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"

  apply (induct xs)

   apply (auto intro: list_eq.intros)

  done

lemma equiv_list_eq:

  shows "EQUIV list_eq"

  unfolding EQUIV_REFL_SYM_TRANS REFL_def SYM_def TRANS_def

  apply(auto intro: list_eq.intros list_eq_refl)

  done

quotient fset = "'a list" / "list_eq"

  apply(rule equiv_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)"

(* fold1_def is not usable, but: *)

thm fold1.simps

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

ML {* val defs = @{thms EMPTY_def IN_def FUNION_def CARD_def INSERT_def fmap_def FOLD_def} *}

lemma memb_rsp:

  fixes z

  assumes a: "list_eq x y"

  shows "(z memb x) = (z memb y)"

  using a by induct auto

lemma ho_memb_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: "(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:

  "(op = ===> op \<approx> ===> op \<approx>) op # op #"

  by (simp add: cons_rsp)

lemma append_rsp_fst:

  assumes a : "list_eq l1 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 : "list_eq l1 r1"

  assumes b : "list_eq l2 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:

  "(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:

  "(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:

  "(op = ===> op = ===> op = ===> op \<approx> ===> op =) fold1 fold1"

  apply (auto simp add: FUN_REL_EQ)

  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

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}

  @ @{thms ho_all_prs ho_ex_prs} *}

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 {* val consts = lookup_quot_consts defs *}

ML {* fun lift_tac_fset lthy t = lift_tac lthy t rel_eqv rel_refl rty quot trans2 rsp_thms reps_same defs *}

lemma "IN x EMPTY = False"

by (tactic {* lift_tac_fset @{context} @{thm m1} 1 *})

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 "\<forall>f g z a x. 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

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

ML {* val aps = findaps rty (prop_of (atomize_thm @{thm list.induct})) *}

ML {* val app_prs_thms = map (applic_prs @{context} rty qty absrep) aps *}

lemma "\<lbrakk>P EMPTY; \<And>a x. P x \<Longrightarrow> P (INSERT a x)\<rbrakk> \<Longrightarrow> P l"

apply (tactic {* lift_tac_fset @{context} @{thm list.induct} 1 *})

apply (tactic {* REPEAT_ALL_NEW (EqSubst.eqsubst_tac @{context} [0] app_prs_thms) 1 *})

apply (tactic {* simp_tac (HOL_ss addsimps [reps_same]) 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:

  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_rec list_rec"

  apply (auto simp add: FUN_REL_EQ)

  apply (erule_tac list_eq.induct)

  apply (simp_all)

  sorry

lemma list_case_rsp:

  "(op = ===> (op = ===> op \<approx> ===> op =) ===> op \<approx> ===> op =) list_case list_case"

  apply (auto simp add: FUN_REL_EQ)

  sorry

ML {* val rsp_thms = @{thms list_rec_rsp list_case_rsp} @ rsp_thms *}

ML {* val defs = @{thms fset_rec_def fset_case_def} @ defs *}

ML {* fun lift_tac_fset lthy t = lift_tac lthy t rel_eqv rel_refl rty quot trans2 rsp_thms reps_same defs *}

ML {* val aps = findaps rty (prop_of (atomize_thm @{thm list.recs(2)})) *}

ML {* val app_prs_thms = map (applic_prs @{context} rty qty absrep) aps *}

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 *})

apply (tactic {* REPEAT_ALL_NEW (EqSubst.eqsubst_tac @{context} [0] app_prs_thms) 1 *})

apply (tactic {* (simp_tac (HOL_ss addsimps

  @{thms eq_reflection[OF FUN_MAP_I] eq_reflection[OF id_apply] id_def_sym prod_fun_id map_id})) 1 *})

ML_prf {* val lower = flat (map (add_lower_defs @{context}) defs) *}

apply (tactic {* REPEAT_ALL_NEW (EqSubst.eqsubst_tac @{context} [0] lower) 1 *})

apply (tactic {* simp_tac (HOL_ss addsimps [reps_same]) 1 *})

done

ML {* val aps = findaps rty (prop_of (atomize_thm @{thm list.cases(2)})) *}

ML {* val app_prs_thms = map (applic_prs @{context} rty qty absrep) aps *}

lemma "fset_case (f1::'t) f2 (INSERT a xa) = f2 a xa"

apply (tactic {* lift_tac_fset @{context} @{thm list.cases(2)} 1 *})

apply (tactic {* REPEAT_ALL_NEW (EqSubst.eqsubst_tac @{context} [0] app_prs_thms) 1 *})