(* 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 Quotient_Product+ −
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 = (\<forall>x. x \<in> set xs \<longleftrightarrow> x \<in> 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 *}+ −
+ −
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> (\<forall>x. x \<in> set xs \<longrightarrow> x \<in> set ys)"+ −
+ −
fun+ −
fcard_raw :: "'a list \<Rightarrow> nat"+ −
where+ −
fcard_raw_nil: "fcard_raw [] = 0"+ −
| fcard_raw_cons: "fcard_raw (x # xs) = (if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"+ −
+ −
primrec+ −
finter_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+ −
where+ −
"finter_raw [] l = []"+ −
| "finter_raw (h # t) l =+ −
(if memb h l then h # (finter_raw t l) else finter_raw t l)"+ −
+ −
primrec+ −
delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"+ −
where+ −
"delete_raw [] x = []"+ −
| "delete_raw (a # xs) x = (if (a = x) then delete_raw xs x else a # (delete_raw xs x))"+ −
+ −
primrec+ −
fminus_raw :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"+ −
where+ −
"fminus_raw l [] = l"+ −
| "fminus_raw l (h # t) = fminus_raw (delete_raw l h) t"+ −
+ −
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 memb a 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 set_in_eq: "(\<forall>e. ((e \<in> xs) \<longleftrightarrow> (e \<in> ys))) \<equiv> xs = ys"+ −
by (rule eq_reflection) auto+ −
+ −
lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"+ −
unfolding list_eq.simps+ −
by (simp only: set_map set_in_eq)+ −
+ −
+ −
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+ −
+ −
text {* Respectfullness *}+ −
+ −
lemma append_rsp[quot_respect]:+ −
shows "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"+ −
apply(simp del: list_eq.simps)+ −
by auto+ −
+ −
lemma append_rsp_unfolded:+ −
"\<lbrakk>x \<approx> y; v \<approx> w\<rbrakk> \<Longrightarrow> x @ v \<approx> y @ w"+ −
by auto+ −
+ −
lemma [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 "[] \<approx> []"+ −
by simp+ −
+ −
lemma cons_rsp[quot_respect]:+ −
shows "(op = ===> op \<approx> ===> op \<approx>) op # op #"+ −
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 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_finter_raw:+ −
"memb x (finter_raw xs ys) \<longleftrightarrow> memb x xs \<and> memb x ys"+ −
by (induct xs) (auto simp add: not_memb_nil memb_cons_iff)+ −
+ −
lemma [quot_respect]:+ −
"(op \<approx> ===> op \<approx> ===> op \<approx>) finter_raw finter_raw"+ −
by (simp add: memb_def[symmetric] memb_finter_raw)+ −
+ −
lemma memb_delete_raw:+ −
"memb x (delete_raw xs y) = (memb x xs \<and> x \<noteq> y)"+ −
by (induct xs arbitrary: x y) (auto simp add: memb_def)+ −
+ −
lemma [quot_respect]:+ −
"(op \<approx> ===> op = ===> op \<approx>) delete_raw delete_raw"+ −
by (simp add: memb_def[symmetric] memb_delete_raw)+ −
+ −
lemma fminus_raw_memb: "memb x (fminus_raw xs ys) = (memb x xs \<and> \<not> memb x ys)"+ −
by (induct ys arbitrary: xs)+ −
(simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)+ −
+ −
lemma [quot_respect]:+ −
"(op \<approx> ===> op \<approx> ===> op \<approx>) fminus_raw fminus_raw"+ −
by (simp add: memb_def[symmetric] fminus_raw_memb)+ −
+ −
lemma fcard_raw_gt_0:+ −
assumes a: "x \<in> set xs"+ −
shows "0 < fcard_raw xs"+ −
using a by (induct xs) (auto simp add: memb_def)+ −
+ −
lemma fcard_raw_delete_one:+ −
shows "fcard_raw ([x \<leftarrow> xs. x \<noteq> y]) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"+ −
by (induct xs) (auto dest: fcard_raw_gt_0 simp add: memb_def)+ −
+ −
lemma fcard_raw_rsp_aux:+ −
assumes a: "xs \<approx> ys"+ −
shows "fcard_raw xs = fcard_raw ys"+ −
using a+ −
proof (induct xs arbitrary: ys)+ −
case Nil+ −
show ?case using Nil.prems by simp+ −
next+ −
case (Cons a xs)+ −
have a: "a # xs \<approx> ys" by fact+ −
have b: "\<And>ys. xs \<approx> ys \<Longrightarrow> fcard_raw xs = fcard_raw ys" by fact+ −
show ?case proof (cases "a \<in> set xs")+ −
assume c: "a \<in> set xs"+ −
have "\<forall>x. (x \<in> set xs) = (x \<in> set ys)"+ −
proof (intro allI iffI)+ −
fix x+ −
assume "x \<in> set xs"+ −
then show "x \<in> set ys" using a by auto+ −
next+ −
fix x+ −
assume d: "x \<in> set ys"+ −
have e: "(x \<in> set (a # xs)) = (x \<in> set ys)" using a by simp+ −
show "x \<in> set xs" using c d e unfolding list_eq.simps by simp blast+ −
qed+ −
then show ?thesis using b c by (simp add: memb_def)+ −
next+ −
assume c: "a \<notin> set xs"+ −
have d: "xs \<approx> [x\<leftarrow>ys . x \<noteq> a] \<Longrightarrow> fcard_raw xs = fcard_raw [x\<leftarrow>ys . x \<noteq> a]" using b by simp+ −
have "Suc (fcard_raw xs) = fcard_raw ys"+ −
proof (cases "a \<in> set ys")+ −
assume e: "a \<in> set ys"+ −
have f: "\<forall>x. (x \<in> set xs) = (x \<in> set ys \<and> x \<noteq> a)" using a c+ −
by (auto simp add: fcard_raw_delete_one)+ −
have "fcard_raw ys = Suc (fcard_raw ys - 1)" by (rule Suc_pred'[OF fcard_raw_gt_0]) (rule e)+ −
then show ?thesis using d e f by (simp_all add: fcard_raw_delete_one memb_def)+ −
next+ −
case False then show ?thesis using a c d by auto+ −
qed+ −
then show ?thesis using a c d by (simp add: memb_def)+ −
qed+ −
qed+ −
+ −
lemma fcard_raw_rsp[quot_respect]:+ −
shows "(op \<approx> ===> op =) fcard_raw fcard_raw"+ −
by (simp add: fcard_raw_rsp_aux)+ −
+ −
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 not_memb_delete_raw_ident:+ −
shows "\<not> memb x xs \<Longrightarrow> delete_raw xs x = xs"+ −
by (induct xs) (auto simp add: memb_def)+ −
+ −
lemma memb_commute_ffold_raw:+ −
"rsp_fold f \<Longrightarrow> memb h b \<Longrightarrow> ffold_raw f z b = f h (ffold_raw f z (delete_raw b h))"+ −
apply (induct b)+ −
apply (simp_all add: not_memb_nil)+ −
apply (auto)+ −
apply (simp_all add: memb_delete_raw not_memb_delete_raw_ident rsp_fold_def memb_cons_iff)+ −
done+ −
+ −
lemma ffold_raw_rsp_pre:+ −
"\<forall>e. memb e a = memb e b \<Longrightarrow> ffold_raw f z a = ffold_raw f z b"+ −
apply (induct a arbitrary: b)+ −
apply (simp add: memb_absorb memb_def none_memb_nil)+ −
apply (simp)+ −
apply (rule conjI)+ −
apply (rule_tac [!] impI)+ −
apply (rule_tac [!] conjI)+ −
apply (rule_tac [!] impI)+ −
apply (subgoal_tac "\<forall>e. memb e a2 = memb e b")+ −
apply (simp)+ −
apply (simp add: memb_cons_iff memb_def)+ −
apply (auto)[1]+ −
apply (drule_tac x="e" in spec)+ −
apply (blast)+ −
apply (case_tac b)+ −
apply (simp_all)+ −
apply (subgoal_tac "ffold_raw f z b = f a1 (ffold_raw f z (delete_raw b a1))")+ −
apply (simp only:)+ −
apply (rule_tac f="f a1" in arg_cong)+ −
apply (subgoal_tac "\<forall>e. memb e a2 = memb e (delete_raw b a1)")+ −
apply (simp)+ −
apply (simp add: memb_delete_raw)+ −
apply (auto simp add: memb_cons_iff)[1]+ −
apply (erule memb_commute_ffold_raw)+ −
apply (drule_tac x="a1" in spec)+ −
apply (simp add: memb_cons_iff)+ −
apply (simp add: memb_cons_iff)+ −
apply (case_tac b)+ −
apply (simp_all)+ −
done+ −
+ −
lemma [quot_respect]:+ −
"(op = ===> op = ===> op \<approx> ===> op =) ffold_raw ffold_raw"+ −
by (simp add: memb_def[symmetric] 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 simp+ −
qed+ −
+ −
+ −
+ −
lemma concat_rsp_unfolded:+ −
"\<lbrakk>list_all2 op \<approx> a ba; ba \<approx> bb; list_all2 op \<approx> bb b\<rbrakk> \<Longrightarrow> concat a \<approx> concat b"+ −
proof -+ −
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 simp+ −
qed+ −
+ −
lemma [quot_respect]:+ −
"((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 memb_def[symmetric] memb_finter_raw)+ −
+ −
lemma sub_list_inter_right:+ −
shows "sub_list (finter_raw x y) y"+ −
by (simp add: sub_list_def memb_def[symmetric] memb_finter_raw)+ −
+ −
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 memb_def[symmetric] memb_finter_raw)+ −
+ −
lemma append_inter_distrib:+ −
"x @ (finter_raw y z) \<approx> finter_raw (x @ y) (x @ z)"+ −
apply (induct x)+ −
apply (simp_all add: memb_def)+ −
apply (simp add: memb_def[symmetric] memb_finter_raw)+ −
apply (auto simp add: memb_def)+ −
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:+ −
"(xs :: 'a fset) < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"+ −
+ −
abbreviation+ −
f_subset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)+ −
where+ −
"xs |\<subset>| ys \<equiv> xs < ys"+ −
+ −
quotient_definition+ −
"sup \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"+ −
is+ −
"(op @) \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"+ −
+ −
abbreviation+ −
funion (infixl "|\<union>|" 65)+ −
where+ −
"xs |\<union>| ys \<equiv> sup (xs :: 'a fset) ys"+ −
+ −
quotient_definition+ −
"inf \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"+ −
is+ −
"finter_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"+ −
+ −
abbreviation+ −
finter (infixl "|\<inter>|" 65)+ −
where+ −
"xs |\<inter>| ys \<equiv> inf (xs :: 'a fset) ys"+ −
+ −
quotient_definition+ −
"minus \<Colon> ('a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset)"+ −
is+ −
"fminus_raw \<Colon> ('a list \<Rightarrow> 'a list \<Rightarrow> 'a list)"+ −
+ −
instance+ −
proof+ −
fix x y z :: "'a fset"+ −
show "(x |\<subset>| y) = (x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x)"+ −
unfolding less_fset by (lifting sub_list_not_eq)+ −
show "x |\<subseteq>| x" by (lifting sub_list_refl)+ −
show "{||} |\<subseteq>| x" by (lifting sub_list_empty)+ −
show "x |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_left)+ −
show "y |\<subseteq>| x |\<union>| y" by (lifting sub_list_append_right)+ −
show "x |\<inter>| y |\<subseteq>| x" by (lifting sub_list_inter_left)+ −
show "x |\<inter>| y |\<subseteq>| y" by (lifting sub_list_inter_right)+ −
show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)" by (lifting 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 (lifting sub_list_trans)+ −
next+ −
fix x y :: "'a fset"+ −
assume a: "x |\<subseteq>| y"+ −
assume b: "y |\<subseteq>| x"+ −
show "x = y" using a b by (lifting sub_list_list_eq)+ −
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 (lifting sub_list_append)+ −
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 (lifting sub_list_inter)+ −
qed+ −
+ −
end+ −
+ −
section {* Finsert and Membership *}+ −
+ −
quotient_definition+ −
"finsert :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"+ −
is "op #"+ −
+ −
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 fset \<Rightarrow> 'a \<Rightarrow> 'a fset"+ −
is "delete_raw"+ −
+ −
quotient_definition+ −
"fset_to_set :: '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"+ −
+ −
thm fconcat_def+ −
+ −
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]:+ −
"(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) op # op #"+ −
apply auto+ −
apply (simp add: set_in_eq)+ −
apply (rule_tac b="x # b" in pred_compI)+ −
apply auto+ −
apply (rule_tac b="x # ba" in pred_compI)+ −
apply auto+ −
done+ −
+ −
lemma insert_preserve2:+ −
shows "((rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op #) =+ −
(id ---> rep_fset ---> abs_fset) op #"+ −
by (simp add: expand_fun_eq abs_o_rep[OF Quotient_fset] map_id Quotient_abs_rep[OF Quotient_fset])+ −
+ −
lemma [quot_preserve]:+ −
"(rep_fset ---> (map rep_fset \<circ> rep_fset) ---> (abs_fset \<circ> map abs_fset)) op # = finsert"+ −
by (simp add: expand_fun_eq 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: expand_fun_eq 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 add: id_simps) auto+ −
+ −
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 memb x ys then fminus_raw xs ys else x # (fminus_raw xs ys))"+ −
by (induct ys arbitrary: xs x)+ −
(simp_all add: not_memb_nil memb_delete_raw memb_cons_iff)+ −
+ −
text {* Cardinality of finite sets *}+ −
+ −
lemma fcard_raw_0:+ −
shows "fcard_raw xs = 0 \<longleftrightarrow> xs \<approx> []"+ −
by (induct xs) (auto simp add: memb_def)+ −
+ −
lemma fcard_raw_not_memb:+ −
shows "\<not> memb x xs \<longleftrightarrow> fcard_raw (x # xs) = Suc (fcard_raw xs)"+ −
by auto+ −
+ −
lemma fcard_raw_suc:+ −
assumes a: "fcard_raw xs = Suc n"+ −
shows "\<exists>x ys. \<not> (memb x ys) \<and> xs \<approx> (x # ys) \<and> fcard_raw ys = n"+ −
using a+ −
by (induct xs) (auto simp add: memb_def split: if_splits)+ −
+ −
lemma singleton_fcard_1:+ −
shows "set xs = {x} \<Longrightarrow> fcard_raw xs = 1"+ −
by (induct xs) (auto simp add: memb_def subset_insert)+ −
+ −
lemma fcard_raw_1:+ −
shows "fcard_raw xs = 1 \<longleftrightarrow> (\<exists>x. xs \<approx> [x])"+ −
apply (auto dest!: fcard_raw_suc)+ −
apply (simp add: fcard_raw_0)+ −
apply (rule_tac x="x" in exI)+ −
apply simp+ −
apply (subgoal_tac "set xs = {x}")+ −
apply (drule singleton_fcard_1)+ −
apply auto+ −
done+ −
+ −
lemma fcard_raw_suc_memb:+ −
assumes a: "fcard_raw A = Suc n"+ −
shows "\<exists>a. memb a A"+ −
using a+ −
by (induct A) (auto simp add: memb_def)+ −
+ −
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+ −
+ −
text {* fmap *}+ −
+ −
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 cons_left_comm:+ −
"x # y # xs \<approx> y # x # xs"+ −
by auto+ −
+ −
lemma cons_left_idem:+ −
"x # x # xs \<approx> x # xs"+ −
by auto+ −
+ −
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 memb_delete_raw_ident:+ −
shows "\<not> memb x (delete_raw xs x)"+ −
by (induct xs) (auto simp add: memb_def)+ −
+ −
lemma fset_raw_delete_raw_cases:+ −
"xs = [] \<or> (\<exists>x. memb x xs \<and> xs \<approx> x # delete_raw xs x)"+ −
by (induct xs) (auto simp add: memb_def)+ −
+ −
lemma fdelete_raw_filter:+ −
"delete_raw xs y = [x \<leftarrow> xs. x \<noteq> y]"+ −
by (induct xs) simp_all+ −
+ −
lemma fcard_raw_delete:+ −
"fcard_raw (delete_raw xs y) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"+ −
by (simp add: fdelete_raw_filter fcard_raw_delete_one)+ −
+ −
lemma finter_raw_empty:+ −
"finter_raw l [] = []"+ −
by (induct l) (simp_all add: not_memb_nil)+ −
+ −
lemma set_cong:+ −
shows "(x \<approx> y) = (set x = set y)"+ −
by auto+ −
+ −
lemma inj_map_eq_iff:+ −
"inj f \<Longrightarrow> (map f l \<approx> map f m) = (l \<approx> m)"+ −
by (simp add: expand_set_eq[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 # (delete_raw 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 not_memb_delete_raw_ident)+ −
done+ −
+ −
lemma memb_delete_list_eq2:+ −
assumes a: "memb e r"+ −
shows "list_eq2 (e # delete_raw r e) r"+ −
using a cons_delete_list_eq2[of e r]+ −
by simp+ −
+ −
lemma delete_raw_rsp:+ −
"xs \<approx> ys \<Longrightarrow> delete_raw xs x \<approx> delete_raw ys x"+ −
by (simp add: memb_def[symmetric] memb_delete_raw)+ −
+ −
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" by (rule fcard_raw_suc_memb)+ −
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 by auto+ −
have f: "fcard_raw (delete_raw l a) = m" using fcard_raw_delete[of l a] e d by simp+ −
have g: "delete_raw l a \<approx> delete_raw r a" using delete_raw_rsp[OF b] by simp+ −
have "list_eq2 (delete_raw l a) (delete_raw r a)" by (rule Suc.hyps[OF f g])+ −
then have h: "list_eq2 (a # delete_raw l a) (a # delete_raw r a)" by (rule list_eq2.intros(5))+ −
have i: "list_eq2 l (a # delete_raw l a)"+ −
by (rule list_eq2.intros(3)[OF memb_delete_list_eq2[OF e]])+ −
have "list_eq2 l (a # delete_raw r a)" 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 {* Set *}+ −
+ −
lemma sub_list_set: "sub_list xs ys = (set xs \<subseteq> set ys)"+ −
by (metis rev_append set_append set_cong set_rev sub_list_append sub_list_append_left sub_list_def sub_list_not_eq subset_Un_eq)+ −
+ −
lemma sub_list_neq_set: "(sub_list xs ys \<and> \<not> list_eq xs ys) = (set xs \<subset> set ys)"+ −
by (auto simp add: sub_list_set)+ −
+ −
lemma fcard_raw_set: "fcard_raw xs = card (set xs)"+ −
by (induct xs) (auto simp add: insert_absorb memb_def card_insert_disjoint finite_set)+ −
+ −
lemma memb_set: "memb x xs = (x \<in> set xs)"+ −
by (simp only: memb_def)+ −
+ −
lemma filter_set: "set (filter P xs) = P \<inter> (set xs)"+ −
by (induct xs, simp)+ −
(metis Int_insert_right_if0 Int_insert_right_if1 List.set.simps(2) filter.simps(2) mem_def)+ −
+ −
lemma delete_raw_set: "set (delete_raw xs x) = set xs - {x}"+ −
by (induct xs) auto+ −
+ −
lemma inter_raw_set: "set (finter_raw xs ys) = set xs \<inter> set ys"+ −
by (induct xs) (simp_all add: memb_def)+ −
+ −
lemma fminus_raw_set: "set (fminus_raw xs ys) = set xs - set ys"+ −
by (induct ys arbitrary: xs)+ −
(simp_all add: fminus_raw.simps delete_raw_set, blast)+ −
+ −
text {* Raw theorems of ffilter *}+ −
+ −
lemma sub_list_filter: "sub_list (filter P xs) (filter Q xs) = (\<forall> x. memb x xs \<longrightarrow> P x \<longrightarrow> Q x)"+ −
unfolding sub_list_def memb_def by auto+ −
+ −
lemma list_eq_filter: "list_eq (filter P xs) (filter Q xs) = (\<forall>x. memb x xs \<longrightarrow> P x = Q x)"+ −
unfolding memb_def by auto+ −
+ −
text {* Lifted theorems *}+ −
+ −
lemma not_fin_fnil: "x |\<notin>| {||}"+ −
by (lifting not_memb_nil)+ −
+ −
lemma fin_finsert_iff[simp]:+ −
"x |\<in>| finsert y S = (x = y \<or> x |\<in>| S)"+ −
by (lifting memb_cons_iff)+ −
+ −
lemma+ −
shows finsertI1: "x |\<in>| finsert x S"+ −
and finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"+ −
by (lifting memb_consI1, lifting memb_consI2)+ −
+ −
lemma finsert_absorb[simp]:+ −
shows "x |\<in>| S \<Longrightarrow> finsert x S = S"+ −
by (lifting memb_absorb)+ −
+ −
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 (lifting cons_left_comm)+ −
+ −
lemma finsert_left_idem:+ −
"finsert x (finsert x S) = finsert x S"+ −
by (lifting cons_left_idem)+ −
+ −
lemma fsingleton_eq[simp]:+ −
shows "{|x|} = {|y|} \<longleftrightarrow> x = y"+ −
by (lifting singleton_list_eq)+ −
+ −
text {* fset_to_set *}+ −
+ −
lemma fset_to_set_simps[simp]:+ −
"fset_to_set {||} = ({} :: 'a set)"+ −
"fset_to_set (finsert (h :: 'a) t) = insert h (fset_to_set t)"+ −
by (lifting set.simps)+ −
+ −
lemma in_fset_to_set:+ −
"x \<in> fset_to_set 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_to_set S = fset_to_set T)"+ −
by (lifting set_cong)+ −
+ −
text {* fcard *}+ −
+ −
lemma fcard_fempty [simp]:+ −
shows "fcard {||} = 0"+ −
by (lifting fcard_raw_nil)+ −
+ −
lemma fcard_finsert_if [simp]:+ −
shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"+ −
by (lifting fcard_raw_cons)+ −
+ −
lemma fcard_0: "(fcard S = 0) = (S = {||})"+ −
by (lifting fcard_raw_0)+ −
+ −
lemma fcard_1:+ −
shows "(fcard S = 1) = (\<exists>x. S = {|x|})"+ −
by (lifting fcard_raw_1)+ −
+ −
lemma fcard_gt_0:+ −
shows "x \<in> fset_to_set S \<Longrightarrow> 0 < fcard S"+ −
by (lifting fcard_raw_gt_0)+ −
+ −
lemma fcard_not_fin:+ −
shows "(x |\<notin>| S) = (fcard (finsert x S) = Suc (fcard S))"+ −
by (lifting fcard_raw_not_memb)+ −
+ −
lemma fcard_suc: "fcard S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = finsert x T \<and> fcard T = n"+ −
by (lifting fcard_raw_suc)+ −
+ −
lemma fcard_delete:+ −
"fcard (fdelete S y) = (if y |\<in>| S then fcard S - 1 else fcard S)"+ −
by (lifting fcard_raw_delete)+ −
+ −
lemma fcard_suc_memb: "fcard A = Suc n \<Longrightarrow> \<exists>a. a |\<in>| A"+ −
by (lifting fcard_raw_suc_memb)+ −
+ −
lemma fin_fcard_not_0: "a |\<in>| A \<Longrightarrow> fcard A \<noteq> 0"+ −
by (lifting memb_card_not_0)+ −
+ −
text {* 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:+ −
"{|a|} |\<union>| S = finsert a S"+ −
by simp+ −
+ −
lemma singleton_union_right:+ −
"S |\<union>| {|a|} = finsert a S"+ −
by (subst sup.commute) simp+ −
+ −
section {* Induction and Cases rules for finite sets *}+ −
+ −
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+ −
+ −
text {* fmap *}+ −
+ −
lemma fmap_simps[simp]:+ −
"fmap (f :: 'a \<Rightarrow> 'b) {||} = {||}"+ −
"fmap f (finsert x S) = finsert (f x) (fmap f S)"+ −
by (lifting map.simps)+ −
+ −
lemma fmap_set_image:+ −
"fset_to_set (fmap f S) = f ` (fset_to_set S)"+ −
by (induct S) simp_all+ −
+ −
lemma inj_fmap_eq_iff:+ −
"inj f \<Longrightarrow> (fmap f S = fmap f T) = (S = T)"+ −
by (lifting inj_map_eq_iff)+ −
+ −
lemma fmap_funion: "fmap f (S |\<union>| T) = fmap f S |\<union>| fmap f T"+ −
by (lifting map_append)+ −
+ −
lemma fin_funion:+ −
"x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"+ −
by (lifting memb_append)+ −
+ −
text {* to_set *}+ −
+ −
lemma fin_set: "(x |\<in>| xs) = (x \<in> fset_to_set xs)"+ −
by (lifting memb_set)+ −
+ −
lemma fnotin_set: "(x |\<notin>| xs) = (x \<notin> fset_to_set xs)"+ −
by (simp add: fin_set)+ −
+ −
lemma fcard_set: "fcard xs = card (fset_to_set xs)"+ −
by (lifting fcard_raw_set)+ −
+ −
lemma fsubseteq_set: "(xs |\<subseteq>| ys) = (fset_to_set xs \<subseteq> fset_to_set ys)"+ −
by (lifting sub_list_set)+ −
+ −
lemma fsubset_set: "(xs |\<subset>| ys) = (fset_to_set xs \<subset> fset_to_set ys)"+ −
unfolding less_fset by (lifting sub_list_neq_set)+ −
+ −
lemma ffilter_set: "fset_to_set (ffilter P xs) = P \<inter> fset_to_set xs"+ −
by (lifting filter_set)+ −
+ −
lemma fdelete_set: "fset_to_set (fdelete xs x) = fset_to_set xs - {x}"+ −
by (lifting delete_raw_set)+ −
+ −
lemma inter_set: "fset_to_set (xs |\<inter>| ys) = fset_to_set xs \<inter> fset_to_set ys"+ −
by (lifting inter_raw_set)+ −
+ −
lemma union_set: "fset_to_set (xs |\<union>| ys) = fset_to_set xs \<union> fset_to_set ys"+ −
by (lifting set_append)+ −
+ −
lemma fminus_set: "fset_to_set (xs - ys) = fset_to_set xs - fset_to_set ys"+ −
by (lifting fminus_raw_set)+ −
+ −
lemmas fset_to_set_trans =+ −
fin_set fnotin_set fcard_set fsubseteq_set fsubset_set+ −
inter_set union_set ffilter_set fset_to_set_simps+ −
fset_cong fdelete_set fmap_set_image fminus_set+ −
+ −
+ −
text {* ffold *}+ −
+ −
lemma ffold_nil: "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 (lifting ffold_raw.simps(2)[where 'a="'b" and 'b="'a"])+ −
+ −
lemma fin_commute_ffold:+ −
"\<lbrakk>rsp_fold f; h |\<in>| b\<rbrakk> \<Longrightarrow> ffold f z b = f h (ffold f z (fdelete b h))"+ −
by (lifting memb_commute_ffold_raw)+ −
+ −
text {* fdelete *}+ −
+ −
lemma fin_fdelete:+ −
shows "x |\<in>| fdelete S y \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"+ −
by (lifting memb_delete_raw)+ −
+ −
lemma fin_fdelete_ident:+ −
shows "x |\<notin>| fdelete S x"+ −
by (lifting memb_delete_raw_ident)+ −
+ −
lemma not_memb_fdelete_ident:+ −
shows "x |\<notin>| S \<Longrightarrow> fdelete S x = S"+ −
by (lifting not_memb_delete_raw_ident)+ −
+ −
lemma fset_fdelete_cases:+ −
shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = finsert x (fdelete S x))"+ −
by (lifting fset_raw_delete_raw_cases)+ −
+ −
text {* inter *}+ −
+ −
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:+ −
"finsert x S |\<inter>| T = (if x |\<in>| T then finsert x (S |\<inter>| T) else S |\<inter>| T)"+ −
by (lifting finter_raw.simps(2))+ −
+ −
lemma fin_finter:+ −
"x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"+ −
by (lifting memb_finter_raw)+ −
+ −
lemma fsubset_finsert:+ −
"(finsert x xs |\<subseteq>| ys) = (x |\<in>| ys \<and> xs |\<subseteq>| ys)"+ −
by (lifting sub_list_cons)+ −
+ −
lemma "xs |\<subseteq>| ys \<equiv> \<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys"+ −
by (lifting sub_list_def[simplified memb_def[symmetric]])+ −
+ −
lemma fsubset_fin: "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"+ −
by (rule meta_eq_to_obj_eq)+ −
(lifting sub_list_def[simplified memb_def[symmetric]])+ −
+ −
lemma fminus_fin: "(x |\<in>| xs - ys) = (x |\<in>| xs \<and> x |\<notin>| ys)"+ −
by (lifting fminus_raw_memb)+ −
+ −
lemma fminus_red: "finsert x xs - ys = (if x |\<in>| ys then xs - ys else finsert x (xs - ys))"+ −
by (lifting fminus_raw_red)+ −
+ −
lemma fminus_red_fin[simp]: "x |\<in>| ys \<Longrightarrow> finsert x xs - ys = xs - ys"+ −
by (simp add: fminus_red)+ −
+ −
lemma fminus_red_fnotin[simp]: "x |\<notin>| ys \<Longrightarrow> finsert x xs - ys = finsert x (xs - ys)"+ −
by (simp add: fminus_red)+ −
+ −
lemma expand_fset_eq:+ −
"(S = T) = (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"+ −
by (lifting list_eq.simps[simplified memb_def[symmetric]])+ −
+ −
(* 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]])+ −
+ −
text {* concat *}+ −
+ −
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 "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"+ −
by (lifting concat_append)+ −
+ −
text {* ffilter *}+ −
+ −
lemma subseteq_filter: "ffilter P xs <= ffilter Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"+ −
by (lifting sub_list_filter)+ −
+ −
lemma eq_ffilter: "(ffilter P xs = ffilter Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"+ −
by (lifting list_eq_filter)+ −
+ −
lemma subset_ffilter: "(\<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 by (auto simp add: subseteq_filter eq_ffilter)+ −
+ −
section {* lemmas transferred from Finite_Set theory *}+ −
+ −
text {* finiteness for finite sets holds *}+ −
lemma finite_fset: "finite (fset_to_set S)"+ −
by (induct S) auto+ −
+ −
lemma fset_choice: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y) \<Longrightarrow> \<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"+ −
unfolding fset_to_set_trans+ −
by (rule finite_set_choice[simplified Ball_def, OF finite_fset])+ −
+ −
lemma fsubseteq_fnil: "xs |\<subseteq>| {||} = (xs = {||})"+ −
unfolding fset_to_set_trans+ −
by (rule subset_empty)+ −
+ −
lemma not_fsubset_fnil: "\<not> xs |\<subset>| {||}"+ −
unfolding fset_to_set_trans+ −
by (rule not_psubset_empty)+ −
+ −
lemma fcard_mono: "xs |\<subseteq>| ys \<Longrightarrow> fcard xs \<le> fcard ys"+ −
unfolding fset_to_set_trans+ −
by (rule card_mono[OF finite_fset])+ −
+ −
lemma fcard_fseteq: "xs |\<subseteq>| ys \<Longrightarrow> fcard ys \<le> fcard xs \<Longrightarrow> xs = ys"+ −
unfolding fset_to_set_trans+ −
by (rule card_seteq[OF finite_fset])+ −
+ −
lemma psubset_fcard_mono: "xs |\<subset>| ys \<Longrightarrow> fcard xs < fcard ys"+ −
unfolding fset_to_set_trans+ −
by (rule psubset_card_mono[OF finite_fset])+ −
+ −
lemma fcard_funion_finter: "fcard xs + fcard ys = fcard (xs |\<union>| ys) + fcard (xs |\<inter>| ys)"+ −
unfolding fset_to_set_trans+ −
by (rule card_Un_Int[OF finite_fset finite_fset])+ −
+ −
lemma fcard_funion_disjoint: "xs |\<inter>| ys = {||} \<Longrightarrow> fcard (xs |\<union>| ys) = fcard xs + fcard ys"+ −
unfolding fset_to_set_trans+ −
by (rule card_Un_disjoint[OF finite_fset finite_fset])+ −
+ −
lemma fcard_delete1_less: "x |\<in>| xs \<Longrightarrow> fcard (fdelete xs x) < fcard xs"+ −
unfolding fset_to_set_trans+ −
by (rule card_Diff1_less[OF finite_fset])+ −
+ −
lemma fcard_delete2_less: "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> fcard (fdelete (fdelete xs x) y) < fcard xs"+ −
unfolding fset_to_set_trans+ −
by (rule card_Diff2_less[OF finite_fset])+ −
+ −
lemma fcard_delete1_le: "fcard (fdelete xs x) <= fcard xs"+ −
unfolding fset_to_set_trans+ −
by (rule card_Diff1_le[OF finite_fset])+ −
+ −
lemma fcard_psubset: "ys |\<subseteq>| xs \<Longrightarrow> fcard ys < fcard xs \<Longrightarrow> ys |\<subset>| xs"+ −
unfolding fset_to_set_trans+ −
by (rule card_psubset[OF finite_fset])+ −
+ −
lemma fcard_fmap_le: "fcard (fmap f xs) \<le> fcard xs"+ −
unfolding fset_to_set_trans+ −
by (rule card_image_le[OF finite_fset])+ −
+ −
lemma fin_fminus_fnotin: "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"+ −
unfolding fset_to_set_trans+ −
by blast+ −
+ −
lemma fin_fnotin_fminus: "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"+ −
unfolding fset_to_set_trans+ −
by blast+ −
+ −
lemma fin_mdef: "x |\<in>| F = ((x |\<notin>| (F - {|x|})) & (F = finsert x (F - {|x|})))"+ −
unfolding fset_to_set_trans+ −
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 fset_to_set_trans+ −
by (rule 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 fset_to_set_trans+ −
by (rule card_Diff_subset[OF finite_fset])+ −
+ −
lemma fcard_fminus_subset_finter:+ −
"fcard (A - B) = fcard A - fcard (A |\<inter>| B)"+ −
unfolding fset_to_set_trans+ −
by (rule card_Diff_subset_Int) (fold inter_set, rule finite_fset)+ −
+ −
+ −
ML {*+ −
fun dest_fsetT (Type (@{type_name fset}, [T])) = T+ −
| dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);+ −
*}+ −
+ −
ML {*+ −
open Quotient_Info;+ −
+ −
exception LIFT_MATCH of string+ −
+ −
+ −
+ −
(*** Aggregate Rep/Abs Function ***)+ −
+ −
+ −
(* The flag RepF is for types in negative position; AbsF is for types+ −
in positive position. Because of this, function types need to be+ −
treated specially, since there the polarity changes.+ −
*)+ −
+ −
datatype flag = AbsF | RepF+ −
+ −
fun negF AbsF = RepF+ −
| negF RepF = AbsF+ −
+ −
fun is_identity (Const (@{const_name "id"}, _)) = true+ −
| is_identity _ = false+ −
+ −
fun mk_identity ty = Const (@{const_name "id"}, ty --> ty)+ −
+ −
fun mk_fun_compose flag (trm1, trm2) =+ −
case flag of+ −
AbsF => Const (@{const_name "comp"}, dummyT) $ trm1 $ trm2+ −
| RepF => Const (@{const_name "comp"}, dummyT) $ trm2 $ trm1+ −
+ −
fun get_mapfun ctxt s =+ −
let+ −
val thy = ProofContext.theory_of ctxt+ −
val exn = LIFT_MATCH ("No map function for type " ^ quote s ^ " found.")+ −
val mapfun = #mapfun (maps_lookup thy s) handle Quotient_Info.NotFound => raise exn+ −
in+ −
Const (mapfun, dummyT)+ −
end+ −
+ −
(* makes a Free out of a TVar *)+ −
fun mk_Free (TVar ((x, i), _)) = Free (unprefix "'" x ^ string_of_int i, dummyT)+ −
+ −
(* produces an aggregate map function for the+ −
rty-part of a quotient definition; abstracts+ −
over all variables listed in vs (these variables+ −
correspond to the type variables in rty)+ −
+ −
for example for: (?'a list * ?'b)+ −
it produces: %a b. prod_map (map a) b+ −
*)+ −
fun mk_mapfun ctxt vs rty =+ −
let+ −
val vs' = map (mk_Free) vs+ −
+ −
fun mk_mapfun_aux rty =+ −
case rty of+ −
TVar _ => mk_Free rty+ −
| Type (_, []) => mk_identity rty+ −
| Type (s, tys) => list_comb (get_mapfun ctxt s, map mk_mapfun_aux tys)+ −
| _ => raise LIFT_MATCH "mk_mapfun (default)"+ −
in+ −
fold_rev Term.lambda vs' (mk_mapfun_aux rty)+ −
end+ −
+ −
(* looks up the (varified) rty and qty for+ −
a quotient definition+ −
*)+ −
fun get_rty_qty ctxt s =+ −
let+ −
val thy = ProofContext.theory_of ctxt+ −
val exn = LIFT_MATCH ("No quotient type " ^ quote s ^ " found.")+ −
val qdata = (quotdata_lookup thy s) handle Quotient_Info.NotFound => raise exn+ −
in+ −
(#rtyp qdata, #qtyp qdata)+ −
end+ −
+ −
(* takes two type-environments and looks+ −
up in both of them the variable v, which+ −
must be listed in the environment+ −
*)+ −
fun double_lookup rtyenv qtyenv v =+ −
let+ −
val v' = fst (dest_TVar v)+ −
in+ −
(snd (the (Vartab.lookup rtyenv v')), snd (the (Vartab.lookup qtyenv v')))+ −
end+ −
+ −
(* matches a type pattern with a type *)+ −
fun match ctxt err ty_pat ty =+ −
let+ −
val thy = ProofContext.theory_of ctxt+ −
in+ −
Sign.typ_match thy (ty_pat, ty) Vartab.empty+ −
handle MATCH_TYPE => err ctxt ty_pat ty+ −
end+ −
+ −
(* produces the rep or abs constant for a qty *)+ −
fun absrep_const flag ctxt qty_str =+ −
let+ −
val qty_name = Long_Name.base_name qty_str+ −
val qualifier = Long_Name.qualifier qty_str+ −
in+ −
case flag of+ −
AbsF => Const (Long_Name.qualify qualifier ("abs_" ^ qty_name), dummyT)+ −
| RepF => Const (Long_Name.qualify qualifier ("rep_" ^ qty_name), dummyT)+ −
end+ −
+ −
(* Lets Nitpick represent elements of quotient types as elements of the raw type *)+ −
fun absrep_const_chk flag ctxt qty_str =+ −
Syntax.check_term ctxt (absrep_const flag ctxt qty_str)+ −
+ −
fun absrep_match_err ctxt ty_pat ty =+ −
let+ −
val ty_pat_str = Syntax.string_of_typ ctxt ty_pat+ −
val ty_str = Syntax.string_of_typ ctxt ty+ −
in+ −
raise LIFT_MATCH (space_implode " "+ −
["absrep_fun (Types ", quote ty_pat_str, "and", quote ty_str, " do not match.)"])+ −
end+ −
+ −
+ −
(** generation of an aggregate absrep function **)+ −
+ −
(* - In case of equal types we just return the identity.+ −
+ −
- In case of TFrees we also return the identity.+ −
+ −
- In case of function types we recurse taking+ −
the polarity change into account.+ −
+ −
- If the type constructors are equal, we recurse for the+ −
arguments and build the appropriate map function.+ −
+ −
- If the type constructors are unequal, there must be an+ −
instance of quotient types:+ −
+ −
- we first look up the corresponding rty_pat and qty_pat+ −
from the quotient definition; the arguments of qty_pat+ −
must be some distinct TVars+ −
- we then match the rty_pat with rty and qty_pat with qty;+ −
if matching fails the types do not correspond -> error+ −
- the matching produces two environments; we look up the+ −
assignments for the qty_pat variables and recurse on the+ −
assignments+ −
- we prefix the aggregate map function for the rty_pat,+ −
which is an abstraction over all type variables+ −
- finally we compose the result with the appropriate+ −
absrep function in case at least one argument produced+ −
a non-identity function /+ −
otherwise we just return the appropriate absrep+ −
function+ −
+ −
The composition is necessary for types like+ −
+ −
('a list) list / ('a foo) foo+ −
+ −
The matching is necessary for types like+ −
+ −
('a * 'a) list / 'a bar+ −
+ −
The test is necessary in order to eliminate superfluous+ −
identity maps.+ −
*)+ −
+ −
fun absrep_fun flag ctxt (rty, qty) =+ −
if rty = qty+ −
then mk_identity rty+ −
else+ −
case (rty, qty) of+ −
(Type ("fun", [ty1, ty2]), Type ("fun", [ty1', ty2'])) =>+ −
let+ −
val arg1 = absrep_fun (negF flag) ctxt (ty1, ty1')+ −
val arg2 = absrep_fun flag ctxt (ty2, ty2')+ −
in+ −
list_comb (get_mapfun ctxt "fun", [arg1, arg2])+ −
end+ −
| (Type (s, tys), Type (s', tys')) =>+ −
if s = s'+ −
then+ −
let+ −
val args = map (absrep_fun flag ctxt) (tys ~~ tys')+ −
in+ −
list_comb (get_mapfun ctxt s, args)+ −
end+ −
else+ −
let+ −
val (rty_pat, qty_pat as Type (_, vs)) = get_rty_qty ctxt s'+ −
val rtyenv = match ctxt absrep_match_err rty_pat rty+ −
val qtyenv = match ctxt absrep_match_err qty_pat qty+ −
val args_aux = map (double_lookup rtyenv qtyenv) vs+ −
val args = map (absrep_fun flag ctxt) args_aux+ −
val map_fun = mk_mapfun ctxt vs rty_pat+ −
val result = list_comb (map_fun, args)+ −
in+ −
(*if forall is_identity args+ −
then absrep_const flag ctxt s'+ −
else*) mk_fun_compose flag (absrep_const flag ctxt s', result)+ −
end+ −
| (TFree x, TFree x') =>+ −
if x = x'+ −
then mk_identity rty+ −
else raise (LIFT_MATCH "absrep_fun (frees)")+ −
| (TVar _, TVar _) => raise (LIFT_MATCH "absrep_fun (vars)")+ −
| _ => raise (LIFT_MATCH "absrep_fun (default)")+ −
+ −
+ −
*}+ −
+ −
ML {*+ −
let+ −
val parser = Args.context -- Scan.lift Args.name_source+ −
fun typ_pat (ctxt, str) =+ −
str |> Syntax.parse_typ ctxt+ −
|> ML_Syntax.print_typ+ −
|> ML_Syntax.atomic+ −
in+ −
ML_Antiquote.inline "typ_pat" (parser >> typ_pat)+ −
end+ −
*}+ −
+ −
ML {*+ −
mk_mapfun @{context} [@{typ_pat "?'a"}] @{typ_pat "(?'a list) * nat"}+ −
|> Syntax.check_term @{context}+ −
|> fastype_of+ −
|> Syntax.string_of_typ @{context}+ −
|> tracing+ −
*}+ −
+ −
ML {*+ −
mk_mapfun @{context} [@{typ_pat "?'a"}] @{typ_pat "(?'a list) * nat"}+ −
|> Syntax.check_term @{context}+ −
|> Syntax.string_of_term @{context}+ −
|> warning+ −
*}+ −
+ −
ML {*+ −
mk_mapfun @{context} [@{typ_pat "?'a"}] @{typ_pat "(?'a list) * nat"}+ −
|> Syntax.check_term @{context}+ −
*}+ −
+ −
term prod_fun+ −
term map+ −
term fun_map+ −
term "op o"+ −
+ −
ML {*+ −
absrep_fun AbsF @{context} (@{typ "('a list) list \<Rightarrow> 'a list"}, @{typ "('a fset) fset \<Rightarrow> 'a fset"})+ −
|> Syntax.string_of_term @{context}+ −
|> writeln+ −
*}+ −
+ −
lemma "(\<lambda> (c::'s \<Rightarrow> bool). \<exists>(x::'s). c = rel x) = {c. \<exists>x. c = rel x}"+ −
apply(auto simp add: mem_def)+ −
done+ −
+ −
lemma ball_reg_right_unfolded: "(\<forall>x. R x \<longrightarrow> P x \<longrightarrow> Q x) \<longrightarrow> (All P \<longrightarrow> Ball R Q)"+ −
apply rule+ −
apply (rule ball_reg_right)+ −
apply auto+ −
done+ −
+ −
lemma list_all2_refl:+ −
assumes q: "equivp R"+ −
shows "(list_all2 R) r r"+ −
by (rule list_all2_refl) (metis equivp_def fset_equivp 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+ −
+ −
no_notation+ −
list_eq (infix "\<approx>" 50)+ −
+ −
+ −
end+ −