diff -r 6088fea1c8b1 -r 8a1c8dc72b5c FSet.thy
--- a/FSet.thy	Mon Dec 07 14:00:36 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,440 +0,0 @@
-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