--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Quot/Examples/FSet.thy	Thu Feb 25 07:57:17 2010 +0100
@@ -0,0 +1,433 @@
+theory FSet
+imports "../Quotient" "../Quotient_List" "../Quotient_Product" List
+begin
+
+inductive
+  list_eq (infix "\<approx>" 50)
+where
+  "a#b#xs \<approx> b#a#xs"
+| "[] \<approx> []"
+| "xs \<approx> ys \<Longrightarrow> ys \<approx> xs"
+| "a#a#xs \<approx> a#xs"
+| "xs \<approx> ys \<Longrightarrow> a#xs \<approx> a#ys"
+| "\<lbrakk>xs1 \<approx> xs2; xs2 \<approx> xs3\<rbrakk> \<Longrightarrow> xs1 \<approx> xs3"
+
+lemma list_eq_refl:
+  shows "xs \<approx> xs"
+  by (induct xs) (auto intro: list_eq.intros)
+
+lemma equivp_list_eq:
+  shows "equivp list_eq"
+  unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def
+  apply(auto intro: list_eq.intros list_eq_refl)
+  done
+
+quotient_type 
+  'a fset = "'a list" / "list_eq"
+  by (rule equivp_list_eq)
+
+quotient_definition
+   "EMPTY :: 'a fset"
+is
+  "[]::'a list"
+
+quotient_definition
+   "INSERT :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+is
+  "op #"
+
+quotient_definition
+   "FUNION :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+is
+  "op @"
+
+fun
+  card1 :: "'a list \<Rightarrow> nat"
+where
+  card1_nil: "(card1 []) = 0"
+| card1_cons: "(card1 (x # xs)) = (if (x mem xs) then (card1 xs) else (Suc (card1 xs)))"
+
+quotient_definition
+   "CARD :: 'a fset \<Rightarrow> nat"
+is
+  "card1"
+
+quotient_definition
+  "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"
+is
+  "concat"
+
+term concat
+term fconcat
+
+text {*
+ Maybe make_const_def should require a theorem that says that the particular lifted function
+ respects the relation. With it such a definition would be impossible:
+ make_const_def CARD @{term "length"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
+*}
+
+lemma card1_0:
+  fixes a :: "'a list"
+  shows "(card1 a = 0) = (a = [])"
+  by (induct a) auto
+
+lemma not_mem_card1:
+  fixes x :: "'a"
+  fixes xs :: "'a list"
+  shows "(~(x mem xs)) = (card1 (x # xs) = Suc (card1 xs))"
+  by auto
+
+lemma mem_cons:
+  fixes x :: "'a"
+  fixes xs :: "'a list"
+  assumes a : "x mem xs"
+  shows "x # xs \<approx> xs"
+  using a by (induct xs) (auto intro: list_eq.intros )
+
+lemma card1_suc:
+  fixes xs :: "'a list"
+  fixes n :: "nat"
+  assumes c: "card1 xs = Suc n"
+  shows "\<exists>a ys. ~(a mem ys) \<and> xs \<approx> (a # ys)"
+  using c
+apply(induct xs)
+apply (metis Suc_neq_Zero card1_0)
+apply (metis FSet.card1_cons list_eq.intros(6) list_eq_refl mem_cons)
+done
+
+definition
+  rsp_fold
+where
+  "rsp_fold f = ((!u v. (f u v = f v u)) \<and> (!u v w. ((f u (f v w) = f (f u v) w))))"
+
+primrec
+  fold1
+where
+  "fold1 f (g :: 'a \<Rightarrow> 'b) (z :: 'b) [] = z"
+| "fold1 f g z (a # A) =
+     (if rsp_fold f
+     then (
+       if (a mem A) then (fold1 f g z A) else (f (g a) (fold1 f g z A))
+     ) else z)"
+
+lemma fs1_strong_cases:
+  fixes X :: "'a list"
+  shows "(X = []) \<or> (\<exists>a. \<exists> Y. (~(a mem Y) \<and> (X \<approx> a # Y)))"
+  apply (induct X)
+  apply (simp)
+  apply (metis List.member.simps(1) list_eq.intros(6) list_eq_refl mem_cons)
+  done
+
+quotient_definition
+   "IN :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool"
+is
+  "op mem"
+
+quotient_definition
+   "FOLD :: ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b fset \<Rightarrow> 'a"
+is
+  "fold1"
+
+quotient_definition
+  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+is
+  "map"
+
+lemma mem_rsp:
+  fixes z
+  assumes a: "x \<approx> y"
+  shows "(z mem x) = (z mem y)"
+  using a by induct auto
+
+lemma ho_memb_rsp[quot_respect]:
+  "(op = ===> (op \<approx> ===> op =)) (op mem) (op mem)"
+  by (simp add: mem_rsp)
+
+lemma card1_rsp:
+  fixes a b :: "'a list"
+  assumes e: "a \<approx> b"
+  shows "card1 a = card1 b"
+  using e by induct (simp_all add: mem_rsp)
+
+lemma ho_card1_rsp[quot_respect]: 
+  "(op \<approx> ===> op =) card1 card1"
+  by (simp add: card1_rsp)
+
+lemma cons_rsp:
+  fixes z
+  assumes a: "xs \<approx> ys"
+  shows "(z # xs) \<approx> (z # ys)"
+  using a by (rule list_eq.intros(5))
+
+lemma ho_cons_rsp[quot_respect]:
+  "(op = ===> op \<approx> ===> op \<approx>) op # op #"
+  by (simp add: cons_rsp)
+
+lemma append_rsp_aux1:
+  assumes a : "l2 \<approx> r2 "
+  shows "(h @ l2) \<approx> (h @ r2)"
+using a
+apply(induct h)
+apply(auto intro: list_eq.intros(5))
+done
+
+lemma append_rsp_aux2:
+  assumes a : "l1 \<approx> r1" "l2 \<approx> r2 "
+  shows "(l1 @ l2) \<approx> (r1 @ r2)"
+using a
+apply(induct arbitrary: l2 r2)
+apply(simp_all)
+apply(blast intro: list_eq.intros append_rsp_aux1)+
+done
+
+lemma append_rsp[quot_respect]:
+  "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
+  by (auto simp add: append_rsp_aux2)
+
+lemma map_rsp:
+  assumes a: "a \<approx> b"
+  shows "map f a \<approx> map f b"
+  using a
+  apply (induct)
+  apply(auto intro: list_eq.intros)
+  done
+
+lemma ho_map_rsp[quot_respect]:
+  "(op = ===> op \<approx> ===> op \<approx>) map map"
+  by (simp add: map_rsp)
+
+lemma map_append:
+  "(map f (a @ b)) \<approx> (map f a) @ (map f b)"
+ by simp (rule list_eq_refl)
+
+lemma ho_fold_rsp[quot_respect]:
+  "(op = ===> op = ===> op = ===> op \<approx> ===> op =) fold1 fold1"
+  apply (auto)
+  apply (case_tac "rsp_fold x")
+  prefer 2
+  apply (erule_tac list_eq.induct)
+  apply (simp_all)
+  apply (erule_tac list_eq.induct)
+  apply (simp_all)
+  apply (auto simp add: mem_rsp rsp_fold_def)
+done
+
+lemma list_equiv_rsp[quot_respect]:
+  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
+by (auto intro: list_eq.intros)
+
+lemma "IN x EMPTY = False"
+apply(lifting member.simps(1))
+done
+
+lemma "IN x (INSERT y xa) = (x = y \<or> IN x xa)"
+apply (lifting member.simps(2))
+done
+
+lemma "INSERT a (INSERT a x) = INSERT a x"
+apply (lifting list_eq.intros(4))
+done
+
+lemma "x = xa \<Longrightarrow> INSERT a x = INSERT a xa"
+apply (lifting list_eq.intros(5))
+done
+
+lemma "CARD x = Suc n \<Longrightarrow> (\<exists>a b. \<not> IN a b & x = INSERT a b)"
+apply (lifting card1_suc)
+done
+
+lemma "(\<not> IN x xa) = (CARD (INSERT x xa) = Suc (CARD xa))"
+apply (lifting not_mem_card1)
+done
+
+lemma "FOLD f g (z::'b) (INSERT a x) =
+  (if rsp_fold f then if IN a x then FOLD f g z x else f (g a) (FOLD f g z x) else z)"
+apply(lifting fold1.simps(2))
+done
+
+lemma "fmap f (FUNION (x::'b fset) (xa::'b fset)) = FUNION (fmap f x) (fmap f xa)"
+apply (lifting map_append)
+done
+
+lemma "FUNION (FUNION x xa) xb = FUNION x (FUNION xa xb)"
+apply (lifting append_assoc)
+done
+
+
+lemma "\<lbrakk>P EMPTY; \<And>a x. P x \<Longrightarrow> P (INSERT a x)\<rbrakk> \<Longrightarrow> P l"
+apply(lifting list.induct)
+done
+
+lemma list_induct_part:
+  assumes a: "P (x :: 'a list) ([] :: 'c list)"
+  assumes b: "\<And>e t. P x t \<Longrightarrow> P x (e # t)"
+  shows "P x l"
+  apply (rule_tac P="P x" in list.induct)
+  apply (rule a)
+  apply (rule b)
+  apply (assumption)
+  done
+
+lemma "P (x :: 'a list) (EMPTY :: 'c fset) \<Longrightarrow> (\<And>e t. P x t \<Longrightarrow> P x (INSERT e t)) \<Longrightarrow> P x l"
+apply (lifting list_induct_part)
+done
+
+lemma "P (x :: 'a fset) (EMPTY :: 'c fset) \<Longrightarrow> (\<And>e t. P x t \<Longrightarrow> P x (INSERT e t)) \<Longrightarrow> P x l"
+apply (lifting list_induct_part)
+done
+
+lemma "P (x :: 'a fset) ([] :: 'c list) \<Longrightarrow> (\<And>e t. P x t \<Longrightarrow> P x (e # t)) \<Longrightarrow> P x l"
+apply (lifting list_induct_part)
+done
+
+quotient_type 'a fset2 = "'a list" / "list_eq"
+  by (rule equivp_list_eq)
+
+quotient_definition
+   "EMPTY2 :: 'a fset2"
+is
+  "[]::'a list"
+
+quotient_definition
+   "INSERT2 :: 'a \<Rightarrow> 'a fset2 \<Rightarrow> 'a fset2"
+is
+  "op #"
+
+lemma "P (x :: 'a fset2) (EMPTY :: 'c fset) \<Longrightarrow> (\<And>e t. P x t \<Longrightarrow> P x (INSERT e t)) \<Longrightarrow> P x l"
+apply (lifting list_induct_part)
+done
+
+lemma "P (x :: 'a fset) (EMPTY2 :: 'c fset2) \<Longrightarrow> (\<And>e t. P x t \<Longrightarrow> P x (INSERT2 e t)) \<Longrightarrow> P x l"
+apply (lifting list_induct_part)
+done
+
+quotient_definition
+  "fset_rec :: 'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
+is
+  "list_rec"
+
+quotient_definition
+  "fset_case :: 'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
+is
+  "list_case"
+
+(* Probably not true without additional assumptions about the function *)
+lemma list_rec_rsp[quot_respect]:
+  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_rec list_rec"
+  apply (auto)
+  apply (erule_tac list_eq.induct)
+  apply (simp_all)
+  sorry
+
+lemma list_case_rsp[quot_respect]:
+  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_case list_case"
+  apply (auto)
+  sorry
+
+lemma "fset_rec (f1::'t) x (INSERT a xa) = x a xa (fset_rec f1 x xa)"
+apply (lifting list.recs(2))
+done
+
+lemma "fset_case (f1::'t) f2 (INSERT a xa) = f2 a xa"
+apply (lifting list.cases(2))
+done
+
+lemma ttt: "((op @) x ((op #) e [])) = (((op #) e x))"
+sorry
+
+lemma "(FUNION x (INSERT e EMPTY)) = ((INSERT e x))"
+apply (lifting ttt)
+done
+
+
+lemma ttt2: "(\<lambda>e. ((op @) x ((op #) e []))) = (\<lambda>e. ((op #) e x))"
+sorry
+
+lemma "(\<lambda>e. (FUNION x (INSERT e EMPTY))) = (\<lambda>e. (INSERT e x))"
+apply(lifting ttt2)
+apply(regularize)
+apply(rule impI)
+apply(simp)
+apply(rule allI)
+apply(rule list_eq_refl)
+done
+
+lemma ttt3: "(\<lambda>x. ((op @) x (e # []))) = (op #) e"
+sorry
+
+lemma "(\<lambda>x. (FUNION x (INSERT e EMPTY))) = INSERT e"
+apply(lifting ttt3)
+apply(regularize)
+apply(auto simp add: cons_rsp)
+done
+lemma hard: "(\<lambda>P. \<lambda>Q. P (Q (x::'a list))) = (\<lambda>P. \<lambda>Q. Q (P (x::'a list)))"
+sorry
+
+lemma eq_imp_rel:
+  shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"
+  by (simp add: equivp_reflp)
+
+
+lemma hard_lift: "(\<lambda>P. \<lambda>Q. P (Q (x::'a fset))) = (\<lambda>P. \<lambda>Q. Q (P (x::'a fset)))"
+apply(lifting hard)
+apply(regularize)
+apply(rule fun_rel_id_asm)
+apply(subst babs_simp)
+apply(tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})
+apply(rule fun_rel_id_asm)
+apply(rule impI)
+apply(rule mp[OF eq_imp_rel[OF fset_equivp]])
+apply(drule fun_cong)
+apply(drule fun_cong)
+apply(assumption)
+done
+
+lemma test: "All (\<lambda>(x::'a list, y). x = y)"
+sorry
+
+lemma  "All (\<lambda>(x::'a fset, y). x = y)"
+apply(lifting test)
+done
+
+lemma test2: "Ex (\<lambda>(x::'a list, y). x = y)"
+sorry
+
+lemma  "Ex (\<lambda>(x::'a fset, y). x = y)"
+apply(lifting test2)
+done
+
+lemma test3: "All (\<lambda> (x :: 'a list, y, z). x = y \<and> y = z)"
+sorry
+
+lemma "All (\<lambda> (x :: 'a fset, y, z). x = y \<and> y = z)"
+apply(lifting test3)
+done
+
+lemma test4: "\<forall> (x :: 'a list, y, z) \<in> Respects(
+  prod_rel (op \<approx>) (prod_rel (op \<approx>) (op \<approx>))
+). x = y \<and> y = z"
+sorry
+
+lemma "All (\<lambda> (x :: 'a fset, y, z). x = y \<and> y = z)"
+apply (lifting test4)
+sorry
+
+lemma test5: "\<forall> (x :: 'a list \<Rightarrow> 'a list, y) \<in> Respects(
+  prod_rel (op \<approx> ===> op \<approx>) (op \<approx> ===> op \<approx>)
+). (op \<approx> ===> op \<approx>) x y"
+sorry
+
+lemma "All (\<lambda> (x :: 'a fset \<Rightarrow> 'a fset, y). x = y)"
+apply (lifting test5)
+done
+
+lemma test6: "\<forall> (x :: 'a list \<Rightarrow> 'a list, y, z) \<in> Respects(
+  prod_rel (op \<approx> ===> op \<approx>) (prod_rel (op \<approx> ===> op \<approx>) (op \<approx> ===> op \<approx>))
+). (op \<approx> ===> op \<approx>) x y \<and> (op \<approx> ===> op \<approx>) y z"
+sorry
+
+lemma "All (\<lambda> (x :: 'a fset \<Rightarrow> 'a fset, y, z). x = y \<and> y = z)"
+apply (lifting test6)
+done
+
+end