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"

ML {* @{term "Abs_fset"} *}

local_setup {*

  make_const_def @{binding EMPTY} @{term "[]"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term Nil

term EMPTY

thm EMPTY_def

local_setup {*

  make_const_def @{binding INSERT} @{term "op #"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term Cons

term INSERT

thm INSERT_def

local_setup {*

  make_const_def @{binding UNION} @{term "op @"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term append

term UNION

thm UNION_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)))"

local_setup {*

  make_const_def @{binding card} @{term "card1"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

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 = [])"

  apply (induct a)

   apply (simp)

  apply (simp_all)

   apply meson

  apply (simp_all)

  done

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

  apply (induct xs)

  apply (auto intro: list_eq.intros)

  done

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

local_setup {*

  make_const_def @{binding IN} @{term "membship"} NoSyn @{typ "'a list"} @{typ "'a fset"} #> snd

*}

term membship

term IN

thm IN_def

ML {*

  val consts = [@{const_name "Nil"}, @{const_name "Cons"},

                @{const_name "membship"}, @{const_name "card1"},

                @{const_name "append"}, @{const_name "fold1"}];

*}

ML {* val fset_defs = @{thms EMPTY_def IN_def UNION_def card_def INSERT_def} *}

ML {* val fset_defs_sym = map (fn t => symmetric (unabs_def @{context} t)) fset_defs *}

thm fun_map.simps

text {* Respectfullness *}

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)"

  apply (simp add: FUN_REL.simps)

  apply (metis memb_rsp)

  done

lemma card1_rsp:

  fixes a b :: "'a list"

  assumes e: "a \<approx> b"

  shows "card1 a = card1 b"

  using e apply induct

  apply (simp_all add:memb_rsp)

  done

lemma ho_card1_rsp: "op \<approx> ===> op = card1 card1"

  apply (simp add: FUN_REL.simps)

  apply (metis card1_rsp)

  done

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 #"

  apply (simp add: FUN_REL.simps)

  apply (metis cons_rsp)

  done

lemma append_rsp_fst:

  assumes a : "list_eq l1 l2"

  shows "list_eq (l1 @ s) (l2 @ s)"

  using a

  apply(induct)

  apply(auto intro: list_eq.intros)

  apply(rule list_eq_refl)

done

lemma "(a @ b) \<approx> (b @ a)"

sorry

lemma (* ho_append_rsp: *)

  "op \<approx> ===> op \<approx> ===> op \<approx> op @ op @"

sorry

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 {*

fun regularize thm rty rel rel_eqv lthy =

  let

    val g = build_regularize_goal thm rty rel lthy;

    fun tac ctxt =

       (asm_full_simp_tac ((Simplifier.context ctxt HOL_ss) addsimps

        [(@{thm equiv_res_forall} OF [rel_eqv]),

         (@{thm equiv_res_exists} OF [rel_eqv])])) 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 []));

    val cthm = Goal.prove lthy [] [] g (fn x => tac (#context x) 1);

  in

    cthm OF [thm]

  end

*}

prove list_induct_r: {*

   build_regularize_goal (atomize_thm @{thm list_induct_hol4}) @{typ "'a List.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

ML {*

fun instantiate_tac thm = Subgoal.FOCUS (fn {concl, ...} =>

let

  val pat = Drule.strip_imp_concl (cprop_of thm)

  val insts = Thm.match (pat, concl)

in

  rtac (Drule.instantiate insts thm) 1

end

handle _ => no_tac

)

*}

ML {*

fun res_forall_rsp_tac ctxt =

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

  THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}

  THEN' instantiate_tac @{thm RES_FORALL_RSP} ctxt THEN'

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

*}

ML {*

fun res_exists_rsp_tac ctxt =

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

  THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI}

  THEN' instantiate_tac @{thm RES_EXISTS_RSP} ctxt THEN'

  (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

*}

ML {*

fun quotient_tac quot_thm =

  REPEAT_ALL_NEW (FIRST' [

    rtac @{thm FUN_QUOTIENT},

    rtac quot_thm,

    rtac @{thm IDENTITY_QUOTIENT}

  ])

*}

ML {*

fun LAMBDA_RES_TAC ctxt i st =

  (case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of

    (_ $ (_ $ (Abs(_,_,_))$(Abs(_,_,_)))) =>

      (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'

      (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})

  | _ => fn _ => no_tac) i st

*}

ML {*

fun WEAK_LAMBDA_RES_TAC ctxt i st =

  (case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of

    (_ $ (_ $ _$(Abs(_,_,_)))) =>

      (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'

      (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})

  | (_ $ (_ $ (Abs(_,_,_))$_)) =>

      (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN'

      (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI})

  | _ => fn _ => no_tac) i st

*}

ML {*

fun r_mk_comb_tac ctxt quot_thm reflex_thm trans_thm rsp_thms =

  (FIRST' [

    rtac @{thm FUN_QUOTIENT},

    rtac quot_thm,

    rtac @{thm IDENTITY_QUOTIENT},

    rtac @{thm ext},

    rtac trans_thm,

    LAMBDA_RES_TAC ctxt,

    res_forall_rsp_tac ctxt,

    res_exists_rsp_tac ctxt,

    (

     (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps (rsp_thms @ @{thms ho_all_prs ho_ex_prs})))

     THEN_ALL_NEW (fn _ => no_tac)

    ),

    (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),

    rtac refl,

    (instantiate_tac @{thm APPLY_RSP} ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),

    rtac reflex_thm, 

    atac,

    (

     (simp_tac ((Simplifier.context @{context} HOL_ss) addsimps @{thms FUN_REL.simps}))

     THEN_ALL_NEW (fn _ => no_tac)

    ),

    WEAK_LAMBDA_RES_TAC ctxt

    ])

*}

ML {*

fun repabs_eq lthy thm constructors rty qty quot_thm reflex_thm trans_thm rsp_thms =

  let

    val rt = build_repabs_term lthy thm constructors rty qty;

    val rg = Logic.mk_equals ((Thm.prop_of thm), rt);

    fun tac ctxt = (ObjectLogic.full_atomize_tac) THEN'

      (REPEAT_ALL_NEW (r_mk_comb_tac ctxt quot_thm reflex_thm trans_thm rsp_thms));

    val cthm = Goal.prove lthy [] [] rg (fn x => tac (#context x) 1);

  in

    (rt, cthm, thm)

  end

*}

ML {*

fun repabs_eq2 lthy (rt, thm, othm) =

  let

    fun tac2 ctxt =

     (simp_tac ((Simplifier.context ctxt empty_ss) addsimps [symmetric thm]))

     THEN' (rtac othm);

    val cthm = Goal.prove lthy [] [] rt (fn x => tac2 (#context x) 1);

  in

    cthm

  end

*}

ML {*

fun r_mk_comb_tac_fset ctxt =

  r_mk_comb_tac ctxt @{thm QUOTIENT_fset} @{thm list_eq_refl} @{thm QUOT_TYPE_I_fset.R_trans2} @{thms ho_memb_rsp ho_cons_rsp ho_card1_rsp}

*}

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 {* 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 lpis = @{thm LAMBDA_PRS} OF [@{thm QUOTIENT_fset}, @{thm IDENTITY_QUOTIENT}];

   val lpist = @{thm "HOL.sym"} OF [lpis];

   val lam_prs = MetaSimplifier.rewrite_rule [@{thm id_apply2}] lpist

*}

text {* the proper way to do it *}

ML {*

  fun findabs rty tm =

    case tm of

      Abs(_, T, b) =>

        let

          val b' = subst_bound ((Free ("x", T)), b);

          val tys = findabs rty b'

          val ty = fastype_of tm

        in if needs_lift rty ty then (ty :: tys) else tys

        end

    | f $ a => (findabs rty f) @ (findabs rty a)

    | _ => []

*}

ML {* findabs @{typ "'a list"} (prop_of (atomize_thm @{thm list_induct_hol4})) *}

ML {*

 val lpi = Drule.instantiate'

   [SOME @{ctyp "'a list"}, NONE, SOME @{ctyp "bool"}, NONE] [] @{thm LAMBDA_PRS};

*}

prove lambda_prs_l_b : {* concl_of lpi *}

apply (tactic {* compose_tac (false, @{thm LAMBDA_PRS}, 2) 1 *})

apply (tactic {* quotient_tac @{thm QUOTIENT_fset} 1 *})

apply (tactic {* quotient_tac @{thm QUOTIENT_fset} 1 *})

done

thm HOL.sym[OF lambda_prs_l_b,simplified]

ML {*

 val lpi = Drule.instantiate'

   [SOME @{ctyp "'a list \<Rightarrow> bool"}, NONE, SOME @{ctyp "bool"}, NONE] [] @{thm LAMBDA_PRS};

*}

prove lambda_prs_lb_b : {* concl_of lpi *}

apply (tactic {* compose_tac (false, @{thm LAMBDA_PRS}, 2) 1 *})

apply (tactic {* quotient_tac @{thm QUOTIENT_fset} 1 *})

apply (tactic {* quotient_tac @{thm QUOTIENT_fset} 1 *})

done

thm HOL.sym[OF lambda_prs_lb_b,simplified]

ML {*

fun eqsubst_thm ctxt thms thm =

  let

    val goalstate = Goal.init (Thm.cprop_of thm)

    val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of

      NONE => error "eqsubst_thm"

    | SOME th => cprem_of th 1

    val tac = (EqSubst.eqsubst_tac ctxt [0] thms 1) THEN simp_tac HOL_ss 1

    val cgoal = cterm_of (ProofContext.theory_of ctxt) (Logic.mk_equals (term_of (Thm.cprop_of thm), term_of a'))

    val rt = Toplevel.program (fn () => Goal.prove_internal [] cgoal (fn _ => tac));

  in

    Simplifier.rewrite_rule [rt] thm

  end

*}

ML {*

  fun simp_lam_prs lthy thm =

    simp_lam_prs lthy (eqsubst_thm lthy

      @{thms HOL.sym[OF lambda_prs_lb_b,simplified] HOL.sym[OF lambda_prs_l_b,simplified]}

    thm)

    handle _ => thm

*}

ML {* val m2_t' = eqsubst_thm @{context} [lam_prs] @{thm m2_t} *}

ML {*

  fun simp_allex_prs lthy thm =

    let

      val rwf = @{thm FORALL_PRS[OF QUOTIENT_fset]};

      val rwfs = @{thm "HOL.sym"} OF [spec OF [rwf]];

      val rwe = @{thm EXISTS_PRS[OF QUOTIENT_fset]};

      val rwes = @{thm "HOL.sym"} OF [spec OF [rwe]]

    in

      (simp_allex_prs lthy (eqsubst_thm lthy [rwfs, rwes] thm))

    end

    handle _ => thm

*}

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

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

ML {* ObjectLogic.rulify rthm *}