--- a/Attic/Quot/Examples/FSet3.thy	Wed Apr 21 12:37:44 2010 +0200
+++ b/Attic/Quot/Examples/FSet3.thy	Wed Apr 21 12:38:20 2010 +0200
@@ -1,291 +1,73 @@
 theory FSet3
-imports "../Quotient" "../Quotient_List" List
+imports "../../../Nominal/FSet"
 begin
 
-ML {*
-structure QuotientRules = Named_Thms
-  (val name = "quot_thm"
-   val description = "Quotient theorems.")
-*}
-
-ML {*
-open QuotientRules
-*}
-
-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 reflp_def symp_def transp_def
-by auto
-
-(* FIXME-TODO: because of beta-reduction, one cannot give the *)
-(* FIXME-TODO: relation as a term or abbreviation             *) 
-quotient_type 
-  'a fset = "'a list" / "list_eq"
-by (rule list_eq_equivp)
-
-
-section {* empty fset, finsert and membership *} 
-
-quotient_definition
-  fempty  ("{||}")
-where
-  "fempty :: 'a fset"
-is "[]::'a list"
-
-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 {||}"
-
-definition 
-  memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"
-where
-  "memb x xs \<equiv> x \<in> set xs"
-
-quotient_definition
-  fin ("_ |\<in>| _" [50, 51] 50)
-where
-  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool"
-is "memb"
-
-abbreviation
-  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" ("_ |\<notin>| _" [50, 51] 50)
-where
-  "a |\<notin>| S \<equiv> \<not>(a |\<in>| S)"
-
-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:
-  "xa \<approx> ya \<Longrightarrow> y # xa \<approx> y # ya"
-  by simp
-
-lemma [quot_respect]:
-  shows "(op = ===> op \<approx> ===> op \<approx>) op # op #"
-  by simp
-
-
-section {* Augmenting a set -- @{const finsert} *}
-
-text {* raw section *}
-
-lemma nil_not_cons:
-  shows "\<not>[] \<approx> x # xs"
-  by auto
-
-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_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 memb_absorb:
-  shows "memb x xs \<Longrightarrow> x # xs \<approx> xs"
-  by (induct xs) (auto simp add: memb_def id_simps)
-
-text {* lifted section *}
-
-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)
-
-
-section {* Singletons *}
-
-text {* raw section *}
-
-lemma singleton_list_eq:
-  shows "[x] \<approx> [y] \<longleftrightarrow> x = y"
-  by (simp add: id_simps) auto
-
-text {* lifted section *}
-
-lemma fempty_not_finsert[simp]:
-  shows "{||} \<noteq> finsert x S"
-  by (lifting nil_not_cons)
-
-lemma fsingleton_eq[simp]:
-  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
-  by (lifting singleton_list_eq)
-
-section {* Union *}
-
-quotient_definition
-  funion  (infixl "|\<union>|" 65)
-where
-  "funion :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-is
-  "op @"
-
-section {* Cardinality of finite sets *}
-
-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))"
-
-quotient_definition
-  "fcard :: 'a fset \<Rightarrow> nat" 
-is "fcard_raw"
-
-text {* raw section *}
-
-lemma fcard_raw_ge_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:
-  "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_ge_0 simp add: memb_def)
-
-lemma fcard_raw_rsp_aux:
-  assumes a: "a \<approx> b"
-  shows "fcard_raw a = fcard_raw b"
-using a
-apply(induct a arbitrary: b)
-apply(auto simp add: memb_def)
-apply(metis)
-apply(drule_tac x="[x \<leftarrow> b. x \<noteq> a1]" in meta_spec)
-apply(simp add: fcard_raw_delete_one)
-apply(metis Suc_pred' fcard_raw_ge_0 fcard_raw_delete_one memb_def)
-done
-
-lemma fcard_raw_rsp[quot_respect]:
-  "(op \<approx> ===> op =) fcard_raw fcard_raw"
-  by (simp add: fcard_raw_rsp_aux)
-
-text {* lifted section *}
-
-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)
-
-
-section {* Induction and Cases rules for finite sets *}
+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 fset_induct[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_induct2[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)
-  case fempty
-  show "P {||}" by (rule prem1)
-next
-  case (finsert x S)
-  have asm: "P S" by fact
-  show "P (finsert x S)"
-  proof(cases "x |\<in>| S")
-    case True
-    have "x |\<in>| S" by fact
-    then show "P (finsert x S)" using asm by simp
+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 False
-    have "x |\<notin>| S" by fact
-    then show "P (finsert x S)" using prem2 asm by simp
+    case (Cons x xs y ys)
+    show ?case using Cons by auto
   qed
 qed
 
-
-section {* fmap and fset comprehension *}
-
-quotient_definition
-  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
-is
- "map"
-
-quotient_definition
-  "fconcat :: ('a fset) fset => 'a fset"
-is
- "concat"
-
-(* PROBLEM: these lemmas needs to be restated, since  *)
-(* concat.simps(1) and concat.simps(2) contain the    *)
-(* type variables ?'a1.0 (which are turned into frees *)
-(* 'a_1                                               *)
-lemma concat1:
-  shows "concat [] \<approx> []"
-by (simp add: id_simps)
-
-lemma concat2:
-  shows "concat (x # xs) \<approx> x @ concat xs"
-by (simp)
-
-lemma concat_rsp:
-  "\<lbrakk>list_rel op \<approx> x x'; x' \<approx> y'; list_rel op \<approx> y' y\<rbrakk> \<Longrightarrow> concat x \<approx> concat y"
-  apply (induct x y arbitrary: x' y' rule: list_induct2')
+lemma concat_rsp_pre:
+  "\<lbrakk>list_rel op \<approx> x x'; x' \<approx> y'; list_rel op \<approx> y' y; \<exists>x\<in>set x. xa \<in> set x\<rbrakk> \<Longrightarrow> 
+    \<exists>x\<in>set y. xa \<in> set x"
+  apply clarify
+  apply (frule list_rel_find_element[of _ "x"])
+  apply assumption
+  apply clarify
+  apply (subgoal_tac "ya \<in> set y'")
+  prefer 2
   apply simp
-  defer defer
-  apply (simp only: concat.simps)
-  sorry
+  apply (frule list_rel_find_element[of _ "y'"])
+  apply assumption
+  apply auto
+  done
 
 lemma [quot_respect]:
   shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"
   apply (simp only: fun_rel_def)
   apply clarify
-  apply (rule concat_rsp)
+  apply (simp (no_asm))
+  apply rule
+  apply rule
+  apply (erule concat_rsp_pre)
   apply assumption+
+  apply (rule concat_rsp_pre)
+  prefer 4
+  apply assumption
+  apply (rule list_rel_symp[OF list_eq_equivp])
+  apply assumption
+  apply (rule equivp_symp[OF list_eq_equivp])
+  apply assumption
+  apply (rule list_rel_symp[OF list_eq_equivp])
+  apply assumption
   done
 
 lemma nil_rsp2[quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"
-  apply (metis FSet3.nil_rsp list_rel.simps(1) pred_comp.intros)
-  done
+  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"
-  apply (rule eq_reflection)
-  apply auto
-  done
+  by (rule eq_reflection) auto
 
 lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
   unfolding list_eq.simps
-  apply(simp only: set_map set_in_eq)
-  done
+  by (simp only: set_map set_in_eq)
 
 lemma quotient_compose_list_pre:
   "(list_rel op \<approx> OOO op \<approx>) r s =
@@ -342,7 +124,7 @@
     (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
   unfolding Quotient_def comp_def
   apply (rule)+
-  apply (simp add: abs_o_rep[OF Quotient_fset] id_simps Quotient_abs_rep[OF Quotient_fset])
+  apply (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)
   apply (rule)
   apply (rule)
   apply (rule)
@@ -359,11 +141,9 @@
 
 lemma fconcat_empty:
   shows "fconcat {||} = {||}"
-  apply(lifting concat1)
+  apply(lifting concat.simps(1))
   apply(cleaning)
-  apply(simp add: comp_def)
-  apply(fold fempty_def[simplified id_simps])
-  apply(rule refl)
+  apply(simp add: comp_def bot_fset_def)
   done
 
 lemma insert_rsp2[quot_respect]:
@@ -373,9 +153,7 @@
   apply (rule_tac b="x # b" in pred_compI)
   apply auto
   apply (rule_tac b="x # ba" in pred_compI)
-  apply (rule cons_rsp)
-  apply simp
-  apply (auto)[1]
+  apply auto
   done
 
 lemma append_rsp[quot_respect]:
@@ -384,30 +162,12 @@
 
 lemma fconcat_insert:
   shows "fconcat (finsert x S) = x |\<union>| fconcat S"
-  apply(lifting concat2)
-  apply(cleaning)
+  apply (lifting concat.simps(2))
+  apply (cleaning)
   apply (simp add: finsert_def fconcat_def comp_def)
-  apply cleaning
+  apply (cleaning)
   done
 
-text {* raw section *}
-
-lemma map_rsp_aux:
-  assumes a: "a \<approx> b"
-  shows "map f a \<approx> map f b"
-  using a
-apply(induct a arbitrary: b)
-apply(auto)
-apply(metis rev_image_eqI)
-done
-
-lemma map_rsp[quot_respect]:
-  shows "(op = ===> op \<approx> ===> op \<approx>) map map"
-by (auto simp add: map_rsp_aux)
-
-
-text {* lifted section *}
-
 (* TBD *)
 
 text {* syntax for fset comprehensions (adapted from lists) *}
@@ -507,210 +267,4 @@
 (* BELOW CONSTRUCTION SITE *)
 
 
-lemma no_mem_nil: 
-  "(\<forall>a. a \<notin> set A) = (A = [])"
-by (induct A) (auto)
-
-lemma none_mem_nil: 
-  "(\<forall>a. a \<notin> set A) = (A \<approx> [])"
-by simp
-
-lemma mem_cons: 
-  "a \<in> set A \<Longrightarrow> a # A \<approx> A"
-by auto
-
-lemma cons_left_comm:
-  "x # y # A \<approx> y # x # A"
-by (auto simp add: id_simps)
-
-lemma cons_left_idem:
-  "x # x # A \<approx> x # A"
-by (auto simp add: id_simps)
-
-lemma finite_set_raw_strong_cases:
-  "(X = []) \<or> (\<exists>a Y. ((a \<notin> set Y) \<and> (X \<approx> a # Y)))"
-  apply (induct X)
-  apply (simp)
-  apply (rule disjI2)
-  apply (erule disjE)
-  apply (rule_tac x="a" in exI)
-  apply (rule_tac x="[]" in exI)
-  apply (simp)
-  apply (erule exE)+
-  apply (case_tac "a = aa")
-  apply (rule_tac x="a" in exI)
-  apply (rule_tac x="Y" in exI)
-  apply (simp)
-  apply (rule_tac x="aa" in exI)
-  apply (rule_tac x="a # Y" in exI)
-  apply (auto)
-  done
-
-fun
-  delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"
-where
-  "delete_raw [] x = []"
-| "delete_raw (a # A) x = (if (a = x) then delete_raw A x else a # (delete_raw A x))"
-
-lemma mem_delete_raw:
-  "x \<in> set (delete_raw A a) = (x \<in> set A \<and> \<not>(x = a))"
-  by (induct A arbitrary: x a) (auto)
-
-lemma mem_delete_raw_ident:
-  "\<not>(a \<in> set (delete_raw A a))"
-by (induct A) (auto)
-
-lemma not_mem_delete_raw_ident:
-  "b \<notin> set A \<Longrightarrow> (delete_raw A b = A)"
-by (induct A) (auto)
-
-lemma delete_raw_RSP:
-  "A \<approx> B \<Longrightarrow> delete_raw A a \<approx> delete_raw B a"
-apply(induct A arbitrary: B a)
-apply(auto)
-sorry
-
-lemma cons_delete_raw:
-  "a # (delete_raw A a) \<approx> (if a \<in> set A then A else (a # A))"
-sorry
-
-lemma mem_cons_delete_raw:
-    "a \<in> set A \<Longrightarrow> a # (delete_raw A a) \<approx> A"
-sorry
-
-lemma finite_set_raw_delete_raw_cases:
-    "X = [] \<or> (\<exists>a. a mem X \<and> X \<approx> a # delete_raw X a)"
-  by (induct X) (auto)
-
-
-
-
-
-lemma list2set_thm:
-  shows "set [] = {}"
-  and "set (h # t) = insert h (set t)"
-  by (auto)
-
-lemma list2set_RSP:
-  "A \<approx> B \<Longrightarrow> set A = set B"
-  by auto
-
-definition
-  rsp_fold
-where
-  "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"
-
-primrec
-  fold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"
-where
-  "fold_raw f z [] = z"
-| "fold_raw f z (a # A) =
-     (if (rsp_fold f) then
-       if a mem A then fold_raw f z A
-       else f a (fold_raw f z A)
-     else z)"
-
-lemma mem_lcommuting_fold_raw:
-  "rsp_fold f \<Longrightarrow> h mem B \<Longrightarrow> fold_raw f z B = f h (fold_raw f z (delete_raw B h))"
-sorry
-
-lemma fold_rsp[quot_respect]:
-  "(op = ===> op = ===> op \<approx> ===> op =) fold_raw fold_raw"
-apply(auto)
-sorry
-
-primrec
-  inter_raw
-where
-  "inter_raw [] B = []"
-| "inter_raw (a # A) B = (if a mem B then a # inter_raw A B else inter_raw A B)"
-
-lemma mem_inter_raw:
-  "x mem (inter_raw A B) = x mem A \<and> x mem B"
-sorry
-
-lemma inter_raw_RSP:
-  "A1 \<approx> A2 \<and> B1 \<approx> B2 \<Longrightarrow> (inter_raw A1 B1) \<approx> (inter_raw A2 B2)"
-sorry
-
-
-(* LIFTING DEFS *)
-
-
-section {* Constants on the Quotient Type *} 
-
-
-quotient_definition
-  "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" 
-  is "delete_raw"
-
-quotient_definition
-  finter ("_ \<inter>f _" [70, 71] 70)
-where
-  "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
-  is "inter_raw"
-
-quotient_definition
-  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" 
-  is "fold_raw"
-
-quotient_definition
-  "fset_to_set :: 'a fset \<Rightarrow> 'a set" 
-  is "set"
-
-
-section {* Lifted Theorems *}
-
-thm list.cases (* ??? *)
-
-thm cons_left_comm
-lemma "finsert a (finsert b S) = finsert b (finsert a S)"
-by (lifting cons_left_comm)
-
-thm cons_left_idem
-lemma "finsert a (finsert a S) = finsert a S"
-by (lifting cons_left_idem)
-
-(* thm MEM:
-  MEM x [] = F
-  MEM x (h::t) = (x=h) \/ MEM x t *)
-thm none_mem_nil
-(*lemma "(\<forall>a. a \<notin>f A) = (A = fempty)"*)
-
-thm mem_cons
-thm finite_set_raw_strong_cases
-(*thm card_raw.simps*)
-(*thm not_mem_card_raw*)
-(*thm card_raw_suc*)
-
-lemma "fcard X = Suc n \<Longrightarrow> (\<exists>a S. a \<notin>f S & X = finsert a S)"
-(*by (lifting card_raw_suc)*)
-sorry
-
-(*thm card_raw_cons_gt_0
-thm mem_card_raw_gt_0
-thm not_nil_equiv_cons
-*)
-thm delete_raw.simps
-(*thm mem_delete_raw*)
-(*thm card_raw_delete_raw*)
-thm cons_delete_raw
-thm mem_cons_delete_raw
-thm finite_set_raw_delete_raw_cases
-thm append.simps
-(* MEM_APPEND: MEM e (APPEND l1 l2) = MEM e l1 \/ MEM e l2 *)
-thm inter_raw.simps
-thm mem_inter_raw
-thm fold_raw.simps
-thm list2set_thm
-thm list_eq_def
-thm list.induct
-lemma "\<lbrakk>P fempty; \<And>a x. P x \<Longrightarrow> P (finsert a x)\<rbrakk> \<Longrightarrow> P l"
-by (lifting list.induct)
-
-(* We also have map and some properties of it in FSet *)
-(* and the following which still lifts ok *)
-lemma "funion (funion x xa) xb = funion x (funion xa xb)"
-by (lifting append_assoc)
-
 end