Synchronizing FSet further.
(* Title: HOL/Quotient_Examples/FSet.thy
Author: Cezary Kaliszyk, TU Munich
Author: Christian Urban, TU Munich
A reasoning infrastructure for the type of finite sets.
*)
theory FSet
imports Quotient_List
begin
text {* Definiton of List relation and the quotient type *}
fun
list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
where
"list_eq xs ys = (set xs = set ys)"
lemma list_eq_equivp:
shows "equivp list_eq"
unfolding equivp_reflp_symp_transp
unfolding reflp_def symp_def transp_def
by auto
quotient_type
'a fset = "'a list" / "list_eq"
by (rule list_eq_equivp)
text {* Raw definitions of membership, sublist, cardinality,
intersection
*}
definition
memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
where
"memb x xs \<equiv> x \<in> set xs"
definition
sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
where
"sub_list xs ys \<equiv> set xs \<subseteq> set ys"
definition
fcard_raw :: "'a list \<Rightarrow> nat"
where
"fcard_raw xs = card (set xs)"
primrec
finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
where
"finter_raw [] ys = []"
| "finter_raw (x # xs) ys =
(if x \<in> set ys then x # (finter_raw xs ys) else finter_raw xs ys)"
primrec
fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
where
"fminus_raw ys [] = ys"
| "fminus_raw ys (x # xs) = fminus_raw (removeAll x ys) xs"
definition
rsp_fold
where
"rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
primrec
ffold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
where
"ffold_raw f z [] = z"
| "ffold_raw f z (a # xs) =
(if (rsp_fold f) then
if a \<in> set xs then ffold_raw f z xs
else f a (ffold_raw f z xs)
else z)"
text {* Composition Quotient *}
lemma list_all2_refl1:
shows "(list_all2 op \<approx>) r r"
by (rule list_all2_refl) (metis equivp_def fset_equivp)
lemma compose_list_refl:
shows "(list_all2 op \<approx> OOO op \<approx>) r r"
proof
have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
show "list_all2 op \<approx> r r" by (rule list_all2_refl1)
with * show "(op \<approx> OO list_all2 op \<approx>) r r" ..
qed
lemma Quotient_fset_list:
shows "Quotient (list_all2 op \<approx>) (map abs_fset) (map rep_fset)"
by (fact list_quotient[OF Quotient_fset])
lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
unfolding list_eq.simps
by (simp only: set_map)
lemma quotient_compose_list[quot_thm]:
shows "Quotient ((list_all2 op \<approx>) OOO (op \<approx>))
(abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
unfolding Quotient_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map abs_fset (map rep_fset (rep_fset a))) = a"
by (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
have b: "list_all2 op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
by (rule list_all2_refl1)
have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
show "(list_all2 op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
by (rule, rule list_all2_refl1) (rule c)
show "(list_all2 op \<approx> OOO op \<approx>) r s = ((list_all2 op \<approx> OOO op \<approx>) r r \<and>
(list_all2 op \<approx> OOO op \<approx>) s s \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"
proof (intro iffI conjI)
show "(list_all2 op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
show "(list_all2 op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
next
assume a: "(list_all2 op \<approx> OOO op \<approx>) r s"
then have b: "map abs_fset r \<approx> map abs_fset s"
proof (elim pred_compE)
fix b ba
assume c: "list_all2 op \<approx> r b"
assume d: "b \<approx> ba"
assume e: "list_all2 op \<approx> ba s"
have f: "map abs_fset r = map abs_fset b"
using Quotient_rel[OF Quotient_fset_list] c by blast
have "map abs_fset ba = map abs_fset s"
using Quotient_rel[OF Quotient_fset_list] e by blast
then have g: "map abs_fset s = map abs_fset ba" by simp
then show "map abs_fset r \<approx> map abs_fset s" using d f map_rel_cong by simp
qed
then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
using Quotient_rel[OF Quotient_fset] by blast
next
assume a: "(list_all2 op \<approx> OOO op \<approx>) r r \<and> (list_all2 op \<approx> OOO op \<approx>) s s
\<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
then have s: "(list_all2 op \<approx> OOO op \<approx>) s s" by simp
have d: "map abs_fset r \<approx> map abs_fset s"
by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
have b: "map rep_fset (map abs_fset r) \<approx> map rep_fset (map abs_fset s)"
by (rule map_rel_cong[OF d])
have y: "list_all2 op \<approx> (map rep_fset (map abs_fset s)) s"
by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_all2_refl1[of s]])
have c: "(op \<approx> OO list_all2 op \<approx>) (map rep_fset (map abs_fset r)) s"
by (rule pred_compI) (rule b, rule y)
have z: "list_all2 op \<approx> r (map rep_fset (map abs_fset r))"
by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_all2_refl1[of r]])
then show "(list_all2 op \<approx> OOO op \<approx>) r s"
using a c pred_compI by simp
qed
qed
lemma set_finter_raw[simp]:
"set (finter_raw xs ys) = set xs \<inter> set ys"
by (induct xs) (auto simp add: memb_def)
lemma set_fminus_raw[simp]:
"set (fminus_raw xs ys) = (set xs - set ys)"
by (induct ys arbitrary: xs) (auto)
text {* Respectfullness *}
lemma append_rsp[quot_respect]:
shows "(op \<approx> ===> op \<approx> ===> op \<approx>) append append"
by (simp)
lemma sub_list_rsp[quot_respect]:
shows "(op \<approx> ===> op \<approx> ===> op =) sub_list sub_list"
by (auto simp add: sub_list_def)
lemma memb_rsp[quot_respect]:
shows "(op = ===> op \<approx> ===> op =) memb memb"
by (auto simp add: memb_def)
lemma nil_rsp[quot_respect]:
shows "(op \<approx>) Nil Nil"
by simp
lemma cons_rsp[quot_respect]:
shows "(op = ===> op \<approx> ===> op \<approx>) Cons Cons"
by simp
lemma map_rsp[quot_respect]:
shows "(op = ===> op \<approx> ===> op \<approx>) map map"
by auto
lemma set_rsp[quot_respect]:
"(op \<approx> ===> op =) set set"
by auto
lemma list_equiv_rsp[quot_respect]:
shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
by auto
lemma finter_raw_rsp[quot_respect]:
shows "(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"
by simp
lemma removeAll_rsp[quot_respect]:
shows "(op = ===> op \<approx> ===> op \<approx>) removeAll removeAll"
by simp
lemma fminus_raw_rsp[quot_respect]:
shows "(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw"
by simp
lemma fcard_raw_rsp[quot_respect]:
shows "(op \<approx> ===> op =) fcard_raw fcard_raw"
by (simp add: fcard_raw_def)
lemma not_memb_nil:
shows "\<not> memb x []"
by (simp add: memb_def)
lemma memb_cons_iff:
shows "memb x (y # xs) = (x = y \<or> memb x xs)"
by (induct xs) (auto simp add: memb_def)
lemma memb_absorb:
shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
by (induct xs) (auto simp add: memb_def)
lemma none_memb_nil:
"(\<forall>x. \<not> memb x xs) = (xs \<approx> [])"
by (simp add: memb_def)
lemma memb_commute_ffold_raw:
"rsp_fold f \<Longrightarrow> h \<in> set b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (removeAll h b))"
apply (induct b)
apply (auto simp add: rsp_fold_def)
done
lemma ffold_raw_rsp_pre:
"set a = set b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"
apply (induct a arbitrary: b)
apply (simp)
apply (simp (no_asm_use))
apply (rule conjI)
apply (rule_tac [!] impI)
apply (rule_tac [!] conjI)
apply (rule_tac [!] impI)
apply (metis insert_absorb)
apply (metis List.insert_def List.set.simps(2) List.set_insert ffold_raw.simps(2))
apply (metis Diff_insert_absorb insertI1 memb_commute_ffold_raw set_removeAll)
apply(drule_tac x="removeAll a1 b" in meta_spec)
apply(auto)
apply(drule meta_mp)
apply(blast)
by (metis List.set.simps(2) emptyE ffold_raw.simps(2) in_listsp_conv_set listsp.simps mem_def)
lemma ffold_raw_rsp[quot_respect]:
shows "(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"
unfolding fun_rel_def
by(auto intro: ffold_raw_rsp_pre)
lemma concat_rsp_pre:
assumes a: "list_all2 op \<approx> x x'"
and b: "x' \<approx> y'"
and c: "list_all2 op \<approx> y' y"
and d: "\<exists>x\<in>set x. xa \<in> set x"
shows "\<exists>x\<in>set y. xa \<in> set x"
proof -
obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
have "ya \<in> set y'" using b h by simp
then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
then show ?thesis using f i by auto
qed
lemma concat_rsp[quot_respect]:
shows "(list_all2 op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
proof (rule fun_relI, elim pred_compE)
fix a b ba bb
assume a: "list_all2 op \<approx> a ba"
assume b: "ba \<approx> bb"
assume c: "list_all2 op \<approx> bb b"
have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
proof
fix x
show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
proof
assume d: "\<exists>xa\<in>set a. x \<in> set xa"
show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
next
assume e: "\<exists>xa\<in>set b. x \<in> set xa"
have a': "list_all2 op \<approx> ba a" by (rule list_all2_symp[OF list_eq_equivp, OF a])
have b': "bb \<approx> ba" by (rule equivp_symp[OF list_eq_equivp, OF b])
have c': "list_all2 op \<approx> b bb" by (rule list_all2_symp[OF list_eq_equivp, OF c])
show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
qed
qed
then show "concat a \<approx> concat b" by auto
qed
lemma [quot_respect]:
shows "((op =) ===> op \<approx> ===> op \<approx>) filter filter"
by auto
text {* Distributive lattice with bot *}
lemma sub_list_not_eq:
"(sub_list x y \<and> \<not> list_eq x y) = (sub_list x y \<and> \<not> sub_list y x)"
by (auto simp add: sub_list_def)
lemma sub_list_refl:
"sub_list x x"
by (simp add: sub_list_def)
lemma sub_list_trans:
"sub_list x y \<Longrightarrow> sub_list y z \<Longrightarrow> sub_list x z"
by (simp add: sub_list_def)
lemma sub_list_empty:
"sub_list [] x"
by (simp add: sub_list_def)
lemma sub_list_append_left:
"sub_list x (x @ y)"
by (simp add: sub_list_def)
lemma sub_list_append_right:
"sub_list y (x @ y)"
by (simp add: sub_list_def)
lemma sub_list_inter_left:
shows "sub_list (finter_raw x y) x"
by (simp add: sub_list_def)
lemma sub_list_inter_right:
shows "sub_list (finter_raw x y) y"
by (simp add: sub_list_def)
lemma sub_list_list_eq:
"sub_list x y \<Longrightarrow> sub_list y x \<Longrightarrow> list_eq x y"
unfolding sub_list_def list_eq.simps by blast
lemma sub_list_append:
"sub_list y x \<Longrightarrow> sub_list z x \<Longrightarrow> sub_list (y @ z) x"
unfolding sub_list_def by auto
lemma sub_list_inter:
"sub_list x y \<Longrightarrow> sub_list x z \<Longrightarrow> sub_list x (finter_raw y z)"
by (simp add: sub_list_def)
lemma append_inter_distrib:
"x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"
apply (induct x)
apply (auto)
done
instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
begin
quotient_definition
"bot :: 'a fset" is "[] :: 'a list"
abbreviation
fempty ("{||}")
where
"{||} \<equiv> bot :: 'a fset"
quotient_definition
"less_eq_fset \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
is
"sub_list \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> bool)"
abbreviation
f_subset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
where
"xs |\<subseteq>| ys \<equiv> xs \<le> ys"
definition
less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
where
"xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
abbreviation
fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
where
"xs |\<subset>| ys \<equiv> xs < ys"
quotient_definition
"sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
"append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
abbreviation
funion (infixl "|\<union>|" 65)
where
"xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
quotient_definition
"inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
"finter_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
abbreviation
finter (infixl "|\<inter>|" 65)
where
"xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
quotient_definition
"minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
"fminus_raw :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
instance
proof
fix x y z :: "'a fset"
show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
unfolding less_fset_def by (lifting sub_list_not_eq)
show "x |\<subseteq>| x" by (descending) (simp add: sub_list_def)
show "{||} |\<subseteq>| x" by (descending) (simp add: sub_list_def)
show "x |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)
show "y |\<subseteq>| x |\<union>| y" by (descending) (simp add: sub_list_def)
show "x |\<inter>| y |\<subseteq>| x"
by (descending) (simp add: sub_list_def memb_def[symmetric])
show "x |\<inter>| y |\<subseteq>| y"
by (descending) (simp add: sub_list_def memb_def[symmetric])
show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
by (descending) (rule append_inter_distrib)
next
fix x y z :: "'a fset"
assume a: "x |\<subseteq>| y"
assume b: "y |\<subseteq>| z"
show "x |\<subseteq>| z" using a b
by (descending) (simp add: sub_list_def)
next
fix x y :: "'a fset"
assume a: "x |\<subseteq>| y"
assume b: "y |\<subseteq>| x"
show "x = y" using a b
by (descending) (unfold sub_list_def list_eq.simps, blast)
next
fix x y z :: "'a fset"
assume a: "y |\<subseteq>| x"
assume b: "z |\<subseteq>| x"
show "y |\<union>| z |\<subseteq>| x" using a b
by (descending) (simp add: sub_list_def)
next
fix x y z :: "'a fset"
assume a: "x |\<subseteq>| y"
assume b: "x |\<subseteq>| z"
show "x |\<subseteq>| y |\<inter>| z" using a b
by (descending) (simp add: sub_list_def memb_def[symmetric])
qed
end
section {* Finsert and Membership *}
quotient_definition
"finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is "Cons"
syntax
"@Finset" :: "args => 'a fset" ("{|(_)|}")
translations
"{|x, xs|}" == "CONST finsert x {|xs|}"
"{|x|}" == "CONST finsert x {||}"
quotient_definition
fin (infix "|\<in>|" 50)
where
"fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" is "memb"
abbreviation
fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
where
"x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
section {* Other constants on the Quotient Type *}
quotient_definition
"fcard :: 'a fset \<Rightarrow> nat"
is
fcard_raw
quotient_definition
"fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
is
map
quotient_definition
"fdelete :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is removeAll
quotient_definition
"fset :: 'a fset \<Rightarrow> 'a set"
is "set"
quotient_definition
"ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
is "ffold_raw"
quotient_definition
"fconcat :: ('a fset) fset \<Rightarrow> 'a fset"
is
"concat"
quotient_definition
"ffilter :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
is
"filter"
text {* Compositional Respectfullness and Preservation *}
lemma [quot_respect]: "(list_all2 op \<approx> OOO op \<approx>) [] []"
by (fact compose_list_refl)
lemma [quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
by simp
lemma [quot_respect]:
shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
apply auto
apply (rule_tac b="x # b" in pred_compI)
apply auto
apply (rule_tac b="x # ba" in pred_compI)
apply auto
done
lemma [quot_preserve]:
"(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"
by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
abs_o_rep[OF Quotient_fset] map_id finsert_def)
lemma [quot_preserve]:
"((map rep_fset \<circ> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op @ = funion"
by (simp add: fun_eq_iff Quotient_abs_rep[OF Quotient_fset]
abs_o_rep[OF Quotient_fset] map_id sup_fset_def)
lemma list_all2_app_l:
assumes a: "reflp R"
and b: "list_all2 R l r"
shows "list_all2 R (z @ l) (z @ r)"
by (induct z) (simp_all add: b rev_iffD1[OF a meta_eq_to_obj_eq[OF reflp_def]])
lemma append_rsp2_pre0:
assumes a:"list_all2 op \<approx> x x'"
shows "list_all2 op \<approx> (x @ z) (x' @ z)"
using a apply (induct x x' rule: list_induct2')
by simp_all (rule list_all2_refl1)
lemma append_rsp2_pre1:
assumes a:"list_all2 op \<approx> x x'"
shows "list_all2 op \<approx> (z @ x) (z @ x')"
using a apply (induct x x' arbitrary: z rule: list_induct2')
apply (rule list_all2_refl1)
apply (simp_all del: list_eq.simps)
apply (rule list_all2_app_l)
apply (simp_all add: reflp_def)
done
lemma append_rsp2_pre:
assumes a:"list_all2 op \<approx> x x'"
and b: "list_all2 op \<approx> z z'"
shows "list_all2 op \<approx> (x @ z) (x' @ z')"
apply (rule list_all2_transp[OF fset_equivp])
apply (rule append_rsp2_pre0)
apply (rule a)
using b apply (induct z z' rule: list_induct2')
apply (simp_all only: append_Nil2)
apply (rule list_all2_refl1)
apply simp_all
apply (rule append_rsp2_pre1)
apply simp
done
lemma [quot_respect]:
"(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op @ op @"
proof (intro fun_relI, elim pred_compE)
fix x y z w x' z' y' w' :: "'a list list"
assume a:"list_all2 op \<approx> x x'"
and b: "x' \<approx> y'"
and c: "list_all2 op \<approx> y' y"
assume aa: "list_all2 op \<approx> z z'"
and bb: "z' \<approx> w'"
and cc: "list_all2 op \<approx> w' w"
have a': "list_all2 op \<approx> (x @ z) (x' @ z')" using a aa append_rsp2_pre by auto
have b': "x' @ z' \<approx> y' @ w'" using b bb by simp
have c': "list_all2 op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
have d': "(op \<approx> OO list_all2 op \<approx>) (x' @ z') (y @ w)"
by (rule pred_compI) (rule b', rule c')
show "(list_all2 op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
by (rule pred_compI) (rule a', rule d')
qed
text {* Raw theorems. Finsert, memb, singleron, sub_list *}
lemma nil_not_cons:
shows "\<not> ([] \<approx> x # xs)"
and "\<not> (x # xs \<approx> [])"
by auto
lemma no_memb_nil:
"(\<forall>x. \<not> memb x xs) = (xs = [])"
by (simp add: memb_def)
lemma memb_consI1:
shows "memb x (x # xs)"
by (simp add: memb_def)
lemma memb_consI2:
shows "memb x xs \<Longrightarrow> memb x (y # xs)"
by (simp add: memb_def)
lemma singleton_list_eq:
shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
by (simp)
lemma sub_list_cons:
"sub_list (x # xs) ys = (memb x ys \<and> sub_list xs ys)"
by (auto simp add: memb_def sub_list_def)
lemma fminus_raw_red:
"fminus_raw (x # xs) ys = (if x \<in> set ys then fminus_raw xs ys else x # (fminus_raw xs ys))"
by (induct ys arbitrary: xs x) (simp_all)
text {* Cardinality of finite sets *}
(* used in memb_card_not_0 *)
lemma fcard_raw_0:
shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []"
unfolding fcard_raw_def
by (induct xs) (auto)
(* used in list_eq2_equiv *)
lemma memb_card_not_0:
assumes a: "memb a A"
shows "\<not>(fcard_raw A = 0)"
proof -
have "\<not>(\<forall>x. \<not> memb x A)" using a by auto
then have "\<not>A \<approx> []" using none_memb_nil[of A] by simp
then show ?thesis using fcard_raw_0[of A] by simp
qed
section {* fmap *}
(* there is another fmap section below *)
lemma map_append:
"map f (xs @ ys) \<approx> (map f xs) @ (map f ys)"
by simp
lemma memb_append:
"memb x (xs @ ys) \<longleftrightarrow> memb x xs \<or> memb x ys"
by (induct xs) (simp_all add: not_memb_nil memb_cons_iff)
lemma fset_raw_strong_cases:
obtains "xs = []"
| x ys where "\<not> memb x ys" and "xs \<approx> x # ys"
proof (induct xs arbitrary: x ys)
case Nil
then show thesis by simp
next
case (Cons a xs)
have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis" by fact
have b: "\<And>x' ys'. \<lbrakk>\<not> memb x' ys'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
have c: "xs = [] \<Longrightarrow> thesis" by (metis no_memb_nil singleton_list_eq b)
have "\<And>x ys. \<lbrakk>\<not> memb x ys; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
proof -
fix x :: 'a
fix ys :: "'a list"
assume d:"\<not> memb x ys"
assume e:"xs \<approx> x # ys"
show thesis
proof (cases "x = a")
assume h: "x = a"
then have f: "\<not> memb a ys" using d by simp
have g: "a # xs \<approx> a # ys" using e h by auto
show thesis using b f g by simp
next
assume h: "x \<noteq> a"
then have f: "\<not> memb x (a # ys)" using d unfolding memb_def by auto
have g: "a # xs \<approx> x # (a # ys)" using e h by auto
show thesis using b f g by simp
qed
qed
then show thesis using a c by blast
qed
section {* deletion *}
lemma fset_raw_removeAll_cases:
"xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # removeAll x xs)"
by (induct xs) (auto simp add: memb_def)
lemma fremoveAll_filter:
"removeAll y xs = [x \<leftarrow> xs. x \<noteq> y]"
by (induct xs) simp_all
lemma fcard_raw_delete:
"fcard_raw (removeAll y xs) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"
by (auto simp add: fcard_raw_def memb_def)
lemma finter_raw_empty:
"finter_raw l [] = []"
by (induct l) (simp_all add: not_memb_nil)
lemma inj_map_eq_iff:
"inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)"
by (simp add: set_eq_iff[symmetric] inj_image_eq_iff)
text {* alternate formulation with a different decomposition principle
and a proof of equivalence *}
inductive
list_eq2
where
"list_eq2 (a # b # xs) (b # a # xs)"
| "list_eq2 [] []"
| "list_eq2 xs ys \<Longrightarrow> list_eq2 ys xs"
| "list_eq2 (a # a # xs) (a # xs)"
| "list_eq2 xs ys \<Longrightarrow> list_eq2 (a # xs) (a # ys)"
| "\<lbrakk>list_eq2 xs1 xs2; list_eq2 xs2 xs3\<rbrakk> \<Longrightarrow> list_eq2 xs1 xs3"
lemma list_eq2_refl:
shows "list_eq2 xs xs"
by (induct xs) (auto intro: list_eq2.intros)
lemma cons_delete_list_eq2:
shows "list_eq2 (a # (removeAll a A)) (if memb a A then A else a # A)"
apply (induct A)
apply (simp add: memb_def list_eq2_refl)
apply (case_tac "memb a (aa # A)")
apply (simp_all only: memb_cons_iff)
apply (case_tac [!] "a = aa")
apply (simp_all)
apply (case_tac "memb a A")
apply (auto simp add: memb_def)[2]
apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
apply (auto simp add: list_eq2_refl memb_def)
done
lemma memb_delete_list_eq2:
assumes a: "memb e r"
shows "list_eq2 (e # removeAll e r) r"
using a cons_delete_list_eq2[of e r]
by simp
lemma list_eq2_equiv:
"(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
proof
show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
next
{
fix n
assume a: "fcard_raw l = n" and b: "l \<approx> r"
have "list_eq2 l r"
using a b
proof (induct n arbitrary: l r)
case 0
have "fcard_raw l = 0" by fact
then have "\<forall>x. \<not> memb x l" using memb_card_not_0[of _ l] by auto
then have z: "l = []" using no_memb_nil by auto
then have "r = []" using `l \<approx> r` by simp
then show ?case using z list_eq2_refl by simp
next
case (Suc m)
have b: "l \<approx> r" by fact
have d: "fcard_raw l = Suc m" by fact
then have "\<exists>a. memb a l"
apply(simp add: fcard_raw_def memb_def)
apply(drule card_eq_SucD)
apply(blast)
done
then obtain a where e: "memb a l" by auto
then have e': "memb a r" using list_eq.simps[simplified memb_def[symmetric], of l r] b
unfolding memb_def by auto
have f: "fcard_raw (removeAll a l) = m" using fcard_raw_delete[of a l] e d by simp
have g: "removeAll a l \<approx> removeAll a r" using removeAll_rsp b by simp
have "list_eq2 (removeAll a l) (removeAll a r)" by (rule Suc.hyps[OF f g])
then have h: "list_eq2 (a # removeAll a l) (a # removeAll a r)" by (rule list_eq2.intros(5))
have i: "list_eq2 l (a # removeAll a l)"
by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])
have "list_eq2 l (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
then show ?case using list_eq2.intros(6)[OF _ memb_delete_list_eq2[OF e']] by simp
qed
}
then show "l \<approx> r \<Longrightarrow> list_eq2 l r" by blast
qed
text {* Lifted theorems *}
lemma not_fin_fnil: "x |\<notin>| {||}"
by (descending) (simp add: memb_def)
lemma fin_finsert_iff[simp]:
"x |\<in>| finsert y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
by (descending) (simp add: memb_def)
lemma
shows finsertI1: "x |\<in>| finsert x S"
and finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"
by (lifting memb_consI1 memb_consI2)
lemma finsert_absorb[simp]:
shows "x |\<in>| S \<Longrightarrow> finsert x S = S"
by (descending) (auto simp add: memb_def)
lemma fempty_not_finsert[simp]:
"{||} \<noteq> finsert x S"
"finsert x S \<noteq> {||}"
by (lifting nil_not_cons)
lemma finsert_left_comm:
"finsert x (finsert y S) = finsert y (finsert x S)"
by (descending) (auto)
lemma finsert_left_idem:
"finsert x (finsert x S) = finsert x S"
by (descending) (auto)
lemma fsingleton_eq[simp]:
shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
by (descending) (auto)
text {* fset *}
lemma fset_simps[simp]:
"fset {||} = ({} :: 'a set)"
"fset (finsert (h :: 'a) t) = insert h (fset t)"
by (lifting set.simps)
lemma in_fset:
"x \<in> fset S \<equiv> x |\<in>| S"
by (lifting memb_def[symmetric])
lemma none_fin_fempty:
"(\<forall>x. x |\<notin>| S) = (S = {||})"
by (lifting none_memb_nil)
lemma fset_cong:
"(S = T) = (fset S = fset T)"
by (lifting list_eq.simps)
section {* fcard *}
lemma fcard_finsert_if [simp]:
shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"
by (descending) (auto simp add: fcard_raw_def memb_def insert_absorb)
lemma fcard_0[simp]:
shows "fcard S = 0 \<longleftrightarrow> S = {||}"
by (descending) (simp add: fcard_raw_def)
lemma fcard_fempty[simp]:
shows "fcard {||} = 0"
by (simp add: fcard_0)
lemma fcard_1:
shows "fcard S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
by (descending) (auto simp add: fcard_raw_def card_Suc_eq)
lemma fcard_gt_0:
shows "x \<in> fset S \<Longrightarrow> 0 < fcard S"
by (descending) (auto simp add: fcard_raw_def card_gt_0_iff)
lemma fcard_not_fin:
assumes a: "x |\<notin>| S"
shows "fcard (finsert x S) = Suc (fcard S)"
using a
by (descending) (simp add: memb_def fcard_raw_def)
lemma fcard_suc:
shows "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"
apply(descending)
apply(simp add: fcard_raw_def memb_def)
apply(drule card_eq_SucD)
apply(auto)
apply(rule_tac x="b" in exI)
apply(rule_tac x="removeAll b S" in exI)
apply(auto)
done
lemma fcard_delete:
"fcard (fdelete y S) = (if y |\<in>| S then fcard S - 1 else fcard S)"
by (lifting fcard_raw_delete)
lemma fcard_suc_memb:
shows "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"
apply(descending)
apply(simp add: fcard_raw_def memb_def)
apply(drule card_eq_SucD)
apply(auto)
done
lemma fin_fcard_not_0:
shows "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"
by (descending) (auto simp add: fcard_raw_def memb_def)
section {* funion *}
lemmas [simp] =
sup_bot_left[where 'a="'a fset", standard]
sup_bot_right[where 'a="'a fset", standard]
lemma funion_finsert[simp]:
shows "finsert x S |\<union>| T = finsert x (S |\<union>| T)"
by (lifting append.simps(2))
lemma singleton_union_left:
shows "{|a|} |\<union>| S = finsert a S"
by simp
lemma singleton_union_right:
shows "S |\<union>| {|a|} = finsert a S"
by (subst sup.commute) simp
section {* Induction and Cases rules for fsets *}
lemma fset_strong_cases:
obtains "xs = {||}"
| x ys where "x |\<notin>| ys" and "xs = finsert x ys"
by (lifting fset_raw_strong_cases)
lemma fset_exhaust[case_names fempty finsert, cases type: fset]:
shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (lifting list.exhaust)
lemma fset_induct_weak[case_names fempty finsert]:
shows "\<lbrakk>P {||}; \<And>x S. P S \<Longrightarrow> P (finsert x S)\<rbrakk> \<Longrightarrow> P S"
by (lifting list.induct)
lemma fset_induct[case_names fempty finsert, induct type: fset]:
assumes prem1: "P {||}"
and prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"
shows "P S"
proof(induct S rule: fset_induct_weak)
case fempty
show "P {||}" by (rule prem1)
next
case (finsert x S)
have asm: "P S" by fact
show "P (finsert x S)"
by (cases "x |\<in>| S") (simp_all add: asm prem2)
qed
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)
apply simp_all
apply (induct_tac xa rule: fset_induct)
apply simp_all
done
lemma fset_fcard_induct:
assumes a: "P {||}"
and b: "\<And>xs ys. Suc (fcard xs) = (fcard ys) \<Longrightarrow> P xs \<Longrightarrow> P ys"
shows "P zs"
proof (induct zs)
show "P {||}" by (rule a)
next
fix x :: 'a and zs :: "'a fset"
assume h: "P zs"
assume "x |\<notin>| zs"
then have H1: "Suc (fcard zs) = fcard (finsert x zs)" using fcard_suc by auto
then show "P (finsert x zs)" using b h by simp
qed
section {* fmap *}
lemma fmap_simps[simp]:
fixes f::"'a \<Rightarrow> 'b"
shows "fmap f {||} = {||}"
and "fmap f (finsert x S) = finsert (f x) (fmap f S)"
by (lifting map.simps)
lemma fmap_set_image:
"fset (fmap f S) = f ` (fset S)"
by (induct S) simp_all
lemma inj_fmap_eq_iff:
"inj f \<Longrightarrow> fmap f S = fmap f T \<longleftrightarrow> S = T"
by (lifting inj_map_eq_iff)
lemma fmap_funion:
shows "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"
by (lifting map_append)
lemma fin_funion:
shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
by (lifting memb_append)
section {* fset *}
lemma fin_set:
shows "x |\<in>| xs \<longleftrightarrow> x \<in> fset xs"
by (lifting memb_def)
lemma fnotin_set:
shows "x |\<notin>| xs \<longleftrightarrow> x \<notin> fset xs"
by (simp add: fin_set)
lemma fcard_set:
shows "fcard xs = card (fset xs)"
by (lifting fcard_raw_def)
lemma fsubseteq_set:
shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
by (lifting sub_list_def)
lemma fsubset_set:
shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
unfolding less_fset_def
by (descending) (auto simp add: sub_list_def)
lemma ffilter_set [simp]:
shows "fset (ffilter P xs) = P \<inter> fset xs"
by (descending) (auto simp add: mem_def)
lemma fdelete_set [simp]:
shows "fset (fdelete x xs) = fset xs - {x}"
by (lifting set_removeAll)
lemma finter_set [simp]:
shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
by (lifting set_finter_raw)
lemma funion_set [simp]:
shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
by (lifting set_append)
lemma fminus_set [simp]:
shows "fset (xs - ys) = fset xs - fset ys"
by (lifting set_fminus_raw)
section {* ffold *}
lemma ffold_nil:
shows "ffold f z {||} = z"
by (lifting ffold_raw.simps(1)[where 'a="'b" and 'b="'a"])
lemma ffold_finsert: "ffold f z (finsert a A) =
(if rsp_fold f then if a |\<in>| A then ffold f z A else f a (ffold f z A) else z)"
by (descending) (simp add: memb_def)
lemma fin_commute_ffold:
"\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete h b))"
by (descending) (simp add: memb_def memb_commute_ffold_raw)
section {* fdelete *}
lemma fin_fdelete:
shows "x |\<in>| fdelete y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
by (descending) (simp add: memb_def)
lemma fnotin_fdelete:
shows "x |\<notin>| fdelete x S"
by (descending) (simp add: memb_def)
lemma fnotin_fdelete_ident:
shows "x |\<notin>| S \<Longrightarrow> fdelete x S = S"
by (descending) (simp add: memb_def)
lemma fset_fdelete_cases:
shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete x S))"
by (lifting fset_raw_removeAll_cases)
section {* finter *}
lemma finter_empty_l: "({||} |\<inter>| S) = {||}"
by (lifting finter_raw.simps(1))
lemma finter_empty_r: "(S |\<inter>| {||}) = {||}"
by (lifting finter_raw_empty)
lemma finter_finsert:
shows "finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"
by (descending) (simp add: memb_def)
lemma fin_finter:
shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
by (descending) (simp add: memb_def)
lemma fsubset_finsert:
shows "(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)"
by (lifting sub_list_cons)
lemma
shows "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"
by (descending) (auto simp add: sub_list_def memb_def)
lemma fsubset_fin:
shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
by (descending) (auto simp add: sub_list_def memb_def)
lemma fminus_fin:
shows "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)"
by (descending) (simp add: memb_def)
lemma fminus_red:
shows "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"
by (descending) (auto simp add: memb_def)
lemma fminus_red_fin[simp]:
shows "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"
by (simp add: fminus_red)
lemma fminus_red_fnotin[simp]:
shows "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"
by (simp add: fminus_red)
lemma fset_eq_iff:
shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
by (descending) (auto simp add: memb_def)
(* We cannot write it as "assumes .. shows" since Isabelle changes
the quantifiers to schematic variables and reintroduces them in
a different order *)
lemma fset_eq_cases:
"\<lbrakk>a1 = a2;
\<And>a b xs. \<lbrakk>a1 = finsert a (finsert b xs); a2 = finsert b (finsert a xs)\<rbrakk> \<Longrightarrow> P;
\<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
\<And>a xs. \<lbrakk>a1 = finsert a (finsert a xs); a2 = finsert a xs\<rbrakk> \<Longrightarrow> P;
\<And>xs ys a. \<lbrakk>a1 = finsert a xs; a2 = finsert a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
\<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
\<Longrightarrow> P"
by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
lemma fset_eq_induct:
assumes "x1 = x2"
and "\<And>a b xs. P (finsert a (finsert b xs)) (finsert b (finsert a xs))"
and "P {||} {||}"
and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
and "\<And>a xs. P (finsert a (finsert a xs)) (finsert a xs)"
and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (finsert a xs) (finsert a ys)"
and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
shows "P x1 x2"
using assms
by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
section {* fconcat *}
lemma fconcat_empty:
shows "fconcat {||} = {||}"
by (lifting concat.simps(1))
lemma fconcat_insert:
shows "fconcat (finsert x S) = x |\<union>| fconcat S"
by (lifting concat.simps(2))
lemma
shows "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"
by (lifting concat_append)
section {* ffilter *}
lemma subseteq_filter:
shows "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
by (descending) (auto simp add: memb_def sub_list_def)
lemma eq_ffilter:
shows "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
by (descending) (auto simp add: memb_def)
lemma subset_ffilter:
shows "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow> ffilter P xs < ffilter Q xs"
unfolding less_fset_def by (auto simp add: subseteq_filter eq_ffilter)
section {* lemmas transferred from Finite_Set theory *}
text {* finiteness for finite sets holds *}
lemma finite_fset [simp]:
shows "finite (fset S)"
by (induct S) auto
lemma fset_choice:
shows "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
apply(descending)
apply(simp add: memb_def)
apply(rule finite_set_choice[simplified Ball_def])
apply(simp_all)
done
lemma fsubseteq_fempty:
shows "xs |\<subseteq>| {||} = (xs = {||})"
by (metis finter_empty_r le_iff_inf)
lemma not_fsubset_fnil:
shows "\<not> xs |\<subset>| {||}"
by (metis fset_simps(1) fsubset_set not_psubset_empty)
lemma fcard_mono:
shows "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"
unfolding fcard_set fsubseteq_set
by (rule card_mono[OF finite_fset])
lemma fcard_fseteq:
shows "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"
unfolding fcard_set fsubseteq_set
by (simp add: card_seteq[OF finite_fset] fset_cong)
lemma psubset_fcard_mono:
shows "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"
unfolding fcard_set fsubset_set
by (rule psubset_card_mono[OF finite_fset])
lemma fcard_funion_finter:
shows "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"
unfolding fcard_set funion_set finter_set
by (rule card_Un_Int[OF finite_fset finite_fset])
lemma fcard_funion_disjoint:
shows "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"
unfolding fcard_set funion_set
apply (rule card_Un_disjoint[OF finite_fset finite_fset])
by (metis finter_set fset_simps(1))
lemma fcard_delete1_less:
shows "x |\<in>| xs \<Longrightarrow> fcard (fdelete x xs) < fcard xs"
unfolding fcard_set fin_set fdelete_set
by (rule card_Diff1_less[OF finite_fset])
lemma fcard_delete2_less:
shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete y (fdelete x xs)) < fcard xs"
unfolding fcard_set fdelete_set fin_set
by (rule card_Diff2_less[OF finite_fset])
lemma fcard_delete1_le:
shows "fcard (fdelete x xs) <= fcard xs"
unfolding fdelete_set fcard_set
by (rule card_Diff1_le[OF finite_fset])
lemma fcard_psubset:
shows "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"
unfolding fcard_set fsubseteq_set fsubset_set
by (rule card_psubset[OF finite_fset])
lemma fcard_fmap_le:
shows "fcard (fmap f xs) \<le> fcard xs"
unfolding fcard_set fmap_set_image
by (rule card_image_le[OF finite_fset])
lemma fin_fminus_fnotin:
shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
unfolding fin_set fminus_set
by blast
lemma fin_fnotin_fminus:
shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
unfolding fin_set fminus_set
by blast
lemma fin_mdef:
shows "x |\<in>| F = ((x |\<notin>| (F - {|x|})) & (F = finsert x (F - {|x|})))"
unfolding fin_set fset_simps fset_cong fminus_set
by blast
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
unfolding fin_set fcard_set fminus_set
by (simp add: card_Diff_insert[OF finite_fset])
lemma fcard_fminus_fsubset:
assumes "B |\<subseteq>| A"
shows "fcard (A - B) = fcard A - fcard B"
using assms
unfolding fsubseteq_set fcard_set fminus_set
by (rule card_Diff_subset[OF finite_fset])
lemma fcard_fminus_subset_finter:
shows "fcard (A - B) = fcard A - fcard (A |\<inter>| B)"
unfolding finter_set fcard_set fminus_set
by (rule card_Diff_subset_Int) (simp)
lemma list_all2_refl:
assumes q: "equivp R"
shows "(list_all2 R) r r"
by (rule list_all2_refl) (metis equivp_def q)
lemma compose_list_refl2:
assumes q: "equivp R"
shows "(list_all2 R OOO op \<approx>) r r"
proof
have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
show "list_all2 R r r" by (rule list_all2_refl[OF q])
with * show "(op \<approx> OO list_all2 R) r r" ..
qed
lemma quotient_compose_list_g:
assumes q: "Quotient R Abs Rep"
and e: "equivp R"
shows "Quotient ((list_all2 R) OOO (op \<approx>))
(abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
unfolding Quotient_def comp_def
proof (intro conjI allI)
fix a r s
show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
by (simp add: abs_o_rep[OF q] Quotient_abs_rep[OF Quotient_fset] map_id)
have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule list_all2_refl[OF e])
have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
by (rule, rule list_all2_refl[OF e]) (rule c)
show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
(list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
proof (intro iffI conjI)
show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl2[OF e])
show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl2[OF e])
next
assume a: "(list_all2 R OOO op \<approx>) r s"
then have b: "map Abs r \<approx> map Abs s"
proof (elim pred_compE)
fix b ba
assume c: "list_all2 R r b"
assume d: "b \<approx> ba"
assume e: "list_all2 R ba s"
have f: "map Abs r = map Abs b"
using Quotient_rel[OF list_quotient[OF q]] c by blast
have "map Abs ba = map Abs s"
using Quotient_rel[OF list_quotient[OF q]] e by blast
then have g: "map Abs s = map Abs ba" by simp
then show "map Abs r \<approx> map Abs s" using d f map_rel_cong by simp
qed
then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
using Quotient_rel[OF Quotient_fset] by blast
next
assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
\<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
then have s: "(list_all2 R OOO op \<approx>) s s" by simp
have d: "map Abs r \<approx> map Abs s"
by (subst Quotient_rel[OF Quotient_fset]) (simp add: a)
have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
by (rule map_rel_cong[OF d])
have y: "list_all2 R (map Rep (map Abs s)) s"
by (fact rep_abs_rsp_left[OF list_quotient[OF q], OF list_all2_refl[OF e, of s]])
have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
by (rule pred_compI) (rule b, rule y)
have z: "list_all2 R r (map Rep (map Abs r))"
by (fact rep_abs_rsp[OF list_quotient[OF q], OF list_all2_refl[OF e, of r]])
then show "(list_all2 R OOO op \<approx>) r s"
using a c pred_compI by simp
qed
qed
ML {*
fun dest_fsetT (Type (@{type_name fset}, [T])) = T
| dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
*}
no_notation
list_eq (infix "\<approx>" 50)
end