# Theory Lifting_Set

```(*  Title:      HOL/Lifting_Set.thy
Author:     Brian Huffman and Ondrej Kuncar
*)

section ‹Setup for Lifting/Transfer for the set type›

theory Lifting_Set
imports Lifting Groups_Big
begin

subsection ‹Relator and predicator properties›

lemma rel_setD1: "⟦ rel_set R A B; x ∈ A ⟧ ⟹ ∃y ∈ B. R x y"
and rel_setD2: "⟦ rel_set R A B; y ∈ B ⟧ ⟹ ∃x ∈ A. R x y"

lemma rel_set_conversep [simp]: "rel_set A¯¯ = (rel_set A)¯¯"
unfolding rel_set_def by auto

lemma rel_set_eq [relator_eq]: "rel_set (=) = (=)"
unfolding rel_set_def fun_eq_iff by auto

lemma rel_set_mono[relator_mono]:
assumes "A ≤ B"
shows "rel_set A ≤ 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)
subgoal for X Z
apply (rule iffI)
apply (rule relcomppI [where b="{y. (∃x∈X. R x y) ∧ (∃z∈Z. S y z)}"])
done
done

lemma Domainp_set[relator_domain]:
"Domainp (rel_set T) = (λA. Ball A (Domainp T))"
unfolding rel_set_def Domainp_iff[abs_def]
apply (intro ext)
apply (rule iffI)
apply blast
subgoal for A by (rule exI [where x="{y. ∃x∈A. T x y}"]) fast
done

lemma left_total_rel_set[transfer_rule]:
"left_total A ⟹ left_total (rel_set A)"
unfolding left_total_def rel_set_def
apply safe
subgoal for X by (rule exI [where x="{y. ∃x∈X. A x y}"]) fast
done

lemma left_unique_rel_set[transfer_rule]:
"left_unique A ⟹ left_unique (rel_set A)"
unfolding left_unique_def rel_set_def
by fast

lemma right_total_rel_set [transfer_rule]:
"right_total A ⟹ right_total (rel_set A)"
using left_total_rel_set[of "A¯¯"] by simp

lemma right_unique_rel_set [transfer_rule]:
"right_unique A ⟹ right_unique (rel_set A)"
unfolding right_unique_def rel_set_def by fast

lemma bi_total_rel_set [transfer_rule]:
"bi_total A ⟹ bi_total (rel_set A)"

lemma bi_unique_rel_set [transfer_rule]:
"bi_unique A ⟹ 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 (λA. 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 "∀x∈X. R x (f x)"
proof
define f where "f x = (THE y. R x y)" for x
{ fix x assume "x ∈ 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 ∈ X›
have  "R x (f x)" "∀x'∈X. R x' (f x) ⟶ x = x'" "∀y∈Y. R x y ⟶ y = f x" "f x ∈ Y"
by (fastforce simp add: bi_unique_def rel_set_def)+ }
note * = this
moreover
{ fix y assume "y ∈ Y"
with ‹rel_set R X Y› *(3) ‹y ∈ Y› have "∃x∈X. y = f x"
by (fastforce simp: rel_set_def) }
ultimately show "∀x∈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 fast
done

subsection ‹Transfer rules for the Transfer package›

subsubsection ‹Unconditional transfer rules›

context includes lifting_syntax
begin

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]: ― ‹TODO deletion candidate›
"(rel_set A ===> (A ===> rel_set B) ===> rel_set B) (λA f. ⋃(f ` A)) (λA f. ⋃(f ` A))"
by transfer_prover

lemma Ball_transfer [transfer_rule]:
"(rel_set A ===> (A ===> (=)) ===> (=)) Ball Ball"
unfolding rel_set_def rel_fun_def by fast

lemma Bex_transfer [transfer_rule]:
"(rel_set A ===> (A ===> (=)) ===> (=)) 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)
apply (rule rel_setI)
subgoal for X Y X'
apply (rule rev_bexI [where x="{y∈Y. ∃x∈X'. A x y}"])
apply clarsimp
apply fast
done
subgoal for X Y Y'
apply (rule rev_bexI [where x="{x∈X. ∃y∈Y'. A x y}"])
apply clarsimp
apply fast
done
done

lemma rel_set_transfer [transfer_rule]:
"((A ===> B ===> (=)) ===> rel_set A ===> rel_set B ===> (=)) 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]: ― ‹TODO deletion candidate›
"(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) (λA f. Inf (f ` A)) (λA f. Inf (f ` A))"
by transfer_prover

lemma SUP_parametric [transfer_rule]: ― ‹TODO deletion candidate›
"(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) (λA f. Sup (f ` A)) (λA f. Sup (f ` A))"
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 ===> (=)) (∈) (∈)"
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 ===> (=)) ===> rel_set A) (λP. Collect (λx. P x ∧ 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 ===> (=)) ===> 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) (-) (-)"
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 ===> (=)) (⊆) (⊆)"
unfolding subset_eq [abs_def] by transfer_prover

context
includes lifting_syntax
begin

lemma strict_subset_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(rel_set A ===> rel_set A ===> (=)) (⊂) (⊂)"
unfolding subset_not_subset_eq by transfer_prover

end

declare right_total_UNIV_transfer[transfer_rule]

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) (λS. uminus S ∩ 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) (λS. ⋂S ∩ 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 ===> (=)) ===> 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 ⟹ (rel_set A ===> (=)) 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 ⟹ (rel_set A ===> (=)) card card"
by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)

context
includes lifting_syntax
begin

lemma vimage_right_total_transfer[transfer_rule]:
assumes [transfer_rule]: "bi_unique B" "right_total A"
shows "((A ===> B) ===> rel_set B ===> rel_set A) (λf X. f -` X ∩ Collect (Domainp A)) vimage"
proof -
let ?vimage = "(λf B. {x. f x ∈ B ∧ Domainp A x})"
have "((A ===> B) ===> rel_set B ===> rel_set A) ?vimage vimage"
unfolding vimage_def
by transfer_prover
also have "?vimage = (λf X. f -` X ∩ Collect (Domainp A))"
by auto
finally show ?thesis .
qed

end

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

lemma Image_parametric [transfer_rule]:
assumes "bi_unique A"
shows "(rel_set (rel_prod A B) ===> rel_set A ===> rel_set B) (``) (``)"
by (intro rel_funI rel_setI)
(force dest: rel_setD1 bi_uniqueDr[OF assms], force dest: rel_setD2 bi_uniqueDl[OF assms])

lemma inj_on_transfer[transfer_rule]:
"((A ===> B) ===> rel_set A ===> (=)) inj_on inj_on"
if [transfer_rule]: "bi_unique A" "bi_unique B"
unfolding inj_on_def
by transfer_prover

end

lemma (in comm_monoid_set) F_parametric [transfer_rule]:
fixes A :: "'b ⇒ 'c ⇒ bool"
assumes "bi_unique A"
shows "rel_fun (rel_fun A (=)) (rel_fun (rel_set A) (=)) F F"
proof (rule rel_funI)+
fix f :: "'b ⇒ 'a" and g S T
assume "rel_fun A (=) f g" "rel_set A S T"
with ‹bi_unique A› obtain i where "bij_betw i S T" "⋀x. x ∈ S ⟹ 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"
qed

lemmas sum_parametric = sum.F_parametric
lemmas prod_parametric = prod.F_parametric

lemma rel_set_UNION:
assumes [transfer_rule]: "rel_set Q A B" "rel_fun Q (rel_set R) f g"
shows "rel_set R (⋃(f ` A)) (⋃(g ` B))"
by transfer_prover

context
includes lifting_syntax
begin

lemma fold_graph_transfer[transfer_rule]:
assumes "bi_unique R" "right_total R"
shows "((R ===> (=) ===> (=)) ===> (=) ===> rel_set R ===> (=) ===> (=)) fold_graph fold_graph"
proof(intro rel_funI)
fix f1 :: "'a ⇒ 'c ⇒ 'c" and f2 :: "'b ⇒ 'c ⇒ 'c"
assume rel_f: "(R ===> (=) ===> (=)) f1 f2"
fix z1 z2 :: 'c assume [simp]: "z1 = z2"
fix A1 A2 assume rel_A: "rel_set R A1 A2"
fix y1 y2 :: 'c assume [simp]: "y1 = y2"

from ‹bi_unique R› ‹right_total R› have The_y: "∀y. ∃!x. R x y"
unfolding bi_unique_def right_total_def by auto
define r where "r ≡ λy. THE x. R x y"

from The_y have r_y: "R (r y) y" for y
unfolding r_def using the_equality by fastforce
with assms rel_A have "inj_on r A2" "A1 = r ` A2"
unfolding r_def rel_set_def inj_on_def bi_unique_def
apply(auto simp: image_iff) by metis+
with ‹bi_unique R› rel_f r_y have "(f1 o r) y = f2 y" for y
unfolding bi_unique_def rel_fun_def by auto
then have "(f1 o r) = f2"
by blast
then show "fold_graph f1 z1 A1 y1 = fold_graph f2 z2 A2 y2"
by (fastforce simp: fold_graph_image[OF ‹inj_on r A2›] ‹A1 = r ` A2›)
qed

lemma fold_transfer[transfer_rule]:
assumes [transfer_rule]: "bi_unique R" "right_total R"
shows "((R ===> (=) ===> (=)) ===> (=) ===> rel_set R ===> (=)) Finite_Set.fold Finite_Set.fold"
unfolding Finite_Set.fold_def
by transfer_prover

end

end
```