theory FSet

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"

as

  "[]::'a list"

quotient_definition

   "INSERT :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"

as

  "op #"

quotient_definition

   "FUNION :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"

as

  "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"

as

  "card1"

quotient_definition

  "fconcat :: ('a fset) fset \<Rightarrow> 'a fset"

as

  "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"

as

  "op mem"

quotient_definition

   "FOLD :: ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b fset \<Rightarrow> 'a"

as

  "fold1"

quotient_definition

  "fmap :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"

as

  "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"

as

  "[]::'a list"

quotient_definition

   "INSERT2 :: 'a \<Rightarrow> 'a fset2 \<Rightarrow> 'a fset2"

as

  "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"

as

  "list_rec"

quotient_definition

  "fset_case :: 'a \<Rightarrow> ('b \<Rightarrow> 'b fset \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a"

as

  "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 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

end