(* Title: HOL/Library/FSet.thy
Author: Ondrej Kuncar, TU Muenchen
Author: Cezary Kaliszyk and Christian Urban
Author: Andrei Popescu, TU Muenchen
Author: Martin Desharnais, MPI-INF Saarbruecken
*)
section \<open>Type of finite sets defined as a subtype of sets\<close>
theory FSet
imports Main Countable
begin
subsection \<open>Definition of the type\<close>
typedef 'a fset = "{A :: 'a set. finite A}" morphisms fset Abs_fset
by auto
setup_lifting type_definition_fset
subsection \<open>Basic operations and type class instantiations\<close>
(* FIXME transfer and right_total vs. bi_total *)
instantiation fset :: (finite) finite
begin
instance by (standard; transfer; simp)
end
instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
begin
lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp
lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer
.
definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
lemma less_fset_transfer[transfer_rule]:
includes lifting_syntax
assumes [transfer_rule]: "bi_unique A"
shows "((pcr_fset A) ===> (pcr_fset A) ===> (=)) (\<subset>) (<)"
unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover
lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer
by simp
lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer
by simp
lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer
by simp
instance
by (standard; transfer; auto)+
end
abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot"
abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys"
abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys"
abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys"
abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys"
instantiation fset :: (equal) equal
begin
definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A"
instance by intro_classes (auto simp add: equal_fset_def)
end
instantiation fset :: (type) conditionally_complete_lattice
begin
context includes lifting_syntax
begin
lemma right_total_Inf_fset_transfer:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
shows "(rel_set (rel_set A) ===> rel_set A)
(\<lambda>S. if finite (\<Inter>S \<inter> Collect (Domainp A)) then \<Inter>S \<inter> Collect (Domainp A) else {})
(\<lambda>S. if finite (Inf S) then Inf S else {})"
by transfer_prover
lemma Inf_fset_transfer:
assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {})
(\<lambda>A. if finite (Inf A) then Inf A else {})"
by transfer_prover
lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}"
parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp
lemma Sup_fset_transfer:
assumes [transfer_rule]: "bi_unique A"
shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {})
(\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover
lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}"
parametric Sup_fset_transfer by simp
lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
by (auto intro: finite_subset)
lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset (=)) ===> (=)) bdd_below bdd_below"
by auto
end
instance
proof
fix x z :: "'a fset"
fix X :: "'a fset set"
{
assume "x \<in> X" "bdd_below X"
then show "Inf X |\<subseteq>| x" by transfer auto
next
assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
next
assume "x \<in> X" "bdd_above X"
then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
by (auto simp: bdd_above_def)
then show "x |\<subseteq>| Sup X"
by transfer (auto intro!: finite_Sup)
next
assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
}
qed
end
instantiation fset :: (finite) complete_lattice
begin
lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer
by simp
instance
by (standard; transfer; auto)
end
instantiation fset :: (finite) complete_boolean_algebra
begin
lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus
parametric right_total_Compl_transfer Compl_transfer by simp
instance
by (standard; transfer) (simp_all add: Inf_Sup Diff_eq)
end
abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top"
abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x"
declare top_fset.rep_eq[simp]
subsection \<open>Other operations\<close>
lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer
by simp
syntax
"_insert_fset" :: "args => 'a fset" ("{|(_)|}")
translations
"{|x, xs|}" == "CONST finsert x {|xs|}"
"{|x|}" == "CONST finsert x {||}"
abbreviation fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) where
"x |\<in>| X \<equiv> x \<in> fset X"
abbreviation not_fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where
"x |\<notin>| X \<equiv> x \<notin> fset X"
context
begin
qualified abbreviation Ball :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
"Ball X \<equiv> Set.Ball (fset X)"
alias fBall = FSet.Ball
qualified abbreviation Bex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where
"Bex X \<equiv> Set.Bex (fset X)"
alias fBex = FSet.Bex
end
syntax (input)
"_fBall" :: "pttrn \<Rightarrow> 'a fset \<Rightarrow> bool \<Rightarrow> bool" ("(3! (_/|:|_)./ _)" [0, 0, 10] 10)
"_fBex" :: "pttrn \<Rightarrow> 'a fset \<Rightarrow> bool \<Rightarrow> bool" ("(3? (_/|:|_)./ _)" [0, 0, 10] 10)
syntax
"_fBall" :: "pttrn \<Rightarrow> 'a fset \<Rightarrow> bool \<Rightarrow> bool" ("(3\<forall>(_/|\<in>|_)./ _)" [0, 0, 10] 10)
"_fBex" :: "pttrn \<Rightarrow> 'a fset \<Rightarrow> bool \<Rightarrow> bool" ("(3\<exists>(_/|\<in>|_)./ _)" [0, 0, 10] 10)
translations
"\<forall>x|\<in>|A. P" \<rightleftharpoons> "CONST FSet.Ball A (\<lambda>x. P)"
"\<exists>x|\<in>|A. P" \<rightleftharpoons> "CONST FSet.Bex A (\<lambda>x. P)"
print_translation \<open>
[Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>fBall\<close> \<^syntax_const>\<open>_fBall\<close>,
Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>fBex\<close> \<^syntax_const>\<open>_fBex\<close>]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
context includes lifting_syntax
begin
lemma fmember_transfer0[transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(A ===> pcr_fset A ===> (=)) (\<in>) (|\<in>|)"
by transfer_prover
lemma fBall_transfer0[transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Ball) (fBall)"
by transfer_prover
lemma fBex_transfer0[transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Bex) (fBex)"
by transfer_prover
lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter
parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp
lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer
by (simp add: finite_subset)
lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer .
lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image
parametric image_transfer by simp
lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem .
lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer
by (simp add: Set.bind_def)
lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp
lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold .
lift_definition fset_of_list :: "'a list \<Rightarrow> 'a fset" is set by (rule finite_set)
lift_definition sorted_list_of_fset :: "'a::linorder fset \<Rightarrow> 'a list" is sorted_list_of_set .
subsection \<open>Transferred lemmas from Set.thy\<close>
lemma fset_eqI: "(\<And>x. (x |\<in>| A) = (x |\<in>| B)) \<Longrightarrow> A = B"
by (rule set_eqI[Transfer.transferred])
lemma fset_eq_iff[no_atp]: "(A = B) = (\<forall>x. (x |\<in>| A) = (x |\<in>| B))"
by (rule set_eq_iff[Transfer.transferred])
lemma fBallI[no_atp]: "(\<And>x. x |\<in>| A \<Longrightarrow> P x) \<Longrightarrow> fBall A P"
by (rule ballI[Transfer.transferred])
lemma fbspec[no_atp]: "fBall A P \<Longrightarrow> x |\<in>| A \<Longrightarrow> P x"
by (rule bspec[Transfer.transferred])
lemma fBallE[no_atp]: "fBall A P \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x |\<notin>| A \<Longrightarrow> Q) \<Longrightarrow> Q"
by (rule ballE[Transfer.transferred])
lemma fBexI[no_atp]: "P x \<Longrightarrow> x |\<in>| A \<Longrightarrow> fBex A P"
by (rule bexI[Transfer.transferred])
lemma rev_fBexI[no_atp]: "x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> fBex A P"
by (rule rev_bexI[Transfer.transferred])
lemma fBexCI[no_atp]: "(fBall A (\<lambda>x. \<not> P x) \<Longrightarrow> P a) \<Longrightarrow> a |\<in>| A \<Longrightarrow> fBex A P"
by (rule bexCI[Transfer.transferred])
lemma fBexE[no_atp]: "fBex A P \<Longrightarrow> (\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q"
by (rule bexE[Transfer.transferred])
lemma fBall_triv[no_atp]: "fBall A (\<lambda>x. P) = ((\<exists>x. x |\<in>| A) \<longrightarrow> P)"
by (rule ball_triv[Transfer.transferred])
lemma fBex_triv[no_atp]: "fBex A (\<lambda>x. P) = ((\<exists>x. x |\<in>| A) \<and> P)"
by (rule bex_triv[Transfer.transferred])
lemma fBex_triv_one_point1[no_atp]: "fBex A (\<lambda>x. x = a) = (a |\<in>| A)"
by (rule bex_triv_one_point1[Transfer.transferred])
lemma fBex_triv_one_point2[no_atp]: "fBex A ((=) a) = (a |\<in>| A)"
by (rule bex_triv_one_point2[Transfer.transferred])
lemma fBex_one_point1[no_atp]: "fBex A (\<lambda>x. x = a \<and> P x) = (a |\<in>| A \<and> P a)"
by (rule bex_one_point1[Transfer.transferred])
lemma fBex_one_point2[no_atp]: "fBex A (\<lambda>x. a = x \<and> P x) = (a |\<in>| A \<and> P a)"
by (rule bex_one_point2[Transfer.transferred])
lemma fBall_one_point1[no_atp]: "fBall A (\<lambda>x. x = a \<longrightarrow> P x) = (a |\<in>| A \<longrightarrow> P a)"
by (rule ball_one_point1[Transfer.transferred])
lemma fBall_one_point2[no_atp]: "fBall A (\<lambda>x. a = x \<longrightarrow> P x) = (a |\<in>| A \<longrightarrow> P a)"
by (rule ball_one_point2[Transfer.transferred])
lemma fBall_conj_distrib: "fBall A (\<lambda>x. P x \<and> Q x) = (fBall A P \<and> fBall A Q)"
by (rule ball_conj_distrib[Transfer.transferred])
lemma fBex_disj_distrib: "fBex A (\<lambda>x. P x \<or> Q x) = (fBex A P \<or> fBex A Q)"
by (rule bex_disj_distrib[Transfer.transferred])
lemma fBall_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x) \<Longrightarrow> fBall A P = fBall B Q"
by (rule ball_cong[Transfer.transferred])
lemma fBex_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x) \<Longrightarrow> fBex A P = fBex B Q"
by (rule bex_cong[Transfer.transferred])
lemma fsubsetI[intro!]: "(\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B) \<Longrightarrow> A |\<subseteq>| B"
by (rule subsetI[Transfer.transferred])
lemma fsubsetD[elim, intro?]: "A |\<subseteq>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B"
by (rule subsetD[Transfer.transferred])
lemma rev_fsubsetD[no_atp,intro?]: "c |\<in>| A \<Longrightarrow> A |\<subseteq>| B \<Longrightarrow> c |\<in>| B"
by (rule rev_subsetD[Transfer.transferred])
lemma fsubsetCE[no_atp,elim]: "A |\<subseteq>| B \<Longrightarrow> (c |\<notin>| A \<Longrightarrow> P) \<Longrightarrow> (c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P"
by (rule subsetCE[Transfer.transferred])
lemma fsubset_eq[no_atp]: "(A |\<subseteq>| B) = fBall A (\<lambda>x. x |\<in>| B)"
by (rule subset_eq[Transfer.transferred])
lemma contra_fsubsetD[no_atp]: "A |\<subseteq>| B \<Longrightarrow> c |\<notin>| B \<Longrightarrow> c |\<notin>| A"
by (rule contra_subsetD[Transfer.transferred])
lemma fsubset_refl: "A |\<subseteq>| A"
by (rule subset_refl[Transfer.transferred])
lemma fsubset_trans: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<subseteq>| C"
by (rule subset_trans[Transfer.transferred])
lemma fset_rev_mp: "c |\<in>| A \<Longrightarrow> A |\<subseteq>| B \<Longrightarrow> c |\<in>| B"
by (rule rev_subsetD[Transfer.transferred])
lemma fset_mp: "A |\<subseteq>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B"
by (rule subsetD[Transfer.transferred])
lemma fsubset_not_fsubset_eq[code]: "(A |\<subset>| B) = (A |\<subseteq>| B \<and> \<not> B |\<subseteq>| A)"
by (rule subset_not_subset_eq[Transfer.transferred])
lemma eq_fmem_trans: "a = b \<Longrightarrow> b |\<in>| A \<Longrightarrow> a |\<in>| A"
by (rule eq_mem_trans[Transfer.transferred])
lemma fsubset_antisym[intro!]: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> A = B"
by (rule subset_antisym[Transfer.transferred])
lemma fequalityD1: "A = B \<Longrightarrow> A |\<subseteq>| B"
by (rule equalityD1[Transfer.transferred])
lemma fequalityD2: "A = B \<Longrightarrow> B |\<subseteq>| A"
by (rule equalityD2[Transfer.transferred])
lemma fequalityE: "A = B \<Longrightarrow> (A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> P) \<Longrightarrow> P"
by (rule equalityE[Transfer.transferred])
lemma fequalityCE[elim]:
"A = B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> (c |\<notin>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> P) \<Longrightarrow> P"
by (rule equalityCE[Transfer.transferred])
lemma eqfset_imp_iff: "A = B \<Longrightarrow> (x |\<in>| A) = (x |\<in>| B)"
by (rule eqset_imp_iff[Transfer.transferred])
lemma eqfelem_imp_iff: "x = y \<Longrightarrow> (x |\<in>| A) = (y |\<in>| A)"
by (rule eqelem_imp_iff[Transfer.transferred])
lemma fempty_iff[simp]: "(c |\<in>| {||}) = False"
by (rule empty_iff[Transfer.transferred])
lemma fempty_fsubsetI[iff]: "{||} |\<subseteq>| x"
by (rule empty_subsetI[Transfer.transferred])
lemma equalsffemptyI: "(\<And>y. y |\<in>| A \<Longrightarrow> False) \<Longrightarrow> A = {||}"
by (rule equals0I[Transfer.transferred])
lemma equalsffemptyD: "A = {||} \<Longrightarrow> a |\<notin>| A"
by (rule equals0D[Transfer.transferred])
lemma fBall_fempty[simp]: "fBall {||} P = True"
by (rule ball_empty[Transfer.transferred])
lemma fBex_fempty[simp]: "fBex {||} P = False"
by (rule bex_empty[Transfer.transferred])
lemma fPow_iff[iff]: "(A |\<in>| fPow B) = (A |\<subseteq>| B)"
by (rule Pow_iff[Transfer.transferred])
lemma fPowI: "A |\<subseteq>| B \<Longrightarrow> A |\<in>| fPow B"
by (rule PowI[Transfer.transferred])
lemma fPowD: "A |\<in>| fPow B \<Longrightarrow> A |\<subseteq>| B"
by (rule PowD[Transfer.transferred])
lemma fPow_bottom: "{||} |\<in>| fPow B"
by (rule Pow_bottom[Transfer.transferred])
lemma fPow_top: "A |\<in>| fPow A"
by (rule Pow_top[Transfer.transferred])
lemma fPow_not_fempty: "fPow A \<noteq> {||}"
by (rule Pow_not_empty[Transfer.transferred])
lemma finter_iff[simp]: "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)"
by (rule Int_iff[Transfer.transferred])
lemma finterI[intro!]: "c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> c |\<in>| A |\<inter>| B"
by (rule IntI[Transfer.transferred])
lemma finterD1: "c |\<in>| A |\<inter>| B \<Longrightarrow> c |\<in>| A"
by (rule IntD1[Transfer.transferred])
lemma finterD2: "c |\<in>| A |\<inter>| B \<Longrightarrow> c |\<in>| B"
by (rule IntD2[Transfer.transferred])
lemma finterE[elim!]: "c |\<in>| A |\<inter>| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P"
by (rule IntE[Transfer.transferred])
lemma funion_iff[simp]: "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)"
by (rule Un_iff[Transfer.transferred])
lemma funionI1[elim?]: "c |\<in>| A \<Longrightarrow> c |\<in>| A |\<union>| B"
by (rule UnI1[Transfer.transferred])
lemma funionI2[elim?]: "c |\<in>| B \<Longrightarrow> c |\<in>| A |\<union>| B"
by (rule UnI2[Transfer.transferred])
lemma funionCI[intro!]: "(c |\<notin>| B \<Longrightarrow> c |\<in>| A) \<Longrightarrow> c |\<in>| A |\<union>| B"
by (rule UnCI[Transfer.transferred])
lemma funionE[elim!]: "c |\<in>| A |\<union>| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> P) \<Longrightarrow> (c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P"
by (rule UnE[Transfer.transferred])
lemma fminus_iff[simp]: "(c |\<in>| A |-| B) = (c |\<in>| A \<and> c |\<notin>| B)"
by (rule Diff_iff[Transfer.transferred])
lemma fminusI[intro!]: "c |\<in>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> c |\<in>| A |-| B"
by (rule DiffI[Transfer.transferred])
lemma fminusD1: "c |\<in>| A |-| B \<Longrightarrow> c |\<in>| A"
by (rule DiffD1[Transfer.transferred])
lemma fminusD2: "c |\<in>| A |-| B \<Longrightarrow> c |\<in>| B \<Longrightarrow> P"
by (rule DiffD2[Transfer.transferred])
lemma fminusE[elim!]: "c |\<in>| A |-| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> P) \<Longrightarrow> P"
by (rule DiffE[Transfer.transferred])
lemma finsert_iff[simp]: "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)"
by (rule insert_iff[Transfer.transferred])
lemma finsertI1: "a |\<in>| finsert a B"
by (rule insertI1[Transfer.transferred])
lemma finsertI2: "a |\<in>| B \<Longrightarrow> a |\<in>| finsert b B"
by (rule insertI2[Transfer.transferred])
lemma finsertE[elim!]: "a |\<in>| finsert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a |\<in>| A \<Longrightarrow> P) \<Longrightarrow> P"
by (rule insertE[Transfer.transferred])
lemma finsertCI[intro!]: "(a |\<notin>| B \<Longrightarrow> a = b) \<Longrightarrow> a |\<in>| finsert b B"
by (rule insertCI[Transfer.transferred])
lemma fsubset_finsert_iff:
"(A |\<subseteq>| finsert x B) = (if x |\<in>| A then A |-| {|x|} |\<subseteq>| B else A |\<subseteq>| B)"
by (rule subset_insert_iff[Transfer.transferred])
lemma finsert_ident: "x |\<notin>| A \<Longrightarrow> x |\<notin>| B \<Longrightarrow> (finsert x A = finsert x B) = (A = B)"
by (rule insert_ident[Transfer.transferred])
lemma fsingletonI[intro!,no_atp]: "a |\<in>| {|a|}"
by (rule singletonI[Transfer.transferred])
lemma fsingletonD[dest!,no_atp]: "b |\<in>| {|a|} \<Longrightarrow> b = a"
by (rule singletonD[Transfer.transferred])
lemma fsingleton_iff: "(b |\<in>| {|a|}) = (b = a)"
by (rule singleton_iff[Transfer.transferred])
lemma fsingleton_inject[dest!]: "{|a|} = {|b|} \<Longrightarrow> a = b"
by (rule singleton_inject[Transfer.transferred])
lemma fsingleton_finsert_inj_eq[iff,no_atp]: "({|b|} = finsert a A) = (a = b \<and> A |\<subseteq>| {|b|})"
by (rule singleton_insert_inj_eq[Transfer.transferred])
lemma fsingleton_finsert_inj_eq'[iff,no_atp]: "(finsert a A = {|b|}) = (a = b \<and> A |\<subseteq>| {|b|})"
by (rule singleton_insert_inj_eq'[Transfer.transferred])
lemma fsubset_fsingletonD: "A |\<subseteq>| {|x|} \<Longrightarrow> A = {||} \<or> A = {|x|}"
by (rule subset_singletonD[Transfer.transferred])
lemma fminus_single_finsert: "A |-| {|x|} |\<subseteq>| B \<Longrightarrow> A |\<subseteq>| finsert x B"
by (rule Diff_single_insert[Transfer.transferred])
lemma fdoubleton_eq_iff: "({|a, b|} = {|c, d|}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
by (rule doubleton_eq_iff[Transfer.transferred])
lemma funion_fsingleton_iff:
"(A |\<union>| B = {|x|}) = (A = {||} \<and> B = {|x|} \<or> A = {|x|} \<and> B = {||} \<or> A = {|x|} \<and> B = {|x|})"
by (rule Un_singleton_iff[Transfer.transferred])
lemma fsingleton_funion_iff:
"({|x|} = A |\<union>| B) = (A = {||} \<and> B = {|x|} \<or> A = {|x|} \<and> B = {||} \<or> A = {|x|} \<and> B = {|x|})"
by (rule singleton_Un_iff[Transfer.transferred])
lemma fimage_eqI[simp, intro]: "b = f x \<Longrightarrow> x |\<in>| A \<Longrightarrow> b |\<in>| f |`| A"
by (rule image_eqI[Transfer.transferred])
lemma fimageI: "x |\<in>| A \<Longrightarrow> f x |\<in>| f |`| A"
by (rule imageI[Transfer.transferred])
lemma rev_fimage_eqI: "x |\<in>| A \<Longrightarrow> b = f x \<Longrightarrow> b |\<in>| f |`| A"
by (rule rev_image_eqI[Transfer.transferred])
lemma fimageE[elim!]: "b |\<in>| f |`| A \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> x |\<in>| A \<Longrightarrow> thesis) \<Longrightarrow> thesis"
by (rule imageE[Transfer.transferred])
lemma Compr_fimage_eq: "{x. x |\<in>| f |`| A \<and> P x} = f ` {x. x |\<in>| A \<and> P (f x)}"
by (rule Compr_image_eq[Transfer.transferred])
lemma fimage_funion: "f |`| (A |\<union>| B) = f |`| A |\<union>| f |`| B"
by (rule image_Un[Transfer.transferred])
lemma fimage_iff: "(z |\<in>| f |`| A) = fBex A (\<lambda>x. z = f x)"
by (rule image_iff[Transfer.transferred])
lemma fimage_fsubset_iff[no_atp]: "(f |`| A |\<subseteq>| B) = fBall A (\<lambda>x. f x |\<in>| B)"
by (rule image_subset_iff[Transfer.transferred])
lemma fimage_fsubsetI: "(\<And>x. x |\<in>| A \<Longrightarrow> f x |\<in>| B) \<Longrightarrow> f |`| A |\<subseteq>| B"
by (rule image_subsetI[Transfer.transferred])
lemma fimage_ident[simp]: "(\<lambda>x. x) |`| Y = Y"
by (rule image_ident[Transfer.transferred])
lemma if_split_fmem1: "((if Q then x else y) |\<in>| b) = ((Q \<longrightarrow> x |\<in>| b) \<and> (\<not> Q \<longrightarrow> y |\<in>| b))"
by (rule if_split_mem1[Transfer.transferred])
lemma if_split_fmem2: "(a |\<in>| (if Q then x else y)) = ((Q \<longrightarrow> a |\<in>| x) \<and> (\<not> Q \<longrightarrow> a |\<in>| y))"
by (rule if_split_mem2[Transfer.transferred])
lemma pfsubsetI[intro!,no_atp]: "A |\<subseteq>| B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A |\<subset>| B"
by (rule psubsetI[Transfer.transferred])
lemma pfsubsetE[elim!,no_atp]: "A |\<subset>| B \<Longrightarrow> (A |\<subseteq>| B \<Longrightarrow> \<not> B |\<subseteq>| A \<Longrightarrow> R) \<Longrightarrow> R"
by (rule psubsetE[Transfer.transferred])
lemma pfsubset_finsert_iff:
"(A |\<subset>| finsert x B) =
(if x |\<in>| B then A |\<subset>| B else if x |\<in>| A then A |-| {|x|} |\<subset>| B else A |\<subseteq>| B)"
by (rule psubset_insert_iff[Transfer.transferred])
lemma pfsubset_eq: "(A |\<subset>| B) = (A |\<subseteq>| B \<and> A \<noteq> B)"
by (rule psubset_eq[Transfer.transferred])
lemma pfsubset_imp_fsubset: "A |\<subset>| B \<Longrightarrow> A |\<subseteq>| B"
by (rule psubset_imp_subset[Transfer.transferred])
lemma pfsubset_trans: "A |\<subset>| B \<Longrightarrow> B |\<subset>| C \<Longrightarrow> A |\<subset>| C"
by (rule psubset_trans[Transfer.transferred])
lemma pfsubsetD: "A |\<subset>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B"
by (rule psubsetD[Transfer.transferred])
lemma pfsubset_fsubset_trans: "A |\<subset>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<subset>| C"
by (rule psubset_subset_trans[Transfer.transferred])
lemma fsubset_pfsubset_trans: "A |\<subseteq>| B \<Longrightarrow> B |\<subset>| C \<Longrightarrow> A |\<subset>| C"
by (rule subset_psubset_trans[Transfer.transferred])
lemma pfsubset_imp_ex_fmem: "A |\<subset>| B \<Longrightarrow> \<exists>b. b |\<in>| B |-| A"
by (rule psubset_imp_ex_mem[Transfer.transferred])
lemma fimage_fPow_mono: "f |`| A |\<subseteq>| B \<Longrightarrow> (|`|) f |`| fPow A |\<subseteq>| fPow B"
by (rule image_Pow_mono[Transfer.transferred])
lemma fimage_fPow_surj: "f |`| A = B \<Longrightarrow> (|`|) f |`| fPow A = fPow B"
by (rule image_Pow_surj[Transfer.transferred])
lemma fsubset_finsertI: "B |\<subseteq>| finsert a B"
by (rule subset_insertI[Transfer.transferred])
lemma fsubset_finsertI2: "A |\<subseteq>| B \<Longrightarrow> A |\<subseteq>| finsert b B"
by (rule subset_insertI2[Transfer.transferred])
lemma fsubset_finsert: "x |\<notin>| A \<Longrightarrow> (A |\<subseteq>| finsert x B) = (A |\<subseteq>| B)"
by (rule subset_insert[Transfer.transferred])
lemma funion_upper1: "A |\<subseteq>| A |\<union>| B"
by (rule Un_upper1[Transfer.transferred])
lemma funion_upper2: "B |\<subseteq>| A |\<union>| B"
by (rule Un_upper2[Transfer.transferred])
lemma funion_least: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<union>| B |\<subseteq>| C"
by (rule Un_least[Transfer.transferred])
lemma finter_lower1: "A |\<inter>| B |\<subseteq>| A"
by (rule Int_lower1[Transfer.transferred])
lemma finter_lower2: "A |\<inter>| B |\<subseteq>| B"
by (rule Int_lower2[Transfer.transferred])
lemma finter_greatest: "C |\<subseteq>| A \<Longrightarrow> C |\<subseteq>| B \<Longrightarrow> C |\<subseteq>| A |\<inter>| B"
by (rule Int_greatest[Transfer.transferred])
lemma fminus_fsubset: "A |-| B |\<subseteq>| A"
by (rule Diff_subset[Transfer.transferred])
lemma fminus_fsubset_conv: "(A |-| B |\<subseteq>| C) = (A |\<subseteq>| B |\<union>| C)"
by (rule Diff_subset_conv[Transfer.transferred])
lemma fsubset_fempty[simp]: "(A |\<subseteq>| {||}) = (A = {||})"
by (rule subset_empty[Transfer.transferred])
lemma not_pfsubset_fempty[iff]: "\<not> A |\<subset>| {||}"
by (rule not_psubset_empty[Transfer.transferred])
lemma finsert_is_funion: "finsert a A = {|a|} |\<union>| A"
by (rule insert_is_Un[Transfer.transferred])
lemma finsert_not_fempty[simp]: "finsert a A \<noteq> {||}"
by (rule insert_not_empty[Transfer.transferred])
lemma fempty_not_finsert: "{||} \<noteq> finsert a A"
by (rule empty_not_insert[Transfer.transferred])
lemma finsert_absorb: "a |\<in>| A \<Longrightarrow> finsert a A = A"
by (rule insert_absorb[Transfer.transferred])
lemma finsert_absorb2[simp]: "finsert x (finsert x A) = finsert x A"
by (rule insert_absorb2[Transfer.transferred])
lemma finsert_commute: "finsert x (finsert y A) = finsert y (finsert x A)"
by (rule insert_commute[Transfer.transferred])
lemma finsert_fsubset[simp]: "(finsert x A |\<subseteq>| B) = (x |\<in>| B \<and> A |\<subseteq>| B)"
by (rule insert_subset[Transfer.transferred])
lemma finsert_inter_finsert[simp]: "finsert a A |\<inter>| finsert a B = finsert a (A |\<inter>| B)"
by (rule insert_inter_insert[Transfer.transferred])
lemma finsert_disjoint[simp,no_atp]:
"(finsert a A |\<inter>| B = {||}) = (a |\<notin>| B \<and> A |\<inter>| B = {||})"
"({||} = finsert a A |\<inter>| B) = (a |\<notin>| B \<and> {||} = A |\<inter>| B)"
by (rule insert_disjoint[Transfer.transferred])+
lemma disjoint_finsert[simp,no_atp]:
"(B |\<inter>| finsert a A = {||}) = (a |\<notin>| B \<and> B |\<inter>| A = {||})"
"({||} = A |\<inter>| finsert b B) = (b |\<notin>| A \<and> {||} = A |\<inter>| B)"
by (rule disjoint_insert[Transfer.transferred])+
lemma fimage_fempty[simp]: "f |`| {||} = {||}"
by (rule image_empty[Transfer.transferred])
lemma fimage_finsert[simp]: "f |`| finsert a B = finsert (f a) (f |`| B)"
by (rule image_insert[Transfer.transferred])
lemma fimage_constant: "x |\<in>| A \<Longrightarrow> (\<lambda>x. c) |`| A = {|c|}"
by (rule image_constant[Transfer.transferred])
lemma fimage_constant_conv: "(\<lambda>x. c) |`| A = (if A = {||} then {||} else {|c|})"
by (rule image_constant_conv[Transfer.transferred])
lemma fimage_fimage: "f |`| g |`| A = (\<lambda>x. f (g x)) |`| A"
by (rule image_image[Transfer.transferred])
lemma finsert_fimage[simp]: "x |\<in>| A \<Longrightarrow> finsert (f x) (f |`| A) = f |`| A"
by (rule insert_image[Transfer.transferred])
lemma fimage_is_fempty[iff]: "(f |`| A = {||}) = (A = {||})"
by (rule image_is_empty[Transfer.transferred])
lemma fempty_is_fimage[iff]: "({||} = f |`| A) = (A = {||})"
by (rule empty_is_image[Transfer.transferred])
lemma fimage_cong: "M = N \<Longrightarrow> (\<And>x. x |\<in>| N \<Longrightarrow> f x = g x) \<Longrightarrow> f |`| M = g |`| N"
by (rule image_cong[Transfer.transferred])
lemma fimage_finter_fsubset: "f |`| (A |\<inter>| B) |\<subseteq>| f |`| A |\<inter>| f |`| B"
by (rule image_Int_subset[Transfer.transferred])
lemma fimage_fminus_fsubset: "f |`| A |-| f |`| B |\<subseteq>| f |`| (A |-| B)"
by (rule image_diff_subset[Transfer.transferred])
lemma finter_absorb: "A |\<inter>| A = A"
by (rule Int_absorb[Transfer.transferred])
lemma finter_left_absorb: "A |\<inter>| (A |\<inter>| B) = A |\<inter>| B"
by (rule Int_left_absorb[Transfer.transferred])
lemma finter_commute: "A |\<inter>| B = B |\<inter>| A"
by (rule Int_commute[Transfer.transferred])
lemma finter_left_commute: "A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)"
by (rule Int_left_commute[Transfer.transferred])
lemma finter_assoc: "A |\<inter>| B |\<inter>| C = A |\<inter>| (B |\<inter>| C)"
by (rule Int_assoc[Transfer.transferred])
lemma finter_ac:
"A |\<inter>| B |\<inter>| C = A |\<inter>| (B |\<inter>| C)"
"A |\<inter>| (A |\<inter>| B) = A |\<inter>| B"
"A |\<inter>| B = B |\<inter>| A"
"A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)"
by (rule Int_ac[Transfer.transferred])+
lemma finter_absorb1: "B |\<subseteq>| A \<Longrightarrow> A |\<inter>| B = B"
by (rule Int_absorb1[Transfer.transferred])
lemma finter_absorb2: "A |\<subseteq>| B \<Longrightarrow> A |\<inter>| B = A"
by (rule Int_absorb2[Transfer.transferred])
lemma finter_fempty_left: "{||} |\<inter>| B = {||}"
by (rule Int_empty_left[Transfer.transferred])
lemma finter_fempty_right: "A |\<inter>| {||} = {||}"
by (rule Int_empty_right[Transfer.transferred])
lemma disjoint_iff_fnot_equal: "(A |\<inter>| B = {||}) = fBall A (\<lambda>x. fBall B ((\<noteq>) x))"
by (rule disjoint_iff_not_equal[Transfer.transferred])
lemma finter_funion_distrib: "A |\<inter>| (B |\<union>| C) = A |\<inter>| B |\<union>| (A |\<inter>| C)"
by (rule Int_Un_distrib[Transfer.transferred])
lemma finter_funion_distrib2: "B |\<union>| C |\<inter>| A = B |\<inter>| A |\<union>| (C |\<inter>| A)"
by (rule Int_Un_distrib2[Transfer.transferred])
lemma finter_fsubset_iff[no_atp, simp]: "(C |\<subseteq>| A |\<inter>| B) = (C |\<subseteq>| A \<and> C |\<subseteq>| B)"
by (rule Int_subset_iff[Transfer.transferred])
lemma funion_absorb: "A |\<union>| A = A"
by (rule Un_absorb[Transfer.transferred])
lemma funion_left_absorb: "A |\<union>| (A |\<union>| B) = A |\<union>| B"
by (rule Un_left_absorb[Transfer.transferred])
lemma funion_commute: "A |\<union>| B = B |\<union>| A"
by (rule Un_commute[Transfer.transferred])
lemma funion_left_commute: "A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)"
by (rule Un_left_commute[Transfer.transferred])
lemma funion_assoc: "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
by (rule Un_assoc[Transfer.transferred])
lemma funion_ac:
"A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)"
"A |\<union>| (A |\<union>| B) = A |\<union>| B"
"A |\<union>| B = B |\<union>| A"
"A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)"
by (rule Un_ac[Transfer.transferred])+
lemma funion_absorb1: "A |\<subseteq>| B \<Longrightarrow> A |\<union>| B = B"
by (rule Un_absorb1[Transfer.transferred])
lemma funion_absorb2: "B |\<subseteq>| A \<Longrightarrow> A |\<union>| B = A"
by (rule Un_absorb2[Transfer.transferred])
lemma funion_fempty_left: "{||} |\<union>| B = B"
by (rule Un_empty_left[Transfer.transferred])
lemma funion_fempty_right: "A |\<union>| {||} = A"
by (rule Un_empty_right[Transfer.transferred])
lemma funion_finsert_left[simp]: "finsert a B |\<union>| C = finsert a (B |\<union>| C)"
by (rule Un_insert_left[Transfer.transferred])
lemma funion_finsert_right[simp]: "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
by (rule Un_insert_right[Transfer.transferred])
lemma finter_finsert_left: "finsert a B |\<inter>| C = (if a |\<in>| C then finsert a (B |\<inter>| C) else B |\<inter>| C)"
by (rule Int_insert_left[Transfer.transferred])
lemma finter_finsert_left_ifffempty[simp]: "a |\<notin>| C \<Longrightarrow> finsert a B |\<inter>| C = B |\<inter>| C"
by (rule Int_insert_left_if0[Transfer.transferred])
lemma finter_finsert_left_if1[simp]: "a |\<in>| C \<Longrightarrow> finsert a B |\<inter>| C = finsert a (B |\<inter>| C)"
by (rule Int_insert_left_if1[Transfer.transferred])
lemma finter_finsert_right:
"A |\<inter>| finsert a B = (if a |\<in>| A then finsert a (A |\<inter>| B) else A |\<inter>| B)"
by (rule Int_insert_right[Transfer.transferred])
lemma finter_finsert_right_ifffempty[simp]: "a |\<notin>| A \<Longrightarrow> A |\<inter>| finsert a B = A |\<inter>| B"
by (rule Int_insert_right_if0[Transfer.transferred])
lemma finter_finsert_right_if1[simp]: "a |\<in>| A \<Longrightarrow> A |\<inter>| finsert a B = finsert a (A |\<inter>| B)"
by (rule Int_insert_right_if1[Transfer.transferred])
lemma funion_finter_distrib: "A |\<union>| (B |\<inter>| C) = A |\<union>| B |\<inter>| (A |\<union>| C)"
by (rule Un_Int_distrib[Transfer.transferred])
lemma funion_finter_distrib2: "B |\<inter>| C |\<union>| A = B |\<union>| A |\<inter>| (C |\<union>| A)"
by (rule Un_Int_distrib2[Transfer.transferred])
lemma funion_finter_crazy:
"A |\<inter>| B |\<union>| (B |\<inter>| C) |\<union>| (C |\<inter>| A) = A |\<union>| B |\<inter>| (B |\<union>| C) |\<inter>| (C |\<union>| A)"
by (rule Un_Int_crazy[Transfer.transferred])
lemma fsubset_funion_eq: "(A |\<subseteq>| B) = (A |\<union>| B = B)"
by (rule subset_Un_eq[Transfer.transferred])
lemma funion_fempty[iff]: "(A |\<union>| B = {||}) = (A = {||} \<and> B = {||})"
by (rule Un_empty[Transfer.transferred])
lemma funion_fsubset_iff[no_atp, simp]: "(A |\<union>| B |\<subseteq>| C) = (A |\<subseteq>| C \<and> B |\<subseteq>| C)"
by (rule Un_subset_iff[Transfer.transferred])
lemma funion_fminus_finter: "A |-| B |\<union>| (A |\<inter>| B) = A"
by (rule Un_Diff_Int[Transfer.transferred])
lemma ffunion_empty[simp]: "ffUnion {||} = {||}"
by (rule Union_empty[Transfer.transferred])
lemma ffunion_mono: "A |\<subseteq>| B \<Longrightarrow> ffUnion A |\<subseteq>| ffUnion B"
by (rule Union_mono[Transfer.transferred])
lemma ffunion_insert[simp]: "ffUnion (finsert a B) = a |\<union>| ffUnion B"
by (rule Union_insert[Transfer.transferred])
lemma fminus_finter2: "A |\<inter>| C |-| (B |\<inter>| C) = A |\<inter>| C |-| B"
by (rule Diff_Int2[Transfer.transferred])
lemma funion_finter_assoc_eq: "(A |\<inter>| B |\<union>| C = A |\<inter>| (B |\<union>| C)) = (C |\<subseteq>| A)"
by (rule Un_Int_assoc_eq[Transfer.transferred])
lemma fBall_funion: "fBall (A |\<union>| B) P = (fBall A P \<and> fBall B P)"
by (rule ball_Un[Transfer.transferred])
lemma fBex_funion: "fBex (A |\<union>| B) P = (fBex A P \<or> fBex B P)"
by (rule bex_Un[Transfer.transferred])
lemma fminus_eq_fempty_iff[simp,no_atp]: "(A |-| B = {||}) = (A |\<subseteq>| B)"
by (rule Diff_eq_empty_iff[Transfer.transferred])
lemma fminus_cancel[simp]: "A |-| A = {||}"
by (rule Diff_cancel[Transfer.transferred])
lemma fminus_idemp[simp]: "A |-| B |-| B = A |-| B"
by (rule Diff_idemp[Transfer.transferred])
lemma fminus_triv: "A |\<inter>| B = {||} \<Longrightarrow> A |-| B = A"
by (rule Diff_triv[Transfer.transferred])
lemma fempty_fminus[simp]: "{||} |-| A = {||}"
by (rule empty_Diff[Transfer.transferred])
lemma fminus_fempty[simp]: "A |-| {||} = A"
by (rule Diff_empty[Transfer.transferred])
lemma fminus_finsertffempty[simp,no_atp]: "x |\<notin>| A \<Longrightarrow> A |-| finsert x B = A |-| B"
by (rule Diff_insert0[Transfer.transferred])
lemma fminus_finsert: "A |-| finsert a B = A |-| B |-| {|a|}"
by (rule Diff_insert[Transfer.transferred])
lemma fminus_finsert2: "A |-| finsert a B = A |-| {|a|} |-| B"
by (rule Diff_insert2[Transfer.transferred])
lemma finsert_fminus_if: "finsert x A |-| B = (if x |\<in>| B then A |-| B else finsert x (A |-| B))"
by (rule insert_Diff_if[Transfer.transferred])
lemma finsert_fminus1[simp]: "x |\<in>| B \<Longrightarrow> finsert x A |-| B = A |-| B"
by (rule insert_Diff1[Transfer.transferred])
lemma finsert_fminus_single[simp]: "finsert a (A |-| {|a|}) = finsert a A"
by (rule insert_Diff_single[Transfer.transferred])
lemma finsert_fminus: "a |\<in>| A \<Longrightarrow> finsert a (A |-| {|a|}) = A"
by (rule insert_Diff[Transfer.transferred])
lemma fminus_finsert_absorb: "x |\<notin>| A \<Longrightarrow> finsert x A |-| {|x|} = A"
by (rule Diff_insert_absorb[Transfer.transferred])
lemma fminus_disjoint[simp]: "A |\<inter>| (B |-| A) = {||}"
by (rule Diff_disjoint[Transfer.transferred])
lemma fminus_partition: "A |\<subseteq>| B \<Longrightarrow> A |\<union>| (B |-| A) = B"
by (rule Diff_partition[Transfer.transferred])
lemma double_fminus: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> B |-| (C |-| A) = A"
by (rule double_diff[Transfer.transferred])
lemma funion_fminus_cancel[simp]: "A |\<union>| (B |-| A) = A |\<union>| B"
by (rule Un_Diff_cancel[Transfer.transferred])
lemma funion_fminus_cancel2[simp]: "B |-| A |\<union>| A = B |\<union>| A"
by (rule Un_Diff_cancel2[Transfer.transferred])
lemma fminus_funion: "A |-| (B |\<union>| C) = A |-| B |\<inter>| (A |-| C)"
by (rule Diff_Un[Transfer.transferred])
lemma fminus_finter: "A |-| (B |\<inter>| C) = A |-| B |\<union>| (A |-| C)"
by (rule Diff_Int[Transfer.transferred])
lemma funion_fminus: "A |\<union>| B |-| C = A |-| C |\<union>| (B |-| C)"
by (rule Un_Diff[Transfer.transferred])
lemma finter_fminus: "A |\<inter>| B |-| C = A |\<inter>| (B |-| C)"
by (rule Int_Diff[Transfer.transferred])
lemma fminus_finter_distrib: "C |\<inter>| (A |-| B) = C |\<inter>| A |-| (C |\<inter>| B)"
by (rule Diff_Int_distrib[Transfer.transferred])
lemma fminus_finter_distrib2: "A |-| B |\<inter>| C = A |\<inter>| C |-| (B |\<inter>| C)"
by (rule Diff_Int_distrib2[Transfer.transferred])
lemma fUNIV_bool[no_atp]: "fUNIV = {|False, True|}"
by (rule UNIV_bool[Transfer.transferred])
lemma fPow_fempty[simp]: "fPow {||} = {|{||}|}"
by (rule Pow_empty[Transfer.transferred])
lemma fPow_finsert: "fPow (finsert a A) = fPow A |\<union>| finsert a |`| fPow A"
by (rule Pow_insert[Transfer.transferred])
lemma funion_fPow_fsubset: "fPow A |\<union>| fPow B |\<subseteq>| fPow (A |\<union>| B)"
by (rule Un_Pow_subset[Transfer.transferred])
lemma fPow_finter_eq[simp]: "fPow (A |\<inter>| B) = fPow A |\<inter>| fPow B"
by (rule Pow_Int_eq[Transfer.transferred])
lemma fset_eq_fsubset: "(A = B) = (A |\<subseteq>| B \<and> B |\<subseteq>| A)"
by (rule set_eq_subset[Transfer.transferred])
lemma fsubset_iff[no_atp]: "(A |\<subseteq>| B) = (\<forall>t. t |\<in>| A \<longrightarrow> t |\<in>| B)"
by (rule subset_iff[Transfer.transferred])
lemma fsubset_iff_pfsubset_eq: "(A |\<subseteq>| B) = (A |\<subset>| B \<or> A = B)"
by (rule subset_iff_psubset_eq[Transfer.transferred])
lemma all_not_fin_conv[simp]: "(\<forall>x. x |\<notin>| A) = (A = {||})"
by (rule all_not_in_conv[Transfer.transferred])
lemma ex_fin_conv: "(\<exists>x. x |\<in>| A) = (A \<noteq> {||})"
by (rule ex_in_conv[Transfer.transferred])
lemma fimage_mono: "A |\<subseteq>| B \<Longrightarrow> f |`| A |\<subseteq>| f |`| B"
by (rule image_mono[Transfer.transferred])
lemma fPow_mono: "A |\<subseteq>| B \<Longrightarrow> fPow A |\<subseteq>| fPow B"
by (rule Pow_mono[Transfer.transferred])
lemma finsert_mono: "C |\<subseteq>| D \<Longrightarrow> finsert a C |\<subseteq>| finsert a D"
by (rule insert_mono[Transfer.transferred])
lemma funion_mono: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| D \<Longrightarrow> A |\<union>| B |\<subseteq>| C |\<union>| D"
by (rule Un_mono[Transfer.transferred])
lemma finter_mono: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| D \<Longrightarrow> A |\<inter>| B |\<subseteq>| C |\<inter>| D"
by (rule Int_mono[Transfer.transferred])
lemma fminus_mono: "A |\<subseteq>| C \<Longrightarrow> D |\<subseteq>| B \<Longrightarrow> A |-| B |\<subseteq>| C |-| D"
by (rule Diff_mono[Transfer.transferred])
lemma fin_mono: "A |\<subseteq>| B \<Longrightarrow> x |\<in>| A \<longrightarrow> x |\<in>| B"
by (rule in_mono[Transfer.transferred])
lemma fthe_felem_eq[simp]: "fthe_elem {|x|} = x"
by (rule the_elem_eq[Transfer.transferred])
lemma fLeast_mono:
"mono f \<Longrightarrow> fBex S (\<lambda>x. fBall S ((\<le>) x)) \<Longrightarrow> (LEAST y. y |\<in>| f |`| S) = f (LEAST x. x |\<in>| S)"
by (rule Least_mono[Transfer.transferred])
lemma fbind_fbind: "fbind (fbind A B) C = fbind A (\<lambda>x. fbind (B x) C)"
by (rule Set.bind_bind[Transfer.transferred])
lemma fempty_fbind[simp]: "fbind {||} f = {||}"
by (rule empty_bind[Transfer.transferred])
lemma nonfempty_fbind_const: "A \<noteq> {||} \<Longrightarrow> fbind A (\<lambda>_. B) = B"
by (rule nonempty_bind_const[Transfer.transferred])
lemma fbind_const: "fbind A (\<lambda>_. B) = (if A = {||} then {||} else B)"
by (rule bind_const[Transfer.transferred])
lemma ffmember_filter[simp]: "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
by (rule member_filter[Transfer.transferred])
lemma fequalityI: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> A = B"
by (rule equalityI[Transfer.transferred])
lemma fset_of_list_simps[simp]:
"fset_of_list [] = {||}"
"fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)"
by (rule set_simps[Transfer.transferred])+
lemma fset_of_list_append[simp]: "fset_of_list (xs @ ys) = fset_of_list xs |\<union>| fset_of_list ys"
by (rule set_append[Transfer.transferred])
lemma fset_of_list_rev[simp]: "fset_of_list (rev xs) = fset_of_list xs"
by (rule set_rev[Transfer.transferred])
lemma fset_of_list_map[simp]: "fset_of_list (map f xs) = f |`| fset_of_list xs"
by (rule set_map[Transfer.transferred])
subsection \<open>Additional lemmas\<close>
subsubsection \<open>\<open>ffUnion\<close>\<close>
lemma ffUnion_funion_distrib[simp]: "ffUnion (A |\<union>| B) = ffUnion A |\<union>| ffUnion B"
by (rule Union_Un_distrib[Transfer.transferred])
subsubsection \<open>\<open>fbind\<close>\<close>
lemma fbind_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> f x = g x) \<Longrightarrow> fbind A f = fbind B g"
by transfer force
subsubsection \<open>\<open>fsingleton\<close>\<close>
lemma fsingletonE: " b |\<in>| {|a|} \<Longrightarrow> (b = a \<Longrightarrow> thesis) \<Longrightarrow> thesis"
by (rule fsingletonD [elim_format])
subsubsection \<open>\<open>femepty\<close>\<close>
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}"
by transfer auto
(* FIXME, transferred doesn't work here *)
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P"
by simp
subsubsection \<open>\<open>fset\<close>\<close>
lemma fset_simps[simp]:
"fset {||} = {}"
"fset (finsert x X) = insert x (fset X)"
by (rule bot_fset.rep_eq finsert.rep_eq)+
lemma finite_fset [simp]:
shows "finite (fset S)"
by transfer simp
lemmas fset_cong = fset_inject
lemma filter_fset [simp]:
shows "fset (ffilter P xs) = Collect P \<inter> fset xs"
by transfer auto
lemma inter_fset[simp]: "fset (A |\<inter>| B) = fset A \<inter> fset B"
by (rule inf_fset.rep_eq)
lemma union_fset[simp]: "fset (A |\<union>| B) = fset A \<union> fset B"
by (rule sup_fset.rep_eq)
lemma minus_fset[simp]: "fset (A |-| B) = fset A - fset B"
by (rule minus_fset.rep_eq)
subsubsection \<open>\<open>ffilter\<close>\<close>
lemma subset_ffilter:
"ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
by transfer auto
lemma eq_ffilter:
"(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
by transfer auto
lemma pfsubset_ffilter:
"(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A \<and> \<not> P x \<and> Q x) \<Longrightarrow>
ffilter P A |\<subset>| ffilter Q A"
unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter)
subsubsection \<open>\<open>fset_of_list\<close>\<close>
lemma fset_of_list_filter[simp]:
"fset_of_list (filter P xs) = ffilter P (fset_of_list xs)"
by transfer (auto simp: Set.filter_def)
lemma fset_of_list_subset[intro]:
"set xs \<subseteq> set ys \<Longrightarrow> fset_of_list xs |\<subseteq>| fset_of_list ys"
by transfer simp
lemma fset_of_list_elem: "(x |\<in>| fset_of_list xs) \<longleftrightarrow> (x \<in> set xs)"
by transfer simp
subsubsection \<open>\<open>finsert\<close>\<close>
(* FIXME, transferred doesn't work here *)
lemma set_finsert:
assumes "x |\<in>| A"
obtains B where "A = finsert x B" and "x |\<notin>| B"
using assms by transfer (metis Set.set_insert finite_insert)
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B"
by (rule exI [where x = "A |-| {|a|}"]) blast
lemma finsert_eq_iff:
assumes "a |\<notin>| A" and "b |\<notin>| B"
shows "(finsert a A = finsert b B) =
(if a = b then A = B else \<exists>C. A = finsert b C \<and> b |\<notin>| C \<and> B = finsert a C \<and> a |\<notin>| C)"
using assms by transfer (force simp: insert_eq_iff)
subsubsection \<open>\<open>fimage\<close>\<close>
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)"
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff)
lemma fimage_strict_mono:
assumes "inj_on f (fset B)" and "A |\<subset>| B"
shows "f |`| A |\<subset>| f |`| B"
\<comment> \<open>TODO: Configure transfer framework to lift @{thm Fun.image_strict_mono}.\<close>
proof (rule pfsubsetI)
from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B"
by (rule pfsubset_imp_fsubset)
thus "f |`| A |\<subseteq>| f |`| B"
by (rule fimage_mono)
next
from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B" and "A \<noteq> B"
by (simp_all add: pfsubset_eq)
have "fset A \<noteq> fset B"
using \<open>A \<noteq> B\<close>
by (simp add: fset_cong)
hence "f ` fset A \<noteq> f ` fset B"
using \<open>A |\<subseteq>| B\<close>
by (simp add: inj_on_image_eq_iff[OF \<open>inj_on f (fset B)\<close>] less_eq_fset.rep_eq)
hence "fset (f |`| A) \<noteq> fset (f |`| B)"
by (simp add: fimage.rep_eq)
thus "f |`| A \<noteq> f |`| B"
by (simp add: fset_cong)
qed
subsubsection \<open>bounded quantification\<close>
lemma bex_simps [simp, no_atp]:
"\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)"
"\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)"
"\<And>P. fBex {||} P = False"
"\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)"
"\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))"
"\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)"
by auto
lemma ball_simps [simp, no_atp]:
"\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)"
"\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)"
"\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)"
"\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)"
"\<And>P. fBall {||} P = True"
"\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)"
"\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))"
"\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)"
by auto
lemma atomize_fBall:
"(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))"
apply (simp only: atomize_all atomize_imp)
apply (rule equal_intr_rule)
by (transfer, simp)+
lemma fBall_mono[mono]: "P \<le> Q \<Longrightarrow> fBall S P \<le> fBall S Q"
by auto
lemma fBex_mono[mono]: "P \<le> Q \<Longrightarrow> fBex S P \<le> fBex S Q"
by auto
end
subsubsection \<open>\<open>fcard\<close>\<close>
(* FIXME: improve transferred to handle bounded meta quantification *)
lemma fcard_fempty:
"fcard {||} = 0"
by transfer (rule card.empty)
lemma fcard_finsert_disjoint:
"x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)"
by transfer (rule card_insert_disjoint)
lemma fcard_finsert_if:
"fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))"
by transfer (rule card_insert_if)
lemma fcard_0_eq [simp, no_atp]:
"fcard A = 0 \<longleftrightarrow> A = {||}"
by transfer (rule card_0_eq)
lemma fcard_Suc_fminus1:
"x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A"
by transfer (rule card_Suc_Diff1)
lemma fcard_fminus_fsingleton:
"x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1"
by transfer (rule card_Diff_singleton)
lemma fcard_fminus_fsingleton_if:
"fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)"
by transfer (rule card_Diff_singleton_if)
lemma fcard_fminus_finsert[simp]:
assumes "a |\<in>| A" and "a |\<notin>| B"
shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1"
using assms by transfer (rule card_Diff_insert)
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))"
by transfer (rule card.insert_remove)
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)"
by transfer (rule card_insert_le)
lemma fcard_mono:
"A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B"
by transfer (rule card_mono)
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B"
by transfer (rule card_seteq)
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B"
by transfer (rule psubset_card_mono)
lemma fcard_funion_finter:
"fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)"
by transfer (rule card_Un_Int)
lemma fcard_funion_disjoint:
"A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B"
by transfer (rule card_Un_disjoint)
lemma fcard_funion_fsubset:
"B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B"
by transfer (rule card_Diff_subset)
lemma diff_fcard_le_fcard_fminus:
"fcard A - fcard B \<le> fcard(A |-| B)"
by transfer (rule diff_card_le_card_Diff)
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A"
by transfer (rule card_Diff1_less)
lemma fcard_fminus2_less:
"x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A"
by transfer (rule card_Diff2_less)
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A"
by transfer (rule card_Diff1_le)
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B"
by transfer (rule card_psubset)
subsubsection \<open>\<open>sorted_list_of_fset\<close>\<close>
lemma sorted_list_of_fset_simps[simp]:
"set (sorted_list_of_fset S) = fset S"
"fset_of_list (sorted_list_of_fset S) = S"
by (transfer, simp)+
subsubsection \<open>\<open>ffold\<close>\<close>
(* FIXME: improve transferred to handle bounded meta quantification *)
context comp_fun_commute
begin
lemma ffold_empty[simp]: "ffold f z {||} = z"
by (rule fold_empty[Transfer.transferred])
lemma ffold_finsert [simp]:
assumes "x |\<notin>| A"
shows "ffold f z (finsert x A) = f x (ffold f z A)"
using assms by (transfer fixing: f) (rule fold_insert)
lemma ffold_fun_left_comm:
"f x (ffold f z A) = ffold f (f x z) A"
by (transfer fixing: f) (rule fold_fun_left_comm)
lemma ffold_finsert2:
"x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A"
by (transfer fixing: f) (rule fold_insert2)
lemma ffold_rec:
assumes "x |\<in>| A"
shows "ffold f z A = f x (ffold f z (A |-| {|x|}))"
using assms by (transfer fixing: f) (rule fold_rec)
lemma ffold_finsert_fremove:
"ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
by (transfer fixing: f) (rule fold_insert_remove)
end
lemma ffold_fimage:
assumes "inj_on g (fset A)"
shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A"
using assms by transfer' (rule fold_image)
lemma ffold_cong:
assumes "comp_fun_commute f" "comp_fun_commute g"
"\<And>x. x |\<in>| A \<Longrightarrow> f x = g x"
and "s = t" and "A = B"
shows "ffold f s A = ffold g t B"
using assms[unfolded comp_fun_commute_def']
by transfer (meson Finite_Set.fold_cong subset_UNIV)
context comp_fun_idem
begin
lemma ffold_finsert_idem:
"ffold f z (finsert x A) = f x (ffold f z A)"
by (transfer fixing: f) (rule fold_insert_idem)
declare ffold_finsert [simp del] ffold_finsert_idem [simp]
lemma ffold_finsert_idem2:
"ffold f z (finsert x A) = ffold f (f x z) A"
by (transfer fixing: f) (rule fold_insert_idem2)
end
subsubsection \<open>@{term fsubset}\<close>
lemma wfP_pfsubset: "wfP (|\<subset>|)"
proof (rule wfp_if_convertible_to_nat)
show "\<And>x y. x |\<subset>| y \<Longrightarrow> fcard x < fcard y"
by (rule pfsubset_fcard_mono)
qed
subsubsection \<open>Group operations\<close>
locale comm_monoid_fset = comm_monoid
begin
sublocale set: comm_monoid_set ..
lift_definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a" is set.F .
lemma cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> g x = h x) \<Longrightarrow> F g A = F h B"
by (rule set.cong[Transfer.transferred])
lemma cong_simp[cong]:
"\<lbrakk> A = B; \<And>x. x |\<in>| B =simp=> g x = h x \<rbrakk> \<Longrightarrow> F g A = F h B"
unfolding simp_implies_def by (auto cong: cong)
end
context comm_monoid_add begin
sublocale fsum: comm_monoid_fset plus 0
rewrites "comm_monoid_set.F plus 0 = sum"
defines fsum = fsum.F
proof -
show "comm_monoid_fset (+) 0" by standard
show "comm_monoid_set.F (+) 0 = sum" unfolding sum_def ..
qed
end
subsubsection \<open>Semilattice operations\<close>
locale semilattice_fset = semilattice
begin
sublocale set: semilattice_set ..
lift_definition F :: "'a fset \<Rightarrow> 'a" is set.F .
lemma eq_fold: "F (finsert x A) = ffold f x A"
by transfer (rule set.eq_fold)
lemma singleton [simp]: "F {|x|} = x"
by transfer (rule set.singleton)
lemma insert_not_elem: "x |\<notin>| A \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A"
by transfer (rule set.insert_not_elem)
lemma in_idem: "x |\<in>| A \<Longrightarrow> x \<^bold>* F A = F A"
by transfer (rule set.in_idem)
lemma insert [simp]: "A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A"
by transfer (rule set.insert)
end
locale semilattice_order_fset = binary?: semilattice_order + semilattice_fset
begin
end
context linorder begin
sublocale fMin: semilattice_order_fset min less_eq less
rewrites "semilattice_set.F min = Min"
defines fMin = fMin.F
proof -
show "semilattice_order_fset min (\<le>) (<)" by standard
show "semilattice_set.F min = Min" unfolding Min_def ..
qed
sublocale fMax: semilattice_order_fset max greater_eq greater
rewrites "semilattice_set.F max = Max"
defines fMax = fMax.F
proof -
show "semilattice_order_fset max (\<ge>) (>)"
by standard
show "semilattice_set.F max = Max"
unfolding Max_def ..
qed
end
lemma mono_fMax_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMax A) = fMax (f |`| A)"
by transfer (rule mono_Max_commute)
lemma mono_fMin_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMin A) = fMin (f |`| A)"
by transfer (rule mono_Min_commute)
lemma fMax_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMax A |\<in>| A"
by transfer (rule Max_in)
lemma fMin_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMin A |\<in>| A"
by transfer (rule Min_in)
lemma fMax_ge[simp]: "x |\<in>| A \<Longrightarrow> x \<le> fMax A"
by transfer (rule Max_ge)
lemma fMin_le[simp]: "x |\<in>| A \<Longrightarrow> fMin A \<le> x"
by transfer (rule Min_le)
lemma fMax_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> y \<le> x) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMax A = x"
by transfer (rule Max_eqI)
lemma fMin_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> x \<le> y) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMin A = x"
by transfer (rule Min_eqI)
lemma fMax_finsert[simp]: "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
by transfer simp
lemma fMin_finsert[simp]: "fMin (finsert x A) = (if A = {||} then x else min x (fMin A))"
by transfer simp
context linorder begin
lemma fset_linorder_max_induct[case_names fempty finsert]:
assumes "P {||}"
and "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y < x; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
shows "P S"
proof -
(* FIXME transfer and right_total vs. bi_total *)
note Domainp_forall_transfer[transfer_rule]
show ?thesis
using assms by (transfer fixing: less) (auto intro: finite_linorder_max_induct)
qed
lemma fset_linorder_min_induct[case_names fempty finsert]:
assumes "P {||}"
and "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y > x; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
shows "P S"
proof -
(* FIXME transfer and right_total vs. bi_total *)
note Domainp_forall_transfer[transfer_rule]
show ?thesis
using assms by (transfer fixing: less) (auto intro: finite_linorder_min_induct)
qed
end
subsection \<open>Choice in fsets\<close>
lemma fset_choice:
assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
using assms by transfer metis
subsection \<open>Induction and Cases rules for fsets\<close>
lemma fset_exhaust [case_names empty insert, cases type: fset]:
assumes fempty_case: "S = {||} \<Longrightarrow> P"
and finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P"
shows "P"
using assms by transfer blast
lemma fset_induct [case_names empty insert]:
assumes fempty_case: "P {||}"
and finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)"
shows "P S"
proof -
(* FIXME transfer and right_total vs. bi_total *)
note Domainp_forall_transfer[transfer_rule]
show ?thesis
using assms by transfer (auto intro: finite_induct)
qed
lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
assumes empty_fset_case: "P {||}"
and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
shows "P S"
proof -
(* FIXME transfer and right_total vs. bi_total *)
note Domainp_forall_transfer[transfer_rule]
show ?thesis
using assms by transfer (auto intro: finite_induct)
qed
lemma fset_card_induct:
assumes empty_fset_case: "P {||}"
and card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T"
shows "P S"
proof (induct S)
case empty
show "P {||}" by (rule empty_fset_case)
next
case (insert x S)
have h: "P S" by fact
have "x |\<notin>| S" by fact
then have "Suc (fcard S) = fcard (finsert x S)"
by transfer auto
then show "P (finsert x S)"
using h card_fset_Suc_case by simp
qed
lemma fset_strong_cases:
obtains "xs = {||}"
| ys x where "x |\<notin>| ys" and "xs = finsert x ys"
by transfer blast
lemma fset_induct2:
"P {||} {||} \<Longrightarrow>
(\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow>
(\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow>
(\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow>
P xsa ysa"
apply (induct xsa arbitrary: ysa)
apply (induct_tac x rule: fset_induct_stronger)
apply simp_all
apply (induct_tac xa rule: fset_induct_stronger)
apply simp_all
done
subsection \<open>Lemmas depending on induction\<close>
lemma ffUnion_fsubset_iff: "ffUnion A |\<subseteq>| B \<longleftrightarrow> fBall A (\<lambda>x. x |\<subseteq>| B)"
by (induction A) simp_all
subsection \<open>Setup for Lifting/Transfer\<close>
subsubsection \<open>Relator and predicator properties\<close>
lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set
parametric rel_set_transfer .
lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y)
\<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))"
apply (rule ext)+
apply transfer'
apply (subst rel_set_def[unfolded fun_eq_iff])
by blast
lemma finite_rel_set:
assumes fin: "finite X" "finite Z"
assumes R_S: "rel_set (R OO S) X Z"
shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z"
proof -
obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)"
apply atomize_elim
apply (subst bchoice_iff[symmetric])
using R_S[unfolded rel_set_def OO_def] by blast
obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
apply atomize_elim
apply (subst bchoice_iff[symmetric])
using R_S[unfolded rel_set_def OO_def] by blast
let ?Y = "f ` X \<union> g ` Z"
have "finite ?Y" by (simp add: fin)
moreover have "rel_set R X ?Y"
unfolding rel_set_def
using f g by clarsimp blast
moreover have "rel_set S ?Y Z"
unfolding rel_set_def
using f g by clarsimp blast
ultimately show ?thesis by metis
qed
subsubsection \<open>Transfer rules for the Transfer package\<close>
text \<open>Unconditional transfer rules\<close>
context includes lifting_syntax
begin
lemma fempty_transfer [transfer_rule]:
"rel_fset A {||} {||}"
by (rule empty_transfer[Transfer.transferred])
lemma finsert_transfer [transfer_rule]:
"(A ===> rel_fset A ===> rel_fset A) finsert finsert"
unfolding rel_fun_def rel_fset_alt_def by blast
lemma funion_transfer [transfer_rule]:
"(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion"
unfolding rel_fun_def rel_fset_alt_def by blast
lemma ffUnion_transfer [transfer_rule]:
"(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion"
unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast)
lemma fimage_transfer [transfer_rule]:
"((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage"
unfolding rel_fun_def rel_fset_alt_def by simp blast
lemma fBall_transfer [transfer_rule]:
"(rel_fset A ===> (A ===> (=)) ===> (=)) fBall fBall"
unfolding rel_fset_alt_def rel_fun_def by blast
lemma fBex_transfer [transfer_rule]:
"(rel_fset A ===> (A ===> (=)) ===> (=)) fBex fBex"
unfolding rel_fset_alt_def rel_fun_def by blast
(* FIXME transfer doesn't work here *)
lemma fPow_transfer [transfer_rule]:
"(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow"
unfolding rel_fun_def
using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred]
by blast
lemma rel_fset_transfer [transfer_rule]:
"((A ===> B ===> (=)) ===> rel_fset A ===> rel_fset B ===> (=))
rel_fset rel_fset"
unfolding rel_fun_def
using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B]
by simp
lemma bind_transfer [transfer_rule]:
"(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind"
unfolding rel_fun_def
using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
text \<open>Rules requiring bi-unique, bi-total or right-total relations\<close>
lemma fmember_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(A ===> rel_fset A ===> (=)) (|\<in>|) (|\<in>|)"
using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis
lemma finter_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter"
using assms unfolding rel_fun_def
using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
lemma fminus_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (|-|) (|-|)"
using assms unfolding rel_fun_def
using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
lemma fsubset_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(rel_fset A ===> rel_fset A ===> (=)) (|\<subseteq>|) (|\<subseteq>|)"
using assms unfolding rel_fun_def
using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
lemma fSup_transfer [transfer_rule]:
"bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup"
unfolding rel_fun_def
apply clarify
apply transfer'
using Sup_fset_transfer[unfolded rel_fun_def] by blast
(* FIXME: add right_total_fInf_transfer *)
lemma fInf_transfer [transfer_rule]:
assumes "bi_unique A" and "bi_total A"
shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf"
using assms unfolding rel_fun_def
apply clarify
apply transfer'
using Inf_fset_transfer[unfolded rel_fun_def] by blast
lemma ffilter_transfer [transfer_rule]:
assumes "bi_unique A"
shows "((A ===> (=)) ===> rel_fset A ===> rel_fset A) ffilter ffilter"
using assms unfolding rel_fun_def
using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
lemma card_transfer [transfer_rule]:
"bi_unique A \<Longrightarrow> (rel_fset A ===> (=)) fcard fcard"
unfolding rel_fun_def
using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast
end
lifting_update fset.lifting
lifting_forget fset.lifting
subsection \<open>BNF setup\<close>
context
includes fset.lifting
begin
lemma rel_fset_alt:
"rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)"
by transfer (simp add: rel_set_def)
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A"
apply (rule f_the_inv_into_f[unfolded inj_on_def])
apply (simp add: fset_inject)
apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+
.
lemma rel_fset_aux:
"(\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>u \<in> fset b. \<exists>t \<in> fset a. R t u) \<longleftrightarrow>
((BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO
BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R")
proof
assume ?L
define R' where "R' =
the_inv fset (Collect (case_prod R) \<inter> (fset a \<times> fset b))" (is "_ = the_inv fset ?L'")
have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+
hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset)
show ?R unfolding Grp_def relcompp.simps conversep.simps
proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl)
from * show "a = fimage fst R'" using conjunct1[OF \<open>?L\<close>]
by (transfer, auto simp add: image_def Int_def split: prod.splits)
from * show "b = fimage snd R'" using conjunct2[OF \<open>?L\<close>]
by (transfer, auto simp add: image_def Int_def split: prod.splits)
qed (auto simp add: *)
next
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps
apply (simp add: subset_eq Ball_def)
apply (rule conjI)
apply (transfer, clarsimp, metis snd_conv)
by (transfer, clarsimp, metis fst_conv)
qed
bnf "'a fset"
map: fimage
sets: fset
bd: natLeq
wits: "{||}"
rel: rel_fset
apply -
apply transfer' apply simp
apply transfer' apply force
apply transfer apply force
apply transfer' apply force
apply (rule natLeq_card_order)
apply (rule natLeq_cinfinite)
apply (rule regularCard_natLeq)
apply transfer apply (metis finite_iff_ordLess_natLeq)
apply (fastforce simp: rel_fset_alt)
apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt
rel_fset_aux[unfolded OO_Grp_alt])
apply transfer apply simp
done
lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2"
by transfer (rule refl)
end
declare
fset.map_comp[simp]
fset.map_id[simp]
fset.set_map[simp]
subsection \<open>Size setup\<close>
context includes fset.lifting begin
lift_definition size_fset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a fset \<Rightarrow> nat" is "\<lambda>f. sum (Suc \<circ> f)" .
end
instantiation fset :: (type) size begin
definition size_fset where
size_fset_overloaded_def: "size_fset = FSet.size_fset (\<lambda>_. 0)"
instance ..
end
lemma size_fset_simps[simp]: "size_fset f X = (\<Sum>x \<in> fset X. Suc (f x))"
by (rule size_fset_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong,
unfolded map_fun_def comp_def id_apply])
lemma size_fset_overloaded_simps[simp]: "size X = (\<Sum>x \<in> fset X. Suc 0)"
by (rule size_fset_simps[of "\<lambda>_. 0", unfolded add_0_left add_0_right,
folded size_fset_overloaded_def])
lemma fset_size_o_map: "inj f \<Longrightarrow> size_fset g \<circ> fimage f = size_fset (g \<circ> f)"
apply (subst fun_eq_iff)
including fset.lifting by transfer (auto intro: sum.reindex_cong subset_inj_on)
setup \<open>
BNF_LFP_Size.register_size_global \<^type_name>\<open>fset\<close> \<^const_name>\<open>size_fset\<close>
@{thm size_fset_overloaded_def} @{thms size_fset_simps size_fset_overloaded_simps}
@{thms fset_size_o_map}
\<close>
lifting_update fset.lifting
lifting_forget fset.lifting
subsection \<open>Advanced relator customization\<close>
text \<open>Set vs. sum relators:\<close>
lemma rel_set_rel_sum[simp]:
"rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow>
rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)"
(is "?L \<longleftrightarrow> ?Rl \<and> ?Rr")
proof safe
assume L: "?L"
show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe
fix l1 assume "Inl l1 \<in> A1"
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2"
using L unfolding rel_set_def by auto
then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto)
thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto
next
fix l2 assume "Inl l2 \<in> A2"
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)"
using L unfolding rel_set_def by auto
then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto)
thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto
qed
show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe
fix r1 assume "Inr r1 \<in> A1"
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2"
using L unfolding rel_set_def by auto
then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto)
thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto
next
fix r2 assume "Inr r2 \<in> A2"
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)"
using L unfolding rel_set_def by auto
then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto)
thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto
qed
next
assume Rl: "?Rl" and Rr: "?Rr"
show ?L unfolding rel_set_def Bex_def vimage_eq proof safe
fix a1 assume a1: "a1 \<in> A1"
show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2"
proof(cases a1)
case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2"
using Rl a1 unfolding rel_set_def by blast
thus ?thesis unfolding Inl by auto
next
case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2"
using Rr a1 unfolding rel_set_def by blast
thus ?thesis unfolding Inr by auto
qed
next
fix a2 assume a2: "a2 \<in> A2"
show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2"
proof(cases a2)
case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2"
using Rl a2 unfolding rel_set_def by blast
thus ?thesis unfolding Inl by auto
next
case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2"
using Rr a2 unfolding rel_set_def by blast
thus ?thesis unfolding Inr by auto
qed
qed
qed
subsubsection \<open>Countability\<close>
lemma exists_fset_of_list: "\<exists>xs. fset_of_list xs = S"
including fset.lifting
by transfer (rule finite_list)
lemma fset_of_list_surj[simp, intro]: "surj fset_of_list"
proof -
have "x \<in> range fset_of_list" for x :: "'a fset"
unfolding image_iff
using exists_fset_of_list by fastforce
thus ?thesis by auto
qed
instance fset :: (countable) countable
proof
obtain to_nat :: "'a list \<Rightarrow> nat" where "inj to_nat"
by (metis ex_inj)
moreover have "inj (inv fset_of_list)"
using fset_of_list_surj by (rule surj_imp_inj_inv)
ultimately have "inj (to_nat \<circ> inv fset_of_list)"
by (rule inj_compose)
thus "\<exists>to_nat::'a fset \<Rightarrow> nat. inj to_nat"
by auto
qed
subsection \<open>Quickcheck setup\<close>
text \<open>Setup adapted from sets.\<close>
notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
context
includes term_syntax
begin
definition [code_unfold]:
"valterm_femptyset = Code_Evaluation.valtermify ({||} :: ('a :: typerep) fset)"
definition [code_unfold]:
"valtermify_finsert x s = Code_Evaluation.valtermify finsert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s"
end
instantiation fset :: (exhaustive) exhaustive
begin
fun exhaustive_fset where
"exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.exhaustive (\<lambda>x. if x |\<in>| A then None else f (finsert x A)) (i - 1)) (i - 1)))"
instance ..
end
instantiation fset :: (full_exhaustive) full_exhaustive
begin
fun full_exhaustive_fset where
"full_exhaustive_fset f i = (if i = 0 then None else (f valterm_femptyset orelse full_exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.full_exhaustive (\<lambda>x. if fst x |\<in>| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))"
instance ..
end
no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55)
instantiation fset :: (random) random
begin
context
includes state_combinator_syntax
begin
fun random_aux_fset :: "natural \<Rightarrow> natural \<Rightarrow> natural \<times> natural \<Rightarrow> ('a fset \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
"random_aux_fset 0 j = Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset)])" |
"random_aux_fset (Code_Numeral.Suc i) j =
Quickcheck_Random.collapse (Random.select_weight
[(1, Pair valterm_femptyset),
(Code_Numeral.Suc i,
Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
lemma [code]:
"random_aux_fset i j =
Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset),
(i, Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset (i - 1) j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])"
proof (induct i rule: natural.induct)
case zero
show ?case by (subst select_weight_drop_zero[symmetric]) (simp add: less_natural_def)
next
case (Suc i)
show ?case by (simp only: random_aux_fset.simps Suc_natural_minus_one)
qed
definition "random_fset i = random_aux_fset i i"
instance ..
end
end
subsection \<open>Code Generation Setup\<close>
text \<open>The following @{attribute code_unfold} lemmas are so the pre-processor of the code generator
will perform conversions like, e.g.,
@{lemma "x |\<in>| fimage f (fset_of_list xs) \<longleftrightarrow> x \<in> f ` set xs"
by (simp only: fimage.rep_eq fset_of_list.rep_eq)}.\<close>
declare
ffilter.rep_eq[code_unfold]
fimage.rep_eq[code_unfold]
finsert.rep_eq[code_unfold]
fset_of_list.rep_eq[code_unfold]
inf_fset.rep_eq[code_unfold]
minus_fset.rep_eq[code_unfold]
sup_fset.rep_eq[code_unfold]
uminus_fset.rep_eq[code_unfold]
end