--- a/src/HOL/Library/Quotient_Set.thy Tue Aug 13 15:59:22 2013 +0200
+++ b/src/HOL/Library/Quotient_Set.thy Tue Aug 13 15:59:22 2013 +0200
@@ -8,273 +8,7 @@
imports Main Quotient_Syntax
begin
-subsection {* Relator for set type *}
-
-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))"
-
-lemma set_relI:
- 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
-
-lemma set_rel_conversep: "set_rel (conversep R) = conversep (set_rel R)"
- unfolding set_rel_def by auto
-
-lemma set_rel_eq [relator_eq]: "set_rel (op =) = (op =)"
- unfolding set_rel_def fun_eq_iff by auto
-
-lemma set_rel_mono[relator_mono]:
- assumes "A \<le> B"
- shows "set_rel A \<le> set_rel B"
-using assms unfolding set_rel_def by blast
-
-lemma set_rel_OO[relator_distr]: "set_rel R OO set_rel S = set_rel (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)
- 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]
-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 reflp_set_rel[reflexivity_rule]: "reflp R \<Longrightarrow> reflp (set_rel R)"
- unfolding reflp_def set_rel_def by fast
-
-lemma left_total_set_rel[reflexivity_rule]:
- "left_total A \<Longrightarrow> left_total (set_rel A)"
- unfolding left_total_def set_rel_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
- by fast
-
-lemma symp_set_rel: "symp R \<Longrightarrow> symp (set_rel R)"
- unfolding symp_def set_rel_def by fast
-
-lemma transp_set_rel: "transp R \<Longrightarrow> transp (set_rel R)"
- unfolding transp_def set_rel_def by fast
-
-lemma equivp_set_rel: "equivp R \<Longrightarrow> equivp (set_rel R)"
- by (blast intro: equivpI reflp_set_rel symp_set_rel transp_set_rel
- elim: equivpE)
-
-lemma right_total_set_rel [transfer_rule]:
- "right_total A \<Longrightarrow> right_total (set_rel A)"
- unfolding right_total_def set_rel_def
- by (rule allI, rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
-
-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 bi_total_set_rel [transfer_rule]:
- "bi_total A \<Longrightarrow> bi_total (set_rel A)"
- unfolding bi_total_def set_rel_def
- apply safe
- apply (rename_tac X, rule_tac x="{y. \<exists>x\<in>X. A x y}" in exI, fast)
- apply (rename_tac Y, rule_tac x="{x. \<exists>y\<in>Y. A x y}" in exI, fast)
- done
-
-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
-
-subsection {* Transfer rules for transfer package *}
-
-subsubsection {* Unconditional transfer rules *}
-
-lemma empty_transfer [transfer_rule]: "(set_rel A) {} {}"
- unfolding set_rel_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
-
-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
-
-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
-
-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
-
-lemma UNION_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> set_rel B) ===> set_rel 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
-
-lemma Bex_transfer [transfer_rule]:
- "(set_rel A ===> (A ===> op =) ===> op =) Bex Bex"
- unfolding set_rel_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)
- 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 (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)
- 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
-
-
-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
-
-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
-
-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
-
-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
-
-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
- 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>)"
- 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
-
-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
-
-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"
- 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"
- 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"
- 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"
- 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
-
-lemma bi_unique_set_rel_lemma:
- assumes "bi_unique R" and "set_rel 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 (drule (1) bspec, clarify)
- apply (rule theI2, assumption)
- apply (simp add: bi_unique_def)
- apply assumption
- done
- from assms show "Y = image ?f X"
- apply safe
- apply (clarsimp simp add: set_rel_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 (frule (1) bspec, clarify)
- apply (rule theI2, assumption)
- apply (clarsimp simp add: bi_unique_def)
- apply (simp add: bi_unique_def, metis)
- done
- show "inj_on ?f X"
- apply (rule inj_onI)
- apply (drule f [rule_format])
- apply (drule f [rule_format])
- apply (simp add: assms(1) [unfolded bi_unique_def])
- done
-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,
- 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)
-
-subsection {* Setup for 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)"
- using assms unfolding Quotient_alt_def4
- apply (simp add: set_rel_OO[symmetric] set_rel_conversep)
- apply (simp add: set_rel_def, fast)
- done
-
-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
-
-subsection {* Contravariant set map (vimage) and set relator *}
+subsection {* Contravariant set map (vimage) and set relator, rules for the Quotient package *}
definition "vset_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"