total_Compl_transfer [transfer_rule]: 
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214 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

215 
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

216 
unfolding Compl_eq [abs_def] 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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diff
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217 
by (subst Collect_conj_eq[symmetric]) transfer_prover 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

218 

cb82606b8215
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219 
lemma Compl_transfer [transfer_rule]: 
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220 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

221 
shows "(set_rel A ===> set_rel A) uminus uminus" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

222 
unfolding Compl_eq [abs_def] by transfer_prover 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

223 

cb82606b8215
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diff
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224 
lemma right_total_Inter_transfer [transfer_rule]: 
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225 
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

226 
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
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diff
changeset

227 
unfolding Inter_eq[abs_def] 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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diff
changeset

228 
by (subst Collect_conj_eq[symmetric]) transfer_prover 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

229 

cb82606b8215
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diff
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230 
lemma Inter_transfer [transfer_rule]: 
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diff
changeset

231 
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

232 
shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

233 
unfolding Inter_eq [abs_def] by transfer_prover 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

234 

cb82606b8215
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235 
lemma filter_transfer [transfer_rule]: 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

236 
assumes [transfer_rule]: "bi_unique A" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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237 
shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

238 
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

239 

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

240 
lemma bi_unique_set_rel_lemma: 
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diff
changeset

241 
assumes "bi_unique R" and "set_rel R X Y" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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diff
changeset

242 
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
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diff
changeset

243 
proof 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

244 
let ?f = "\<lambda>x. THE y. R x y" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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diff
changeset

245 
from assms show f: "\<forall>x\<in>X. R x (?f x)" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

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

247 
apply (drule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

248 
apply (rule theI2, assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

249 
apply (simp add: bi_unique_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

250 
apply assumption 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

251 
done 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

252 
from assms show "Y = image ?f X" 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

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

254 
apply (clarsimp simp add: set_rel_def) 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

255 
apply (drule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

256 
apply (rule image_eqI) 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

257 
apply (rule the_equality [symmetric], assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

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

259 
apply assumption 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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diff
changeset

260 
apply (clarsimp simp add: set_rel_def) 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

261 
apply (frule (1) bspec, clarify) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

262 
apply (rule theI2, assumption) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

263 
apply (clarsimp simp add: bi_unique_def) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

264 
apply (simp add: bi_unique_def, metis) 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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diff
changeset

265 
done 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

266 
show "inj_on ?f X" 
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diff
changeset

267 
apply (rule inj_onI) 
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268 
apply (drule f [rule_format]) 
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269 
apply (drule f [rule_format]) 
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270 
apply (simp add: assms(1) [unfolded bi_unique_def]) 
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diff
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271 
done 
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272 
qed 
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273 

cb82606b8215
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274 
lemma finite_transfer [transfer_rule]: 
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275 
"bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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276 
by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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277 
auto dest: finite_imageD) 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
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changeset

278 

cb82606b8215
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279 
lemma card_transfer [transfer_rule]: 
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280 
"bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card" 
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move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
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281 
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
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282 

53927  283 
lemma vimage_parametric [transfer_rule]: 
284 
assumes [transfer_rule]: "bi_total A" "bi_unique B" 

285 
shows "((A ===> B) ===> set_rel B ===> set_rel A) vimage vimage" 

286 
unfolding vimage_def[abs_def] by transfer_prover 

287 

288 
lemma setsum_parametric [transfer_rule]: 

289 
assumes "bi_unique A" 

290 
shows "((A ===> op =) ===> set_rel A ===> op =) setsum setsum" 

291 
proof(rule fun_relI)+ 

292 
fix f :: "'a \<Rightarrow> 'c" and g S T 

293 
assume fg: "(A ===> op =) f g" 

294 
and ST: "set_rel A S T" 

295 
show "setsum f S = setsum g T" 

296 
proof(rule setsum_reindex_cong) 

297 
let ?f = "\<lambda>t. THE s. A s t" 

298 
show "S = ?f ` T" 

299 
by(blast dest: set_relD1[OF ST] set_relD2[OF ST] bi_uniqueDl[OF assms] 

300 
intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"]) 

301 

302 
show "inj_on ?f T" 

303 
proof(rule inj_onI) 

304 
fix t1 t2 

305 
assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2" 

306 
from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: set_relD2) 

307 
hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms]) 

308 
moreover 

309 
from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: set_relD2) 

310 
hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms]) 

311 
ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp 

312 
with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms]) 

313 
qed 

314 

315 
fix t 

316 
assume "t \<in> T" 

317 
with ST obtain s where "A s t" "s \<in> S" by(auto dest: set_relD2) 

318 
hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms]) 

319 
moreover from fg `A s t` have "f s = g t" by(rule fun_relD) 

320 
ultimately show "g t = f (?f t)" by simp 

321 
qed 

322 
qed 

323 

53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

324 
end 
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset

325 

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

326 
end 