(* Title: HOL/Lifting_Set.thy
Author: Brian Huffman and Ondrej Kuncar
*)
header {* Setup for Lifting/Transfer for the set type *}
theory Lifting_Set
imports Lifting
begin
subsection {* Relator and predicator properties *}
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 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 "rel_set R A B"
using assms unfolding rel_set_def by simp
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 rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
unfolding rel_set_def by auto
lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
unfolding rel_set_def fun_eq_iff by auto
lemma rel_set_mono[relator_mono]:
assumes "A \<le> B"
shows "rel_set A \<le> rel_set B"
using assms unfolding rel_set_def by blast
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: rel_set_def, fast)
apply (simp add: rel_set_def, fast)
apply (simp add: rel_set_def, fast)
done
lemma Domainp_set[relator_domain]:
"Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
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_rel_set[transfer_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_rel_set[transfer_rule]:
"left_unique A \<Longrightarrow> left_unique (rel_set A)"
unfolding left_unique_def rel_set_def
by fast
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_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_rel_set [transfer_rule]:
"bi_total A \<Longrightarrow> bi_total (rel_set A)"
by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)
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_relator_eq_onp [relator_eq_onp]:
"rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
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
def f \<equiv> "\<lambda>x. THE y. R x y"
{ fix x assume "x \<in> X"
with `rel_set R X Y` `bi_unique R` have "R x (f x)"
by (simp add: bi_unique_def rel_set_def f_def) (metis theI)
with assms `x \<in> X`
have "R x (f x)" "\<forall>x'\<in>X. R x' (f x) \<longrightarrow> x = x'" "\<forall>y\<in>Y. R x y \<longrightarrow> y = f x" "f x \<in> Y"
by (fastforce simp add: bi_unique_def rel_set_def)+ }
note * = this
moreover
{ fix y assume "y \<in> Y"
with `rel_set R X Y` *(3) `y \<in> Y` have "\<exists>x\<in>X. y = f x"
by (fastforce simp: rel_set_def) }
ultimately show "\<forall>x\<in>X. R x (f x)" "Y = image f X" "inj_on f X"
by (auto simp: inj_on_def image_iff)
qed
subsection {* Quotient theorem for the Lifting package *}
lemma Quotient_set[quot_map]:
assumes "Quotient R Abs Rep T"
shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
using assms unfolding Quotient_alt_def4
apply (simp add: rel_set_OO[symmetric])
apply (simp add: rel_set_def, fast)
done
subsection {* Transfer rules for the Transfer package *}
subsubsection {* Unconditional transfer rules *}
context
begin
interpretation lifting_syntax .
lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
unfolding rel_set_def by simp
lemma insert_transfer [transfer_rule]:
"(A ===> rel_set A ===> rel_set A) insert insert"
unfolding rel_fun_def rel_set_def by auto
lemma union_transfer [transfer_rule]:
"(rel_set A ===> rel_set A ===> rel_set A) union union"
unfolding rel_fun_def rel_set_def by auto
lemma Union_transfer [transfer_rule]:
"(rel_set (rel_set A) ===> rel_set A) Union Union"
unfolding rel_fun_def rel_set_def by simp fast
lemma image_transfer [transfer_rule]:
"((A ===> B) ===> rel_set A ===> rel_set B) image image"
unfolding rel_fun_def rel_set_def by simp fast
lemma UNION_transfer [transfer_rule]:
"(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
lemma Ball_transfer [transfer_rule]:
"(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
unfolding rel_set_def rel_fun_def by fast
lemma Bex_transfer [transfer_rule]:
"(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
unfolding rel_set_def rel_fun_def by fast
lemma Pow_transfer [transfer_rule]:
"(rel_set A ===> rel_set (rel_set A)) Pow Pow"
apply (rule rel_funI, 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: 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: rel_set_def, fast)
done
lemma rel_set_transfer [transfer_rule]:
"((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
unfolding rel_fun_def rel_set_def by fast
lemma bind_transfer [transfer_rule]:
"(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
unfolding bind_UNION [abs_def] by transfer_prover
lemma INF_parametric [transfer_rule]:
"(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
unfolding INF_def [abs_def] by transfer_prover
lemma SUP_parametric [transfer_rule]:
"(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
unfolding SUP_def [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 ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
lemma right_total_Collect_transfer[transfer_rule]:
assumes "right_total A"
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 rel_fun_def Domainp_iff by fast
lemma Collect_transfer [transfer_rule]:
assumes "bi_total A"
shows "((A ===> op =) ===> rel_set A) Collect Collect"
using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
lemma inter_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
lemma Diff_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
using assms unfolding rel_fun_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 "(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 "(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 "(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 "(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 "(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 "(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 "(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=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
lemma finite_transfer [transfer_rule]:
"bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
(auto dest: finite_imageD)
lemma card_transfer [transfer_rule]:
"bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
by (rule rel_funI, 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) ===> rel_set B ===> rel_set A) vimage vimage"
unfolding vimage_def[abs_def] by transfer_prover
end
lemma (in comm_monoid_set) F_parametric [transfer_rule]:
fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
assumes "bi_unique A"
shows "rel_fun (rel_fun A (op =)) (rel_fun (rel_set A) (op =)) F F"
proof(rule rel_funI)+
fix f :: "'b \<Rightarrow> 'a" and g S T
assume "rel_fun A (op =) f g" "rel_set A S T"
with `bi_unique A` obtain i where "bij_betw i S T" "\<And>x. x \<in> S \<Longrightarrow> f x = g (i x)"
by (auto elim: bi_unique_rel_set_lemma simp: rel_fun_def bij_betw_def)
then show "F f S = F g T"
by (simp add: reindex_bij_betw)
qed
lemmas setsum_parametric = setsum.F_parametric
lemmas setprod_parametric = setprod.F_parametric
end