--- a/Attic/Quot/Examples/FSet3.thy Thu Apr 22 06:44:24 2010 +0200
+++ b/Attic/Quot/Examples/FSet3.thy Thu Apr 22 12:33:51 2010 +0200
@@ -2,246 +2,6 @@
imports "../../../Nominal/FSet"
begin
-notation
- list_eq (infix "\<approx>" 50)
-
-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 list_rel_find_element:
- assumes a: "x \<in> set a"
- and b: "list_rel R a b"
- shows "\<exists>y. (y \<in> set b \<and> R x y)"
-proof -
- have "length a = length b" using b by (rule list_rel_len)
- then show ?thesis using a b proof (induct a b rule: list_induct2)
- case Nil
- show ?case using Nil.prems by simp
- next
- case (Cons x xs y ys)
- show ?case using Cons by auto
- qed
-qed
-
-lemma concat_rsp_pre:
- assumes a: "list_rel op \<approx> x x'"
- and b: "x' \<approx> y'"
- and c: "list_rel 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_rel_find_element[OF e a])
- then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
- have j: "ya \<in> set y'" using b h by simp
- have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" by (rule list_rel_find_element[OF j c])
- then show ?thesis using f i by auto
-qed
-
-lemma fun_relI [intro]:
- assumes "\<And>a b. P a b \<Longrightarrow> Q (x a) (y b)"
- shows "(P ===> Q) x y"
- using assms by (simp add: fun_rel_def)
-
-lemma [quot_respect]:
- shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
-proof (rule fun_relI, elim pred_compE)
- fix a b ba bb
- assume a: "list_rel op \<approx> a ba"
- assume b: "ba \<approx> bb"
- assume c: "list_rel 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_rel op \<approx> ba a" by (rule list_rel_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_rel op \<approx> b bb" by (rule list_rel_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 nil_rsp2[quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
- by (metis nil_rsp list_rel.simps(1) pred_compI)
-
-lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"
- 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 compose_list_refl:
- shows "(list_rel op \<approx> OOO op \<approx>) r r"
-proof
- show c: "list_rel op \<approx> r r" by (rule list_rel_refl) (metis equivp_def fset_equivp)
- have d: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
- show b: "(op \<approx> OO list_rel op \<approx>) r r" by (rule pred_compI) (rule d, rule c)
-qed
-
-lemma list_rel_refl:
- shows "(list_rel op \<approx>) r r"
- by (rule list_rel_refl)(metis equivp_def fset_equivp)
-
-lemma Quotient_fset_list:
- shows "Quotient (list_rel op \<approx>) (map abs_fset) (map rep_fset)"
- by (fact list_quotient[OF Quotient_fset])
-
-lemma quotient_compose_list[quot_thm]:
- shows "Quotient ((list_rel 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_rel op \<approx> (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
- by (rule list_rel_refl)
- have c: "(op \<approx> OO list_rel 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_rel op \<approx> OOO op \<approx>) (map rep_fset (rep_fset a)) (map rep_fset (rep_fset a))"
- by (rule, rule list_rel_refl) (rule c)
- show "(list_rel op \<approx> OOO op \<approx>) r s = ((list_rel op \<approx> OOO op \<approx>) r r \<and>
- (list_rel 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_rel op \<approx> OOO op \<approx>) r r" by (rule compose_list_refl)
- show "(list_rel op \<approx> OOO op \<approx>) s s" by (rule compose_list_refl)
- next
- assume a: "(list_rel 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_rel op \<approx> r b"
- assume d: "b \<approx> ba"
- assume e: "list_rel op \<approx> ba s"
- have f: "map abs_fset r = map abs_fset b"
- by (metis Quotient_rel[OF Quotient_fset_list] c)
- have g: "map abs_fset s = map abs_fset ba"
- by (metis Quotient_rel[OF Quotient_fset_list] e)
- show "map abs_fset r \<approx> map abs_fset s" using d f g map_rel_cong by simp
- qed
- then show "abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
- by (metis Quotient_rel[OF Quotient_fset])
- next
- assume a: "(list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s
- \<and> abs_fset (map abs_fset r) = abs_fset (map abs_fset s)"
- then have s: "(list_rel 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_rel op \<approx> (map rep_fset (map abs_fset s)) s"
- by (fact rep_abs_rsp_left[OF Quotient_fset_list, OF list_rel_refl[of s]])
- have c: "(op \<approx> OO list_rel op \<approx>) (map rep_fset (map abs_fset r)) s"
- by (rule pred_compI) (rule b, rule y)
- have z: "list_rel op \<approx> r (map rep_fset (map abs_fset r))"
- by (fact rep_abs_rsp[OF Quotient_fset_list, OF list_rel_refl[of r]])
- then show "(list_rel op \<approx> OOO op \<approx>) r s"
- using a c pred_compI by simp
- qed
-qed
-
-lemma nil_prs2[quot_preserve]: "(abs_fset \<circ> map f) [] = abs_fset []"
- by simp
-
-lemma fconcat_empty:
- shows "fconcat {||} = {||}"
- by (lifting concat.simps(1))
-
-lemma insert_rsp2[quot_respect]:
- "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel 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 append_rsp[quot_respect]:
- "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
- by (auto)
-
-lemma insert_prs2[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 fconcat_insert:
- shows "fconcat (finsert x S) = x |\<union>| fconcat S"
- by (lifting concat.simps(2))
-
-lemma append_prs2[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_rel_app_l:
- assumes a: "reflp R"
- and b: "list_rel R l r"
- shows "list_rel R (z @ l) (z @ r)"
- by (induct z) (simp_all add: b, metis a reflp_def)
-
-lemma append_rsp2_pre0:
- assumes a:"list_rel op \<approx> x x'"
- shows "list_rel op \<approx> (x @ z) (x' @ z)"
- using a apply (induct x x' rule: list_induct2')
- apply simp_all
- apply (rule list_rel_refl)
- done
-
-lemma append_rsp2_pre1:
- assumes a:"list_rel op \<approx> x x'"
- shows "list_rel op \<approx> (z @ x) (z @ x')"
- using a apply (induct x x' arbitrary: z rule: list_induct2')
- apply (rule list_rel_refl)
- apply (simp_all del: list_eq.simps)
- apply (rule list_rel_app_l)
- apply (simp_all add: reflp_def)
- done
-
-lemma append_rsp2_pre:
- assumes a:"list_rel op \<approx> x x'"
- and b: "list_rel op \<approx> z z'"
- shows "list_rel op \<approx> (x @ z) (x' @ z')"
- apply (rule list_rel_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_rel_refl)
- apply simp_all
- apply (rule append_rsp2_pre1)
- apply simp
- done
-
-lemma append_rsp2[quot_respect]:
- "(list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel 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_rel op \<approx> x x'"
- and b: "x' \<approx> y'"
- and c: "list_rel op \<approx> y' y"
- assume aa: "list_rel op \<approx> z z'"
- and bb: "z' \<approx> w'"
- and cc: "list_rel op \<approx> w' w"
- have a': "list_rel 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_rel op \<approx> (y' @ w') (y @ w)" using c cc append_rsp2_pre by auto
- have d': "(op \<approx> OO list_rel op \<approx>) (x' @ z') (y @ w)"
- by (rule pred_compI) (rule b', rule c')
- show "(list_rel op \<approx> OOO op \<approx>) (x @ z) (y @ w)"
- by (rule pred_compI) (rule a', rule d')
-qed
-
-lemma "fconcat (xs |\<union>| ys) = fconcat xs |\<union>| fconcat ys"
- by (lifting concat_append)
-
(* TBD *)
text {* syntax for fset comprehensions (adapted from lists) *}