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 (for "'a fset")
  EMPTY :: "'a fset"
where
  "EMPTY \<equiv> ([]::'a list)"

term Nil
term EMPTY
thm EMPTY_def

quotient_def (for "'a fset")
  INSERT :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fet"
where
  "INSERT \<equiv> op #"

term Cons
term INSERT
thm INSERT_def

quotient_def (for "'a fset")
  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 (for "'a fset")
  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) \<Longrightarrow> card1 (x # xs) = Suc (card1 xs)"
  by simp

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

primrec
  fold1
where
  "fold1 f (g :: 'a \<Rightarrow> 'b) (z :: 'b) [] = z"
| "fold1 f g z (a # A) =
     (if ((!u v. (f u v = f v u))
      \<and> (!u v w. ((f u (f v w) = f (f u v) w))))
     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 (for "'a fset")
  IN :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool"
where
  "IN \<equiv> membship"

term membship
term IN
thm IN_def

(* FIXME: does not work yet 
quotient_def (for "'a fset")
  FOLD :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b"
where
  "FOLD \<equiv> fold1"
*)
local_setup {*
  old_make_const_def @{binding fold} @{term "fold1::('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd
*}

term fold1
term fold
thm fold_def

(* FIXME: does not work yet for all types*)
quotient_def (for "'a fset")
  fmap::"('a \<Rightarrow> 'a) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
where
  "fmap \<equiv> (map::('a \<Rightarrow> 'a) \<Rightarrow> 'a list \<Rightarrow> 'a list)"

term map
term fmap
thm fmap_def

ML {* val fset_defs = @{thms EMPTY_def IN_def FUNION_def card_def INSERT_def fmap_def fold_def} *}
(* ML {* val consts = map (fst o dest_Const o fst o Logic.dest_equals o concl_of) fset_defs *} *)

ML {*
  val consts = [@{const_name "Nil"}, @{const_name "Cons"},
                @{const_name "membship"}, @{const_name "card1"},
                @{const_name "append"}, @{const_name "fold1"},
                @{const_name "map"}];
*}

(* FIXME: does not work anymore :o( *)
ML {* val fset_defs_sym = add_lower_defs @{context} fset_defs *}

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 fun_rel_id:
  "op = ===> op = \<equiv> op ="
  apply (rule eq_reflection)
  apply (rule ext)
  apply (rule ext)
  apply (simp)
  apply (auto)
  apply (rule ext)
  apply (simp)
  done

lemma ho_map_rsp:
  "op = ===> op = ===> op \<approx> ===> op \<approx> map map"
  by (simp add: fun_rel_id map_rsp)

lemma map_append :
  "(map f ((a::'a list) @ b)) \<approx>
  ((map f a) ::'a list) @ (map f b)"
 by simp (rule list_eq_refl)

thm list.induct
lemma list_induct_hol4:
  fixes P :: "'a list \<Rightarrow> bool"
  assumes a: "((P []) \<and> (\<forall>t. (P t) \<longrightarrow> (\<forall>h. (P (h # t)))))"
  shows "\<forall>l. (P l)"
  using a
  apply (rule_tac allI)
  apply (induct_tac "l")
  apply (simp)
  apply (metis)
  done

ML {* (atomize_thm @{thm list_induct_hol4}) *}

ML {* regularise (prop_of (atomize_thm @{thm list_induct_hol4})) @{typ "'a list"} @{term "op \<approx>"} @{context} *}

prove list_induct_r: {*
   build_regularize_goal (atomize_thm @{thm list_induct_hol4}) @{typ "'a list"} @{term "op \<approx>"} @{context} *}
  apply (simp only: equiv_res_forall[OF equiv_list_eq])
  thm RIGHT_RES_FORALL_REGULAR
  apply (rule RIGHT_RES_FORALL_REGULAR)
  prefer 2
  apply (assumption)
  apply (metis)
  done

(* The all_prs and ex_prs should be proved for the instance... *)
ML {*
fun r_mk_comb_tac_fset ctxt =
  r_mk_comb_tac ctxt @{typ "'a list"} @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}
  (@{thms ho_memb_rsp ho_cons_rsp ho_card1_rsp ho_map_rsp ho_append_rsp} @ @{thms ho_all_prs ho_ex_prs})
*}


ML {* val thm = @{thm list_induct_r} OF [atomize_thm @{thm list_induct_hol4}] *}
ML {* val trm_r = build_repabs_goal @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}
ML {* val trm = build_repabs_term @{context} thm consts @{typ "'a list"} @{typ "'a fset"} *}

ML {* val rty = @{typ "'a list"} *}

ML {*
fun r_mk_comb_tac_fset ctxt =
  r_mk_comb_tac ctxt rty @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}
  (@{thms ho_memb_rsp ho_cons_rsp ho_card1_rsp ho_map_rsp ho_append_rsp} @ @{thms ho_all_prs ho_ex_prs})
*}


ML {* trm_r *}
prove list_induct_tr: trm_r
apply (atomize(full))
apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
done

prove list_induct_t: trm
apply (simp only: list_induct_tr[symmetric])
apply (tactic {* rtac thm 1 *})
done

thm m2
ML {* atomize_thm @{thm m2} *}

prove m2_r_p: {*
   build_regularize_goal (atomize_thm @{thm m2}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}
  apply (simp add: equiv_res_forall[OF equiv_list_eq])
done

ML {* val m2_r = @{thm m2_r_p} OF [atomize_thm @{thm m2}] *}
ML {* val m2_t_g = build_repabs_goal @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}
ML {* val m2_t = build_repabs_term @{context} m2_r consts @{typ "'a list"} @{typ "'a fset"} *}
prove m2_t_p: m2_t_g
apply (atomize(full))
apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
done

prove m2_t: m2_t
apply (simp only: m2_t_p[symmetric])
apply (tactic {* rtac m2_r 1 *})
done

lemma id_apply2 [simp]: "id x \<equiv> x"
  by (simp add: id_def)

ML {* val quot = @{thm QUOTIENT_fset} *}
ML {* val abs = findabs @{typ "'a list"} (prop_of (atomize_thm @{thm list_induct_hol4})) *}
ML {* val simp_lam_prs_thms = map (make_simp_lam_prs_thm @{context} quot) abs *}

ML {*
  fun simp_lam_prs lthy thm =
    simp_lam_prs lthy (eqsubst_thm lthy simp_lam_prs_thms thm)
    handle _ => thm
*}

ML {* val m2_t' = simp_lam_prs @{context} @{thm m2_t} *}

ML {* val ithm = simp_allex_prs @{context} quot m2_t' *}

ML {* val rthm = MetaSimplifier.rewrite_rule fset_defs_sym ithm *}
ML {* ObjectLogic.rulify rthm *}


ML {* val card1_suc_a = atomize_thm @{thm card1_suc} *}

prove card1_suc_r_p: {*
   build_regularize_goal (atomize_thm @{thm card1_suc}) @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}
  apply (simp add: equiv_res_forall[OF equiv_list_eq] equiv_res_exists[OF equiv_list_eq])
done

ML {* val card1_suc_r = @{thm card1_suc_r_p} OF [atomize_thm @{thm card1_suc}] *}
ML {* val card1_suc_t_g = build_repabs_goal @{context} card1_suc_r consts @{typ "'a list"} @{typ "'a fset"} *}
ML {* val card1_suc_t = build_repabs_term @{context} card1_suc_r consts @{typ "'a list"} @{typ "'a fset"} *}
prove card1_suc_t_p: card1_suc_t_g
apply (atomize(full))
apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
done

prove card1_suc_t: card1_suc_t
apply (simp only: card1_suc_t_p[symmetric])
apply (tactic {* rtac card1_suc_r 1 *})
done

ML {* val card1_suc_t_n = @{thm card1_suc_t} *}
ML {* val card1_suc_t' = simp_lam_prs @{context} @{thm card1_suc_t} *}
ML {* val ithm = simp_allex_prs @{context} quot card1_suc_t' *}
ML {* val rthm = MetaSimplifier.rewrite_rule fset_defs_sym ithm *}
ML {* val qthm = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same} rthm *}
ML {* ObjectLogic.rulify qthm *}

thm fold1.simps(2)
thm list.recs(2)
thm map_append

ML {* val ind_r_a = atomize_thm @{thm list.induct} *}
(*  prove {* build_regularize_goal ind_r_a @{typ "'a List.list"} @{term "op \<approx>"} @{context} *}
  ML_prf {*  fun tac ctxt =
       (asm_full_simp_tac ((Simplifier.context ctxt HOL_ss) addsimps
        [(@{thm equiv_res_forall} OF [@{thm equiv_list_eq}]),
         (@{thm equiv_res_exists} OF [@{thm equiv_list_eq}])])) THEN_ALL_NEW
         (((rtac @{thm RIGHT_RES_FORALL_REGULAR}) THEN' (RANGE [fn _ => all_tac, atac]) THEN'
         (MetisTools.metis_tac ctxt [])) ORELSE' (MetisTools.metis_tac ctxt [])); *}
  apply (tactic {* tac @{context} 1 *}) *)
ML {* val ind_r_r = regularize ind_r_a @{typ "'a List.list"} @{term "op \<approx>"} @{thm equiv_list_eq} @{context} *}
ML {*
  val rt = build_repabs_term @{context} ind_r_r consts @{typ "'a list"} @{typ "'a fset"}
  val rg = Logic.mk_equals ((Thm.prop_of ind_r_r), rt);
*}

prove rg
apply(atomize(full))
apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_fset @{context}) 1 *})
done

ML {* val ind_r_t =
  Toplevel.program (fn () =>
  repabs @{context} ind_r_r consts @{typ "'a list"} @{typ "'a fset"}
   @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}
   (@{thms ho_memb_rsp ho_cons_rsp ho_card1_rsp ho_map_rsp ho_append_rsp} @ @{thms ho_all_prs ho_ex_prs})
  )
*}
ML {* val ind_r_l = simp_lam_prs @{context} ind_r_t *}
lemma app_prs_for_induct: "(ABS_fset ---> id) f (REP_fset T1) = f T1"
  apply (simp add: fun_map.simps QUOT_TYPE_I_fset.thm10)
done

ML {* val ind_r_l1 = eqsubst_thm @{context} @{thms app_prs_for_induct} ind_r_l *}
ML {* val ind_r_l2 = eqsubst_thm @{context} @{thms app_prs_for_induct} ind_r_l1 *}
ML {* val ind_r_l3 = eqsubst_thm @{context} @{thms app_prs_for_induct} ind_r_l2 *}
ML {* val ind_r_l4 = eqsubst_thm @{context} @{thms app_prs_for_induct} ind_r_l3 *}
ML {* val ind_r_a = simp_allex_prs @{context} quot ind_r_l4 *}
ML {* val thm = @{thm FORALL_PRS[OF FUN_QUOTIENT[OF QUOTIENT_fset IDENTITY_QUOTIENT], symmetric]} *}
ML {* val ind_r_a1 = eqsubst_thm @{context} [thm] ind_r_a *}
ML {* val ind_r_d = repeat_eqsubst_thm @{context} fset_defs_sym ind_r_a1 *}
ML {* val ind_r_s = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same} ind_r_d *}
ML {* ObjectLogic.rulify ind_r_s *}

ML {*
fun lift thm =
let
  val ind_r_a = atomize_thm thm;
  val ind_r_r = regularize ind_r_a @{typ "'a List.list"} @{term "op \<approx>"} @{thm equiv_list_eq} @{context};
  val ind_r_t =
    repabs @{context} ind_r_r consts @{typ "'a list"} @{typ "'a fset"}
     @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2}
     (@{thms ho_memb_rsp ho_cons_rsp  ho_card1_rsp ho_map_rsp ho_append_rsp} @ @{thms ho_all_prs ho_ex_prs});
  val ind_r_l = simp_lam_prs @{context} ind_r_t;
  val ind_r_a = simp_allex_prs @{context} quot ind_r_l;
  val ind_r_d = repeat_eqsubst_thm @{context} fset_defs_sym ind_r_a;
  val ind_r_s = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same} ind_r_d
in
  ObjectLogic.rulify ind_r_s
end
*}
ML fset_defs

lemma eq_r: "a = b \<Longrightarrow> a \<approx> b"
by (simp add: list_eq_refl)


ML {* lift @{thm m2} *}
ML {* lift @{thm m1} *}
ML {* lift @{thm list_eq.intros(4)} *}
ML {* lift @{thm list_eq.intros(5)} *}
ML {* lift @{thm card1_suc} *}
ML {* lift @{thm map_append} *}
ML {* lift @{thm eq_r[OF append_assoc]} *}


(*notation ( output) "prop" ("#_" [1000] 1000) *)
notation ( output) "Trueprop" ("#_" [1000] 1000)

(*
ML {*
  fun lift_theorem_fset_aux thm lthy =
    let
      val ((_, [novars]), lthy2) = Variable.import true [thm] lthy;
      val goal = build_repabs_goal @{context} novars consts @{typ "'a list"} @{typ "'a fset"};
      val cgoal = cterm_of @{theory} goal;
      val cgoal2 = Thm.rhs_of (MetaSimplifier.rewrite true fset_defs_sym cgoal);
      val tac = transconv_fset_tac' @{context};
      val cthm = Goal.prove_internal [] cgoal2 (fn _ => tac);
      val nthm = MetaSimplifier.rewrite_rule [symmetric cthm] (snd (no_vars (Context.Theory @{theory}, thm)))
      val nthm2 = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_fset.REPS_same QUOT_TYPE_I_fset.thm10} nthm;
      val [nthm3] = ProofContext.export lthy2 lthy [nthm2]
    in
      nthm3
    end
*}
*)

(*
ML {* lift_theorem_fset_aux @{thm m1} @{context} *}
ML {*
  fun lift_theorem_fset name thm lthy =
    let
      val lifted_thm = lift_theorem_fset_aux thm lthy;
      val (_, lthy2) = note (name, lifted_thm) lthy;
    in
      lthy2
    end;
*}
*)

local_setup {* lift_theorem_fset @{binding "m1_lift"} @{thm m1} *}
local_setup {* lift_theorem_fset @{binding "leqi4_lift"} @{thm list_eq.intros(4)} *}
local_setup {* lift_theorem_fset @{binding "leqi5_lift"} @{thm list_eq.intros(5)} *}
local_setup {* lift_theorem_fset @{binding "m2_lift"} @{thm m2} *}
thm m1_lift
thm leqi4_lift
thm leqi5_lift
thm m2_lift
ML {* @{thm card1_suc_r} OF [card1_suc_f] *}
(*ML {* Toplevel.program (fn () => lift_theorem_fset @{binding "card_suc"}
     (@{thm card1_suc_r} OF [card1_suc_f]) @{context}) *}*)
(*local_setup {* lift_theorem_fset @{binding "card_suc"} @{thm card1_suc} *}*)

thm leqi4_lift
ML {*
  val (nam, typ) = hd (Term.add_vars (prop_of @{thm leqi4_lift}) [])
  val (_, l) = dest_Type typ
  val t = Type ("FSet.fset", l)
  val v = Var (nam, t)
  val cv = cterm_of @{theory} ((term_of @{cpat "REP_fset"}) $ v)
*}


end