--- a/src/HOL/Lifting_Set.thy Thu Mar 06 14:25:55 2014 +0100
+++ b/src/HOL/Lifting_Set.thy Thu Mar 06 14:57:14 2014 +0100
@@ -10,90 +10,90 @@
subsection {* Relator and predicator properties *}
-definition set_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
- 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))"
+definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
+ where "rel_set 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))"
-lemma set_relI:
+lemma rel_setI:
assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
- shows "set_rel R A B"
- using assms unfolding set_rel_def by simp
+ shows "rel_set R A B"
+ using assms unfolding rel_set_def by simp
-lemma set_relD1: "\<lbrakk> set_rel R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
- and set_relD2: "\<lbrakk> set_rel R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
-by(simp_all add: set_rel_def)
+lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
+ and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
+by(simp_all add: rel_set_def)
-lemma set_rel_conversep [simp]: "set_rel A\<inverse>\<inverse> = (set_rel A)\<inverse>\<inverse>"
- unfolding set_rel_def by auto
+lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
+ unfolding rel_set_def by auto
-lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
- unfolding set_rel_def fun_eq_iff by auto
+lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
+ unfolding rel_set_def fun_eq_iff by auto
-lemma set_rel_mono[relator_mono]:
+lemma rel_set_mono[relator_mono]:
assumes "A \<le> B"
- shows "set_rel A \<le> set_rel B"
-using assms unfolding set_rel_def by blast
+ shows "rel_set A \<le> rel_set B"
+using assms unfolding rel_set_def by blast
-lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (R OO S)"
+lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
apply (rule sym)
apply (intro ext, rename_tac X Z)
apply (rule iffI)
apply (rule_tac b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}" in relcomppI)
- apply (simp add: set_rel_def, fast)
- apply (simp add: set_rel_def, fast)
- apply (simp add: set_rel_def, fast)
+ apply (simp add: rel_set_def, fast)
+ apply (simp add: rel_set_def, fast)
+ apply (simp add: rel_set_def, fast)
done
lemma Domainp_set[relator_domain]:
assumes "Domainp T = R"
- shows "Domainp (set_rel T) = (\<lambda>A. Ball A R)"
-using assms unfolding set_rel_def Domainp_iff[abs_def]
+ shows "Domainp (rel_set T) = (\<lambda>A. Ball A R)"
+using assms unfolding rel_set_def Domainp_iff[abs_def]
apply (intro ext)
apply (rule iffI)
apply blast
apply (rename_tac A, rule_tac x="{y. \<exists>x\<in>A. T x y}" in exI, fast)
done
-lemma left_total_set_rel[reflexivity_rule]:
- "left_total A \<Longrightarrow> left_total (set_rel A)"
- unfolding left_total_def set_rel_def
+lemma left_total_rel_set[reflexivity_rule]:
+ "left_total A \<Longrightarrow> left_total (rel_set A)"
+ unfolding left_total_def rel_set_def
apply safe
apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
done
-lemma left_unique_set_rel[reflexivity_rule]:
- "left_unique A \<Longrightarrow> left_unique (set_rel A)"
- unfolding left_unique_def set_rel_def
+lemma left_unique_rel_set[reflexivity_rule]:
+ "left_unique A \<Longrightarrow> left_unique (rel_set A)"
+ unfolding left_unique_def rel_set_def
by fast
-lemma right_total_set_rel [transfer_rule]:
- "right_total A \<Longrightarrow> right_total (set_rel A)"
-using left_total_set_rel[of "A\<inverse>\<inverse>"] by simp
+lemma right_total_rel_set [transfer_rule]:
+ "right_total A \<Longrightarrow> right_total (rel_set A)"
+using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
-lemma right_unique_set_rel [transfer_rule]:
- "right_unique A \<Longrightarrow> right_unique (set_rel A)"
- unfolding right_unique_def set_rel_def by fast
+lemma right_unique_rel_set [transfer_rule]:
+ "right_unique A \<Longrightarrow> right_unique (rel_set A)"
+ unfolding right_unique_def rel_set_def by fast
-lemma bi_total_set_rel [transfer_rule]:
- "bi_total A \<Longrightarrow> bi_total (set_rel A)"
-by(simp add: bi_total_conv_left_right left_total_set_rel right_total_set_rel)
+lemma bi_total_rel_set [transfer_rule]:
+ "bi_total A \<Longrightarrow> bi_total (rel_set A)"
+by(simp add: bi_total_conv_left_right left_total_rel_set right_total_rel_set)
-lemma bi_unique_set_rel [transfer_rule]:
- "bi_unique A \<Longrightarrow> bi_unique (set_rel A)"
- unfolding bi_unique_def set_rel_def by fast
+lemma bi_unique_rel_set [transfer_rule]:
+ "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
+ unfolding bi_unique_def rel_set_def by fast
lemma set_invariant_commute [invariant_commute]:
- "set_rel (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
- unfolding fun_eq_iff set_rel_def Lifting.invariant_def Ball_def by fast
+ "rel_set (Lifting.invariant P) = Lifting.invariant (\<lambda>A. Ball A P)"
+ unfolding fun_eq_iff rel_set_def Lifting.invariant_def Ball_def by fast
subsection {* Quotient theorem for the Lifting package *}
lemma Quotient_set[quot_map]:
assumes "Quotient R Abs Rep T"
- shows "Quotient (set_rel R) (image Abs) (image Rep) (set_rel T)"
+ shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
using assms unfolding Quotient_alt_def4
- apply (simp add: set_rel_OO[symmetric])
- apply (simp add: set_rel_def, fast)
+ apply (simp add: rel_set_OO[symmetric])
+ apply (simp add: rel_set_def, fast)
done
subsection {* Transfer rules for the Transfer package *}
@@ -104,143 +104,143 @@
begin
interpretation lifting_syntax .
-lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
- unfolding set_rel_def by simp
+lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
+ unfolding rel_set_def by simp
lemma insert_transfer [transfer_rule]:
- "(A ===> set_rel A ===> set_rel A) insert insert"
- unfolding fun_rel_def set_rel_def by auto
+ "(A ===> rel_set A ===> rel_set A) insert insert"
+ unfolding fun_rel_def rel_set_def by auto
lemma union_transfer [transfer_rule]:
- "(set_rel A ===> set_rel A ===> set_rel A) union union"
- unfolding fun_rel_def set_rel_def by auto
+ "(rel_set A ===> rel_set A ===> rel_set A) union union"
+ unfolding fun_rel_def rel_set_def by auto
lemma Union_transfer [transfer_rule]:
- "(set_rel (set_rel A) ===> set_rel A) Union Union"
- unfolding fun_rel_def set_rel_def by simp fast
+ "(rel_set (rel_set A) ===> rel_set A) Union Union"
+ unfolding fun_rel_def rel_set_def by simp fast
lemma image_transfer [transfer_rule]:
- "((A ===> B) ===> set_rel A ===> set_rel B) image image"
- unfolding fun_rel_def set_rel_def by simp fast
+ "((A ===> B) ===> rel_set A ===> rel_set B) image image"
+ unfolding fun_rel_def rel_set_def by simp fast
lemma UNION_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) UNION UNION"
+ "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
unfolding SUP_def [abs_def] by transfer_prover
lemma Ball_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> op =) ===> op =) Ball Ball"
- unfolding set_rel_def fun_rel_def by fast
+ "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
+ unfolding rel_set_def fun_rel_def by fast
lemma Bex_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
- unfolding set_rel_def fun_rel_def by fast
+ "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
+ unfolding rel_set_def fun_rel_def by fast
lemma Pow_transfer [transfer_rule]:
- "(set_rel A ===> set_rel (set_rel A)) Pow Pow"
- apply (rule fun_relI, rename_tac X Y, rule set_relI)
+ "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
+ apply (rule fun_relI, rename_tac X Y, rule rel_setI)
apply (rename_tac X', rule_tac x="{y\<in>Y. \<exists>x\<in>X'. A x y}" in rev_bexI, clarsimp)
- apply (simp add: set_rel_def, fast)
+ apply (simp add: rel_set_def, fast)
apply (rename_tac Y', rule_tac x="{x\<in>X. \<exists>y\<in>Y'. A x y}" in rev_bexI, clarsimp)
- apply (simp add: set_rel_def, fast)
+ apply (simp add: rel_set_def, fast)
done
-lemma set_rel_transfer [transfer_rule]:
- "((A ===> B ===> op =) ===> set_rel A ===> set_rel B ===> op =)
- set_rel set_rel"
- unfolding fun_rel_def set_rel_def by fast
+lemma rel_set_transfer [transfer_rule]:
+ "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =)
+ rel_set rel_set"
+ unfolding fun_rel_def rel_set_def by fast
lemma SUPR_parametric [transfer_rule]:
- "(set_rel R ===> (R ===> op =) ===> op =) SUPR (SUPR :: _ \<Rightarrow> _ \<Rightarrow> _::complete_lattice)"
+ "(rel_set R ===> (R ===> op =) ===> op =) SUPR (SUPR :: _ \<Rightarrow> _ \<Rightarrow> _::complete_lattice)"
proof(rule fun_relI)+
fix A B f and g :: "'b \<Rightarrow> 'c"
- assume AB: "set_rel R A B"
+ assume AB: "rel_set R A B"
and fg: "(R ===> op =) f g"
show "SUPR A f = SUPR B g"
- by(rule SUPR_eq)(auto 4 4 dest: set_relD1[OF AB] set_relD2[OF AB] fun_relD[OF fg] intro: rev_bexI)
+ by(rule SUPR_eq)(auto 4 4 dest: rel_setD1[OF AB] rel_setD2[OF AB] fun_relD[OF fg] intro: rev_bexI)
qed
lemma bind_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> set_rel B) ===> set_rel B) Set.bind Set.bind"
+ "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
unfolding bind_UNION[abs_def] by transfer_prover
subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
lemma member_transfer [transfer_rule]:
assumes "bi_unique A"
- shows "(A ===> set_rel A ===> op =) (op \<in>) (op \<in>)"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+ shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
+ using assms unfolding fun_rel_def rel_set_def bi_unique_def by fast
lemma right_total_Collect_transfer[transfer_rule]:
assumes "right_total A"
- shows "((A ===> op =) ===> set_rel A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
- using assms unfolding right_total_def set_rel_def fun_rel_def Domainp_iff by fast
+ shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
+ using assms unfolding right_total_def rel_set_def fun_rel_def Domainp_iff by fast
lemma Collect_transfer [transfer_rule]:
assumes "bi_total A"
- shows "((A ===> op =) ===> set_rel A) Collect Collect"
- using assms unfolding fun_rel_def set_rel_def bi_total_def by fast
+ shows "((A ===> op =) ===> rel_set A) Collect Collect"
+ using assms unfolding fun_rel_def rel_set_def bi_total_def by fast
lemma inter_transfer [transfer_rule]:
assumes "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> set_rel A) inter inter"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def by fast
+ shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
+ using assms unfolding fun_rel_def rel_set_def bi_unique_def by fast
lemma Diff_transfer [transfer_rule]:
assumes "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> set_rel A) (op -) (op -)"
- using assms unfolding fun_rel_def set_rel_def bi_unique_def
+ shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
+ using assms unfolding fun_rel_def rel_set_def bi_unique_def
unfolding Ball_def Bex_def Diff_eq
by (safe, simp, metis, simp, metis)
lemma subset_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "(set_rel A ===> set_rel A ===> op =) (op \<subseteq>) (op \<subseteq>)"
+ shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
unfolding subset_eq [abs_def] by transfer_prover
lemma right_total_UNIV_transfer[transfer_rule]:
assumes "right_total A"
- shows "(set_rel A) (Collect (Domainp A)) UNIV"
- using assms unfolding right_total_def set_rel_def Domainp_iff by blast
+ shows "(rel_set A) (Collect (Domainp A)) UNIV"
+ using assms unfolding right_total_def rel_set_def Domainp_iff by blast
lemma UNIV_transfer [transfer_rule]:
assumes "bi_total A"
- shows "(set_rel A) UNIV UNIV"
- using assms unfolding set_rel_def bi_total_def by simp
+ shows "(rel_set A) UNIV UNIV"
+ using assms unfolding rel_set_def bi_total_def by simp
lemma right_total_Compl_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
- shows "(set_rel A ===> set_rel A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
+ shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
unfolding Compl_eq [abs_def]
by (subst Collect_conj_eq[symmetric]) transfer_prover
lemma Compl_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
- shows "(set_rel A ===> set_rel A) uminus uminus"
+ shows "(rel_set A ===> rel_set A) uminus uminus"
unfolding Compl_eq [abs_def] by transfer_prover
lemma right_total_Inter_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
- shows "(set_rel (set_rel A) ===> set_rel A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
+ shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
unfolding Inter_eq[abs_def]
by (subst Collect_conj_eq[symmetric]) transfer_prover
lemma Inter_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
- shows "(set_rel (set_rel A) ===> set_rel A) Inter Inter"
+ shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
unfolding Inter_eq [abs_def] by transfer_prover
lemma filter_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
- shows "((A ===> op=) ===> set_rel A ===> set_rel A) Set.filter Set.filter"
- unfolding Set.filter_def[abs_def] fun_rel_def set_rel_def by blast
+ shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
+ unfolding Set.filter_def[abs_def] fun_rel_def rel_set_def by blast
-lemma bi_unique_set_rel_lemma:
- assumes "bi_unique R" and "set_rel R X Y"
+lemma bi_unique_rel_set_lemma:
+ assumes "bi_unique R" and "rel_set R X Y"
obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
proof
let ?f = "\<lambda>x. THE y. R x y"
from assms show f: "\<forall>x\<in>X. R x (?f x)"
- apply (clarsimp simp add: set_rel_def)
+ apply (clarsimp simp add: rel_set_def)
apply (drule (1) bspec, clarify)
apply (rule theI2, assumption)
apply (simp add: bi_unique_def)
@@ -248,13 +248,13 @@
done
from assms show "Y = image ?f X"
apply safe
- apply (clarsimp simp add: set_rel_def)
+ apply (clarsimp simp add: rel_set_def)
apply (drule (1) bspec, clarify)
apply (rule image_eqI)
apply (rule the_equality [symmetric], assumption)
apply (simp add: bi_unique_def)
apply assumption
- apply (clarsimp simp add: set_rel_def)
+ apply (clarsimp simp add: rel_set_def)
apply (frule (1) bspec, clarify)
apply (rule theI2, assumption)
apply (clarsimp simp add: bi_unique_def)
@@ -269,41 +269,41 @@
qed
lemma finite_transfer [transfer_rule]:
- "bi_unique A \<Longrightarrow> (set_rel A ===> op =) finite finite"
- by (rule fun_relI, erule (1) bi_unique_set_rel_lemma,
+ "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
+ by (rule fun_relI, erule (1) bi_unique_rel_set_lemma,
auto dest: finite_imageD)
lemma card_transfer [transfer_rule]:
- "bi_unique A \<Longrightarrow> (set_rel A ===> op =) card card"
- by (rule fun_relI, erule (1) bi_unique_set_rel_lemma, simp add: card_image)
+ "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
+ by (rule fun_relI, erule (1) bi_unique_rel_set_lemma, simp add: card_image)
lemma vimage_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_total A" "bi_unique B"
- shows "((A ===> B) ===> set_rel B ===> set_rel A) vimage vimage"
+ shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
unfolding vimage_def[abs_def] by transfer_prover
lemma setsum_parametric [transfer_rule]:
assumes "bi_unique A"
- shows "((A ===> op =) ===> set_rel A ===> op =) setsum setsum"
+ shows "((A ===> op =) ===> rel_set A ===> op =) setsum setsum"
proof(rule fun_relI)+
fix f :: "'a \<Rightarrow> 'c" and g S T
assume fg: "(A ===> op =) f g"
- and ST: "set_rel A S T"
+ and ST: "rel_set A S T"
show "setsum f S = setsum g T"
proof(rule setsum_reindex_cong)
let ?f = "\<lambda>t. THE s. A s t"
show "S = ?f ` T"
- by(blast dest: set_relD1[OF ST] set_relD2[OF ST] bi_uniqueDl[OF assms]
+ by(blast dest: rel_setD1[OF ST] rel_setD2[OF ST] bi_uniqueDl[OF assms]
intro: rev_image_eqI the_equality[symmetric] subst[rotated, where P="\<lambda>x. x \<in> S"])
show "inj_on ?f T"
proof(rule inj_onI)
fix t1 t2
assume "t1 \<in> T" "t2 \<in> T" "?f t1 = ?f t2"
- from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: set_relD2)
+ from ST `t1 \<in> T` obtain s1 where "A s1 t1" "s1 \<in> S" by(auto dest: rel_setD2)
hence "?f t1 = s1" by(auto dest: bi_uniqueDl[OF assms])
moreover
- from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: set_relD2)
+ from ST `t2 \<in> T` obtain s2 where "A s2 t2" "s2 \<in> S" by(auto dest: rel_setD2)
hence "?f t2 = s2" by(auto dest: bi_uniqueDl[OF assms])
ultimately have "s1 = s2" using `?f t1 = ?f t2` by simp
with `A s1 t1` `A s2 t2` show "t1 = t2" by(auto dest: bi_uniqueDr[OF assms])
@@ -311,7 +311,7 @@
fix t
assume "t \<in> T"
- with ST obtain s where "A s t" "s \<in> S" by(auto dest: set_relD2)
+ with ST obtain s where "A s t" "s \<in> S" by(auto dest: rel_setD2)
hence "?f t = s" by(auto dest: bi_uniqueDl[OF assms])
moreover from fg `A s t` have "f s = g t" by(rule fun_relD)
ultimately show "g t = f (?f t)" by simp