
diff -r 8f6e68dc6cbc -r 266abc3ee228 Attic/Quot/Examples/FSet3.thy
--- 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) *}




