--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Quot/Examples/FSet.thy	Mon Dec 07 14:09:50 2009 +0100
@@ -0,0 +1,440 @@
+theory FSet
+imports "../QuotMain"
+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 fset = "'a list" / "list_eq"
+  apply(rule equivp_list_eq)
+  done
+
+print_theorems
+
+typ "'a fset"
+thm "Rep_fset"
+thm "ABS_fset_def"
+
+quotient_def 
+  EMPTY :: "'a fset"
+where
+  "EMPTY \<equiv> ([]::'a list)"
+
+term Nil
+term EMPTY
+thm EMPTY_def
+
+quotient_def 
+  INSERT :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+where
+  "INSERT \<equiv> op #"
+
+term Cons
+term INSERT
+thm INSERT_def
+
+quotient_def 
+  FUNION :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+where
+  "FUNION \<equiv> (op @)"
+
+term append
+term FUNION
+thm FUNION_def
+
+thm Quotient_fset
+
+thm QUOT_TYPE_I_fset.thm11
+
+
+fun
+  membship :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infix "memb" 100)
+where
+  m1: "(x memb []) = False"
+| m2: "(x memb (y#xs)) = ((x=y) \<or> (x memb xs))"
+
+fun
+  card1 :: "'a list \<Rightarrow> nat"
+where
+  card1_nil: "(card1 []) = 0"
+| card1_cons: "(card1 (x # xs)) = (if (x memb xs) then (card1 xs) else (Suc (card1 xs)))"
+
+quotient_def 
+  CARD :: "'a fset \<Rightarrow> nat"
+where
+  "CARD \<equiv> card1"
+
+term card1
+term CARD
+thm CARD_def
+
+(* 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 @{binding 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 memb xs)) = (card1 (x # xs) = Suc (card1 xs))"
+  by auto
+
+lemma mem_cons:
+  fixes x :: "'a"
+  fixes xs :: "'a list"
+  assumes a : "x memb 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 memb ys) \<and> xs \<approx> (a # ys)"
+  using c
+apply(induct xs)
+apply (metis Suc_neq_Zero card1_0)
+apply (metis QUOT_TYPE_I_fset.R_trans card1_cons 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 memb 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 memb Y) \<and> (X \<approx> a # Y)))"
+  apply (induct X)
+  apply (simp)
+  apply (metis QUOT_TYPE_I_fset.thm11 list_eq_refl mem_cons m1)
+  done
+
+quotient_def
+  IN :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool"
+where
+  "IN \<equiv> membship"
+
+term membship
+term IN
+thm IN_def
+
+term fold1
+quotient_def 
+  FOLD :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b fset \<Rightarrow> 'a"
+where
+  "FOLD \<equiv> fold1"
+
+term fold1
+term fold
+thm fold_def
+
+quotient_def 
+  fmap::"('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+where
+  "fmap \<equiv> map"
+
+term map
+term fmap
+thm fmap_def
+
+lemma memb_rsp:
+  fixes z
+  assumes a: "x \<approx> y"
+  shows "(z memb x) = (z memb y)"
+  using a by induct auto
+
+lemma ho_memb_rsp[quotient_rsp]:
+  "(op = ===> (op \<approx> ===> op =)) (op memb) (op memb)"
+  by (simp add: memb_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:memb_rsp)
+
+lemma ho_card1_rsp[quotient_rsp]: 
+  "(op \<approx> ===> op =) card1 card1"
+  by (simp add: card1_rsp)
+
+lemma cons_rsp[quotient_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[quotient_rsp]:
+  "(op = ===> op \<approx> ===> op \<approx>) op # op #"
+  by (simp add: cons_rsp)
+
+lemma append_rsp_fst:
+  assumes a : "l1 \<approx> l2"
+  shows "(l1 @ s) \<approx> (l2 @ s)"
+  using a
+  by (induct) (auto intro: list_eq.intros list_eq_refl)
+
+lemma append_end:
+  shows "(e # l) \<approx> (l @ [e])"
+  apply (induct l)
+  apply (auto intro: list_eq.intros list_eq_refl)
+  done
+
+lemma rev_rsp:
+  shows "a \<approx> rev a"
+  apply (induct a)
+  apply simp
+  apply (rule list_eq_refl)
+  apply simp_all
+  apply (rule list_eq.intros(6))
+  prefer 2
+  apply (rule append_rsp_fst)
+  apply assumption
+  apply (rule append_end)
+  done
+
+lemma append_sym_rsp:
+  shows "(a @ b) \<approx> (b @ a)"
+  apply (rule list_eq.intros(6))
+  apply (rule append_rsp_fst)
+  apply (rule rev_rsp)
+  apply (rule list_eq.intros(6))
+  apply (rule rev_rsp)
+  apply (simp)
+  apply (rule append_rsp_fst)
+  apply (rule list_eq.intros(3))
+  apply (rule rev_rsp)
+  done
+
+lemma append_rsp:
+  assumes a : "l1 \<approx> r1"
+  assumes b : "l2 \<approx> r2 "
+  shows "(l1 @ l2) \<approx> (r1 @ r2)"
+  apply (rule list_eq.intros(6))
+  apply (rule append_rsp_fst)
+  using a apply (assumption)
+  apply (rule list_eq.intros(6))
+  apply (rule append_sym_rsp)
+  apply (rule list_eq.intros(6))
+  apply (rule append_rsp_fst)
+  using b apply (assumption)
+  apply (rule append_sym_rsp)
+  done
+
+lemma ho_append_rsp[quotient_rsp]:
+  "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"
+  by (simp add: append_rsp)
+
+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[quotient_rsp]:
+  "(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[quotient_rsp]:
+  "(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: memb_rsp rsp_fold_def)
+done
+
+lemma list_equiv_rsp[quotient_rsp]:
+  shows "(op \<approx> ===> op \<approx> ===> op =) op \<approx> op \<approx>"
+by (auto intro: list_eq.intros)
+
+print_quotients
+
+ML {* val qty = @{typ "'a fset"} *}
+ML {* val rsp_thms =
+  @{thms ho_memb_rsp ho_cons_rsp ho_card1_rsp ho_map_rsp ho_append_rsp ho_fold_rsp} *}
+
+ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}
+ML {* val (trans2, reps_same, absrep, quot) = lookup_quot_thms @{context} "fset"; *}
+ML {* fun lift_tac_fset lthy t = lift_tac lthy t *}
+
+lemma "IN x EMPTY = False"
+apply(tactic {* procedure_tac @{context} @{thm m1} 1 *})
+apply(tactic {* regularize_tac @{context} 1 *})
+apply(tactic {* all_inj_repabs_tac @{context} [rel_refl] [trans2] 1 *})
+apply(tactic {* clean_tac @{context} 1*})
+done
+
+lemma "IN x (INSERT y xa) = (x = y \<or> IN x xa)"
+by (tactic {* lift_tac_fset @{context} @{thm m2} 1 *})
+
+lemma "INSERT a (INSERT a x) = INSERT a x"
+apply (tactic {* lift_tac_fset @{context} @{thm list_eq.intros(4)} 1 *})
+done
+
+lemma "x = xa \<Longrightarrow> INSERT a x = INSERT a xa"
+apply (tactic {* lift_tac_fset @{context} @{thm list_eq.intros(5)} 1 *})
+done
+
+lemma "CARD x = Suc n \<Longrightarrow> (\<exists>a b. \<not> IN a b & x = INSERT a b)"
+apply (tactic {* lift_tac_fset @{context} @{thm card1_suc} 1 *})
+done
+
+lemma "(\<not> IN x xa) = (CARD (INSERT x xa) = Suc (CARD xa))"
+apply (tactic {* lift_tac_fset @{context} @{thm not_mem_card1} 1 *})
+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(tactic {* lift_tac_fset @{context} @{thm fold1.simps(2)} 1 *})
+done
+
+ML {* fun inj_repabs_tac_fset lthy = inj_repabs_tac lthy [rel_refl] [trans2] *}
+
+lemma "fmap f (FUNION (x::'b fset) (xa::'b fset)) = FUNION (fmap f x) (fmap f xa)"
+apply (tactic {* lift_tac_fset @{context} @{thm map_append} 1 *})
+done
+
+lemma "FUNION (FUNION x xa) xb = FUNION x (FUNION xa xb)"
+apply (tactic {* lift_tac_fset @{context} @{thm append_assoc} 1 *})
+done
+
+
+lemma "\<lbrakk>P EMPTY; \<And>a x. P x \<Longrightarrow> P (INSERT a x)\<rbrakk> \<Longrightarrow> P l"
+apply (tactic {* (ObjectLogic.full_atomize_tac THEN' gen_frees_tac @{context}) 1 *})
+apply(tactic {* procedure_tac @{context} @{thm list.induct} 1 *})
+apply(tactic {* regularize_tac @{context} 1 *})
+defer
+apply(tactic {* clean_tac @{context} 1 *})
+apply(tactic {* inj_repabs_tac_fset @{context} 1*})+
+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
+
+ML {* quot *}
+thm quotient_thm
+
+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 (tactic {* lift_tac_fset @{context} @{thm list_induct_part} 1 *})
+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 (tactic {* lift_tac_fset @{context} @{thm list_induct_part} 1 *})
+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 (tactic {* lift_tac_fset @{context} @{thm list_induct_part} 1 *})
+done
+
+quotient fset2 = "'a list" / "list_eq"
+  apply(rule equivp_list_eq)
+  done
+
+quotient_def
+  EMPTY2 :: "'a fset2"
+where
+  "EMPTY2 \<equiv> ([]::'a list)"
+
+quotient_def
+  INSERT2 :: "'a \<Rightarrow> 'a fset2 \<Rightarrow> 'a fset2"
+where
+  "INSERT2 \<equiv> op #"
+
+ML {* val quot = @{thms Quotient_fset Quotient_fset2} *}
+ML {* fun inj_repabs_tac_fset lthy = inj_repabs_tac lthy [rel_refl] [trans2] *}
+ML {* fun lift_tac_fset lthy t = lift_tac lthy t  *}
+
+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 (tactic {* lift_tac_fset @{context} @{thm list_induct_part} 1 *})
+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 (tactic {* lift_tac_fset @{context} @{thm list_induct_part} 1 *})
+done
+
+quotient_def
+  fset_rec::"'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
+where
+  "fset_rec \<equiv> list_rec"
+
+quotient_def
+  fset_case::"'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"
+where
+  "fset_case \<equiv> list_case"
+
+(* Probably not true without additional assumptions about the function *)
+lemma list_rec_rsp[quotient_rsp]:
+  "(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[quotient_rsp]:
+  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_case list_case"
+  apply (auto)
+  sorry
+
+ML {* val rsp_thms = @{thms list_rec_rsp list_case_rsp} @ rsp_thms *}
+ML {* fun lift_tac_fset lthy t = lift_tac lthy t *}
+
+lemma "fset_rec (f1::'t) x (INSERT a xa) = x a xa (fset_rec f1 x xa)"
+apply (tactic {* lift_tac_fset @{context} @{thm list.recs(2)} 1 *})
+done
+
+lemma "fset_case (f1::'t) f2 (INSERT a xa) = f2 a xa"
+apply (tactic {* lift_tac_fset @{context} @{thm list.cases(2)} 1 *})
+done
+
+
+end