(*  Title:      Pure/drule.ML

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1993  University of Cambridge

Derived rules and other operations on theorems.

*)

infix 0 RS RSN RL RLN MRS MRL OF COMP;

signature BASIC_DRULE =

sig

  val mk_implies        : cterm * cterm -> cterm

  val list_implies      : cterm list * cterm -> cterm

  val dest_implies      : cterm -> cterm * cterm

  val dest_equals       : cterm -> cterm * cterm

  val skip_flexpairs    : cterm -> cterm

  val strip_imp_prems   : cterm -> cterm list

  val strip_imp_concl   : cterm -> cterm

  val cprems_of         : thm -> cterm list

  val read_insts        :

          Sign.sg -> (indexname -> typ option) * (indexname -> sort option)

                  -> (indexname -> typ option) * (indexname -> sort option)

                  -> string list -> (string*string)list

                  -> (indexname*ctyp)list * (cterm*cterm)list

  val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)

  val strip_shyps_warning : thm -> thm

  val forall_intr_list  : cterm list -> thm -> thm

  val forall_intr_frees : thm -> thm

  val forall_intr_vars  : thm -> thm

  val forall_elim_list  : cterm list -> thm -> thm

  val forall_elim_var   : int -> thm -> thm

  val forall_elim_vars  : int -> thm -> thm

  val forall_elim_vars_safe  : thm -> thm

  val freeze_thaw       : thm -> thm * (thm -> thm)

  val implies_elim_list : thm -> thm list -> thm

  val implies_intr_list : cterm list -> thm -> thm

  val instantiate       :

    (indexname * ctyp) list * (cterm * cterm) list -> thm -> thm

  val zero_var_indexes  : thm -> thm

  val standard          : thm -> thm

  val rotate_prems      : int -> thm -> thm

  val assume_ax         : theory -> string -> thm

  val RSN               : thm * (int * thm) -> thm

  val RS                : thm * thm -> thm

  val RLN               : thm list * (int * thm list) -> thm list

  val RL                : thm list * thm list -> thm list

  val MRS               : thm list * thm -> thm

  val MRL               : thm list list * thm list -> thm list

  val OF                : thm * thm list -> thm

  val compose           : thm * int * thm -> thm list

  val COMP              : thm * thm -> thm

  val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm

  val read_instantiate  : (string*string)list -> thm -> thm

  val cterm_instantiate : (cterm*cterm)list -> thm -> thm

  val weak_eq_thm       : thm * thm -> bool

  val eq_thm_sg         : thm * thm -> bool

  val size_of_thm       : thm -> int

  val reflexive_thm     : thm

  val symmetric_thm     : thm

  val transitive_thm    : thm

  val refl_implies      : thm

  val symmetric_fun     : thm -> thm

  val imp_cong          : thm

  val swap_prems_eq     : thm

  val equal_abs_elim    : cterm  -> thm -> thm

  val equal_abs_elim_list: cterm list -> thm -> thm

  val flexpair_abs_elim_list: cterm list -> thm -> thm

  val asm_rl            : thm

  val cut_rl            : thm

  val revcut_rl         : thm

  val thin_rl           : thm

  val triv_forall_equality: thm

  val swap_prems_rl     : thm

  val equal_intr_rule   : thm

  val inst              : string -> string -> thm -> thm

  val instantiate'      : ctyp option list -> cterm option list -> thm -> thm

  val incr_indexes_wrt  : int list -> ctyp list -> cterm list -> thm list -> thm -> thm

end;

signature DRULE =

sig

  include BASIC_DRULE

  val rule_attribute    : ('a -> thm -> thm) -> 'a attribute

  val tag_rule          : tag -> thm -> thm

  val untag_rule        : string -> thm -> thm

  val tag               : tag -> 'a attribute

  val untag             : string -> 'a attribute

  val tag_lemma         : 'a attribute

  val tag_assumption    : 'a attribute

  val tag_internal      : 'a attribute

  val has_internal	: tag list -> bool

  val close_derivation  : thm -> thm

  val compose_single    : thm * int * thm -> thm

  val add_rules		: thm list -> thm list -> thm list

  val del_rules		: thm list -> thm list -> thm list

  val merge_rules	: thm list * thm list -> thm list

  val norm_hhf_eq	: thm

  val triv_goal         : thm

  val rev_triv_goal     : thm

  val freeze_all        : thm -> thm

  val mk_triv_goal      : cterm -> thm

  val mk_cgoal          : cterm -> cterm

  val assume_goal       : cterm -> thm

  val tvars_of_terms    : term list -> (indexname * sort) list

  val vars_of_terms     : term list -> (indexname * typ) list

  val tvars_of          : thm -> (indexname * sort) list

  val vars_of           : thm -> (indexname * typ) list

  val unvarifyT         : thm -> thm

  val unvarify          : thm -> thm

  val tvars_intr_list	: string list -> thm -> thm

end;

structure Drule: DRULE =

struct

(** some cterm->cterm operations: much faster than calling cterm_of! **)

(** SAME NAMES as in structure Logic: use compound identifiers! **)

(*dest_implies for cterms. Note T=prop below*)

fun dest_implies ct =

    case term_of ct of

        (Const("==>", _) $ _ $ _) =>

            let val (ct1,ct2) = dest_comb ct

            in  (#2 (dest_comb ct1), ct2)  end

      | _ => raise TERM ("dest_implies", [term_of ct]) ;

fun dest_equals ct =

    case term_of ct of

        (Const("==", _) $ _ $ _) =>

            let val (ct1,ct2) = dest_comb ct

            in  (#2 (dest_comb ct1), ct2)  end

      | _ => raise TERM ("dest_equals", [term_of ct]) ;

(*Discard flexflex pairs; return a cterm*)

fun skip_flexpairs ct =

    case term_of ct of

        (Const("==>", _) $ (Const("=?=",_)$_$_) $ _) =>

            skip_flexpairs (#2 (dest_implies ct))

      | _ => ct;

(* A1==>...An==>B  goes to  [A1,...,An], where B is not an implication *)

fun strip_imp_prems ct =

    let val (cA,cB) = dest_implies ct

    in  cA :: strip_imp_prems cB  end

    handle TERM _ => [];

(* A1==>...An==>B  goes to B, where B is not an implication *)

fun strip_imp_concl ct =

    case term_of ct of (Const("==>", _) $ _ $ _) =>

        strip_imp_concl (#2 (dest_comb ct))

  | _ => ct;

(*The premises of a theorem, as a cterm list*)

val cprems_of = strip_imp_prems o skip_flexpairs o cprop_of;

val proto_sign = Theory.sign_of ProtoPure.thy;

val implies = cterm_of proto_sign Term.implies;

(*cterm version of mk_implies*)

fun mk_implies(A,B) = capply (capply implies A) B;

(*cterm version of list_implies: [A1,...,An], B  goes to [|A1;==>;An|]==>B *)

fun list_implies([], B) = B

  | list_implies(A::AS, B) = mk_implies (A, list_implies(AS,B));

(** reading of instantiations **)

fun absent ixn =

  error("No such variable in term: " ^ Syntax.string_of_vname ixn);

fun inst_failure ixn =

  error("Instantiation of " ^ Syntax.string_of_vname ixn ^ " fails");

fun read_insts sign (rtypes,rsorts) (types,sorts) used insts =

let

    fun split([],tvs,vs) = (tvs,vs)

      | split((sv,st)::l,tvs,vs) = (case Symbol.explode sv of

                  "'"::cs => split(l,(Syntax.indexname cs,st)::tvs,vs)

                | cs => split(l,tvs,(Syntax.indexname cs,st)::vs));

    val (tvs,vs) = split(insts,[],[]);

    fun readT((a,i),st) =

        let val ixn = ("'" ^ a,i);

            val S = case rsorts ixn of Some S => S | None => absent ixn;

            val T = Sign.read_typ (sign,sorts) st;

        in if Sign.typ_instance sign (T, TVar(ixn,S)) then (ixn,T)

           else inst_failure ixn

        end

    val tye = map readT tvs;

    fun mkty(ixn,st) = (case rtypes ixn of

                          Some T => (ixn,(st,typ_subst_TVars tye T))

                        | None => absent ixn);

    val ixnsTs = map mkty vs;

    val ixns = map fst ixnsTs

    and sTs  = map snd ixnsTs

    val (cts,tye2) = read_def_cterms(sign,types,sorts) used false sTs;

    fun mkcVar(ixn,T) =

        let val U = typ_subst_TVars tye2 T

        in cterm_of sign (Var(ixn,U)) end

    val ixnTs = ListPair.zip(ixns, map snd sTs)

in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) (tye2 @ tye),

    ListPair.zip(map mkcVar ixnTs,cts))

end;

(*** Find the type (sort) associated with a (T)Var or (T)Free in a term

     Used for establishing default types (of variables) and sorts (of

     type variables) when reading another term.

     Index -1 indicates that a (T)Free rather than a (T)Var is wanted.

***)

fun types_sorts thm =

    let val {prop,hyps,...} = rep_thm thm;

        val big = list_comb(prop,hyps); (* bogus term! *)

        val vars = map dest_Var (term_vars big);

        val frees = map dest_Free (term_frees big);

        val tvars = term_tvars big;

        val tfrees = term_tfrees big;

        fun typ(a,i) = if i<0 then assoc(frees,a) else assoc(vars,(a,i));

        fun sort(a,i) = if i<0 then assoc(tfrees,a) else assoc(tvars,(a,i));

    in (typ,sort) end;

(** basic attributes **)

(* dependent rules *)

fun rule_attribute f (x, thm) = (x, (f x thm));

(* add / delete tags *)

fun map_tags f thm =

  Thm.put_name_tags (Thm.name_of_thm thm, f (#2 (Thm.get_name_tags thm))) thm;

fun tag_rule tg = map_tags (fn tgs => if tg mem tgs then tgs else tgs @ [tg]);

fun untag_rule s = map_tags (filter_out (equal s o #1));

fun tag tg x = rule_attribute (K (tag_rule tg)) x;

fun untag s x = rule_attribute (K (untag_rule s)) x;

fun simple_tag name x = tag (name, []) x;

fun tag_lemma x = simple_tag "lemma" x;

fun tag_assumption x = simple_tag "assumption" x;

val internal_tag = ("internal", []);

fun tag_internal x = tag internal_tag x;

fun has_internal tags = exists (equal internal_tag) tags;

(** Standardization of rules **)

(*Strip extraneous shyps as far as possible*)

fun strip_shyps_warning thm =

  let

    val str_of_sort = Sign.str_of_sort (Thm.sign_of_thm thm);

    val thm' = Thm.strip_shyps thm;

    val xshyps = Thm.extra_shyps thm';

  in

    if null xshyps then ()

    else warning ("Pending sort hypotheses: " ^ commas (map str_of_sort xshyps));

    thm'

  end;

(*Generalization over a list of variables, IGNORING bad ones*)

fun forall_intr_list [] th = th

  | forall_intr_list (y::ys) th =

        let val gth = forall_intr_list ys th

        in  forall_intr y gth   handle THM _ =>  gth  end;

(*Generalization over all suitable Free variables*)

fun forall_intr_frees th =

    let val {prop,sign,...} = rep_thm th

    in  forall_intr_list

         (map (cterm_of sign) (sort (make_ord atless) (term_frees prop)))

         th

    end;

val forall_elim_var = PureThy.forall_elim_var;

val forall_elim_vars = PureThy.forall_elim_vars;

fun forall_elim_vars_safe th =

  forall_elim_vars_safe (forall_elim_var (#maxidx (Thm.rep_thm th) + 1) th)

    handle THM _ => th;

(*Specialization over a list of cterms*)

fun forall_elim_list cts th = foldr (uncurry forall_elim) (rev cts, th);

(* maps [A1,...,An], B   to   [| A1;...;An |] ==> B  *)

fun implies_intr_list cAs th = foldr (uncurry implies_intr) (cAs,th);

(* maps [| A1;...;An |] ==> B and [A1,...,An]   to   B *)

fun implies_elim_list impth ths = foldl (uncurry implies_elim) (impth,ths);

(*Reset Var indexes to zero, renaming to preserve distinctness*)

fun zero_var_indexes th =

    let val {prop,sign,...} = rep_thm th;

        val vars = term_vars prop

        val bs = foldl add_new_id ([], map (fn Var((a,_),_)=>a) vars)

        val inrs = add_term_tvars(prop,[]);

        val nms' = rev(foldl add_new_id ([], map (#1 o #1) inrs));

        val tye = ListPair.map (fn ((v,rs),a) => (v, TVar((a,0),rs)))

                     (inrs, nms')

        val ctye = map (fn (v,T) => (v,ctyp_of sign T)) tye;

        fun varpairs([],[]) = []

          | varpairs((var as Var(v,T)) :: vars, b::bs) =

                let val T' = typ_subst_TVars tye T

                in (cterm_of sign (Var(v,T')),

                    cterm_of sign (Var((b,0),T'))) :: varpairs(vars,bs)

                end

          | varpairs _ = raise TERM("varpairs", []);

    in Thm.instantiate (ctye, varpairs(vars,rev bs)) th end;

(*Standard form of object-rule: no hypotheses, Frees, or outer quantifiers;

    all generality expressed by Vars having index 0.*)

fun close_derivation thm =

  if Thm.get_name_tags thm = ("", []) then Thm.name_thm ("", thm)

  else thm;

fun standard th =

  let val {maxidx,...} = rep_thm th in

    th

    |> implies_intr_hyps

    |> forall_intr_frees |> forall_elim_vars (maxidx + 1)

    |> strip_shyps_warning

    |> zero_var_indexes |> Thm.varifyT |> Thm.compress |> close_derivation

  end;

(*Convert all Vars in a theorem to Frees.  Also return a function for

  reversing that operation.  DOES NOT WORK FOR TYPE VARIABLES.

  Similar code in type/freeze_thaw*)

fun freeze_thaw th =

 let val fth = freezeT th

     val {prop,sign,...} = rep_thm fth

 in

   case term_vars prop of

       [] => (fth, fn x => x)

     | vars =>

         let fun newName (Var(ix,_), (pairs,used)) =

                   let val v = variant used (string_of_indexname ix)

                   in  ((ix,v)::pairs, v::used)  end;

             val (alist, _) = foldr newName

                                (vars, ([], add_term_names (prop, [])))

             fun mk_inst (Var(v,T)) =

                 (cterm_of sign (Var(v,T)),

                  cterm_of sign (Free(the (assoc(alist,v)), T)))

             val insts = map mk_inst vars

             fun thaw th' =

                 th' |> forall_intr_list (map #2 insts)

                     |> forall_elim_list (map #1 insts)

         in  (Thm.instantiate ([],insts) fth, thaw)  end

 end;

(*Rotates a rule's premises to the left by k*)

val rotate_prems = permute_prems 0;

(*Assume a new formula, read following the same conventions as axioms.

  Generalizes over Free variables,

  creates the assumption, and then strips quantifiers.

  Example is [| ALL x:?A. ?P(x) |] ==> [| ?P(?a) |]

             [ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ]    *)

fun assume_ax thy sP =

    let val sign = Theory.sign_of thy

        val prop = Logic.close_form (term_of (read_cterm sign (sP, propT)))

    in forall_elim_vars 0 (assume (cterm_of sign prop))  end;

(*Resolution: exactly one resolvent must be produced.*)

fun tha RSN (i,thb) =

  case Seq.chop (2, biresolution false [(false,tha)] i thb) of

      ([th],_) => th

    | ([],_)   => raise THM("RSN: no unifiers", i, [tha,thb])

    |      _   => raise THM("RSN: multiple unifiers", i, [tha,thb]);

(*resolution: P==>Q, Q==>R gives P==>R. *)

fun tha RS thb = tha RSN (1,thb);

(*For joining lists of rules*)

fun thas RLN (i,thbs) =

  let val resolve = biresolution false (map (pair false) thas) i

      fun resb thb = Seq.list_of (resolve thb) handle THM _ => []

  in  List.concat (map resb thbs)  end;

fun thas RL thbs = thas RLN (1,thbs);

(*Resolve a list of rules against bottom_rl from right to left;

  makes proof trees*)

fun rls MRS bottom_rl =

  let fun rs_aux i [] = bottom_rl

        | rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)

  in  rs_aux 1 rls  end;

(*As above, but for rule lists*)

fun rlss MRL bottom_rls =

  let fun rs_aux i [] = bottom_rls

        | rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)

  in  rs_aux 1 rlss  end;

(*A version of MRS with more appropriate argument order*)

fun bottom_rl OF rls = rls MRS bottom_rl;

(*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R

  with no lifting or renaming!  Q may contain ==> or meta-quants

  ALWAYS deletes premise i *)

fun compose(tha,i,thb) =

    Seq.list_of (bicompose false (false,tha,0) i thb);

fun compose_single (tha,i,thb) =

  (case compose (tha,i,thb) of

    [th] => th

  | _ => raise THM ("compose: unique result expected", i, [tha,thb]));

(*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)

fun tha COMP thb =

    case compose(tha,1,thb) of

        [th] => th

      | _ =>   raise THM("COMP", 1, [tha,thb]);

(** theorem equality **)

(*Do the two theorems have the same signature?*)

fun eq_thm_sg (th1,th2) = Sign.eq_sg(#sign(rep_thm th1), #sign(rep_thm th2));

(*Useful "distance" function for BEST_FIRST*)

val size_of_thm = size_of_term o #prop o rep_thm;

(*maintain lists of theorems --- preserving canonical order*)

fun del_rules rs rules = Library.gen_rems Thm.eq_thm (rules, rs);

fun add_rules rs rules = rs @ del_rules rs rules;

fun merge_rules (rules1, rules2) = Library.generic_merge Thm.eq_thm I I rules1 rules2;

(** Mark Staples's weaker version of eq_thm: ignores variable renaming and

    (some) type variable renaming **)

 (* Can't use term_vars, because it sorts the resulting list of variable names.

    We instead need the unique list noramlised by the order of appearance

    in the term. *)

fun term_vars' (t as Var(v,T)) = [t]

  | term_vars' (Abs(_,_,b)) = term_vars' b

  | term_vars' (f$a) = (term_vars' f) @ (term_vars' a)

  | term_vars' _ = [];

fun forall_intr_vars th =

  let val {prop,sign,...} = rep_thm th;

      val vars = distinct (term_vars' prop);

  in forall_intr_list (map (cterm_of sign) vars) th end;

fun weak_eq_thm (tha,thb) =

    eq_thm(forall_intr_vars (freezeT tha), forall_intr_vars (freezeT thb));

(*** Meta-Rewriting Rules ***)

fun read_prop s = read_cterm proto_sign (s, propT);

fun store_thm name thm = hd (PureThy.smart_store_thms (name, [thm]));

fun store_standard_thm name thm = store_thm name (standard thm);

val reflexive_thm =

  let val cx = cterm_of proto_sign (Var(("x",0),TVar(("'a",0),logicS)))

  in store_standard_thm "reflexive" (Thm.reflexive cx) end;

val symmetric_thm =

  let val xy = read_prop "x::'a::logic == y"

  in store_standard_thm "symmetric" (Thm.implies_intr_hyps (Thm.symmetric (Thm.assume xy))) end;

val transitive_thm =

  let val xy = read_prop "x::'a::logic == y"

      val yz = read_prop "y::'a::logic == z"

      val xythm = Thm.assume xy and yzthm = Thm.assume yz

  in store_standard_thm "transitive" (Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;

fun symmetric_fun thm = thm RS symmetric_thm;

val imp_cong =

  let

    val ABC = read_prop "PROP A ==> PROP B == PROP C"

    val AB = read_prop "PROP A ==> PROP B"

    val AC = read_prop "PROP A ==> PROP C"

    val A = read_prop "PROP A"

  in

    store_standard_thm "imp_cong2" (implies_intr ABC (equal_intr

      (implies_intr AB (implies_intr A

        (equal_elim (implies_elim (assume ABC) (assume A))

          (implies_elim (assume AB) (assume A)))))

      (implies_intr AC (implies_intr A

        (equal_elim (symmetric (implies_elim (assume ABC) (assume A)))

          (implies_elim (assume AC) (assume A)))))))

  end;

val swap_prems_eq =

  let

    val ABC = read_prop "PROP A ==> PROP B ==> PROP C"

    val BAC = read_prop "PROP B ==> PROP A ==> PROP C"

    val A = read_prop "PROP A"

    val B = read_prop "PROP B"

  in

    store_standard_thm "swap_prems_eq" (equal_intr

      (implies_intr ABC (implies_intr B (implies_intr A

        (implies_elim (implies_elim (assume ABC) (assume A)) (assume B)))))

      (implies_intr BAC (implies_intr A (implies_intr B

        (implies_elim (implies_elim (assume BAC) (assume B)) (assume A))))))

  end;

val refl_implies = reflexive implies;

(*** Some useful meta-theorems ***)

(*The rule V/V, obtains assumption solving for eresolve_tac*)

val asm_rl = store_standard_thm "asm_rl" (Thm.trivial (read_prop "PROP ?psi"));

val _ = store_thm "_" asm_rl;

(*Meta-level cut rule: [| V==>W; V |] ==> W *)

val cut_rl =

  store_standard_thm "cut_rl"

    (Thm.trivial (read_prop "PROP ?psi ==> PROP ?theta"));

(*Generalized elim rule for one conclusion; cut_rl with reversed premises:

     [| PROP V;  PROP V ==> PROP W |] ==> PROP W *)

val revcut_rl =

  let val V = read_prop "PROP V"

      and VW = read_prop "PROP V ==> PROP W";

  in

    store_standard_thm "revcut_rl"

      (implies_intr V (implies_intr VW (implies_elim (assume VW) (assume V))))

  end;

(*for deleting an unwanted assumption*)

val thin_rl =

  let val V = read_prop "PROP V"

      and W = read_prop "PROP W";

  in  store_standard_thm "thin_rl" (implies_intr V (implies_intr W (assume W)))

  end;

(* (!!x. PROP ?V) == PROP ?V       Allows removal of redundant parameters*)

val triv_forall_equality =

  let val V  = read_prop "PROP V"

      and QV = read_prop "!!x::'a. PROP V"

      and x  = read_cterm proto_sign ("x", TypeInfer.logicT);

  in

    store_standard_thm "triv_forall_equality"

      (standard (equal_intr (implies_intr QV (forall_elim x (assume QV)))

        (implies_intr V  (forall_intr x (assume V)))))

  end;

(* (PROP ?PhiA ==> PROP ?PhiB ==> PROP ?Psi) ==>

   (PROP ?PhiB ==> PROP ?PhiA ==> PROP ?Psi)

   `thm COMP swap_prems_rl' swaps the first two premises of `thm'

*)

val swap_prems_rl =

  let val cmajor = read_prop "PROP PhiA ==> PROP PhiB ==> PROP Psi";

      val major = assume cmajor;

      val cminor1 = read_prop "PROP PhiA";

      val minor1 = assume cminor1;

      val cminor2 = read_prop "PROP PhiB";

      val minor2 = assume cminor2;

  in store_standard_thm "swap_prems_rl"

       (implies_intr cmajor (implies_intr cminor2 (implies_intr cminor1

         (implies_elim (implies_elim major minor1) minor2))))

  end;

(* [| PROP ?phi ==> PROP ?psi; PROP ?psi ==> PROP ?phi |]

   ==> PROP ?phi == PROP ?psi

   Introduction rule for == as a meta-theorem.

*)

val equal_intr_rule =

  let val PQ = read_prop "PROP phi ==> PROP psi"

      and QP = read_prop "PROP psi ==> PROP phi"

  in

    store_standard_thm "equal_intr_rule"

      (implies_intr PQ (implies_intr QP (equal_intr (assume PQ) (assume QP))))

  end;

(*(PROP ?phi ==> (!!x. PROP ?psi(x))) == (!!x. PROP ?phi ==> PROP ?psi(x))

  Rewrite rule for HHF normalization.

  Note: the syntax of ProtoPure is insufficient to handle this

  statement; storing it would be asking for trouble, e.g. when someone

  tries to print the theory later.

*)

val norm_hhf_eq =

  let

    val cert = Thm.cterm_of proto_sign;

    val aT = TFree ("'a", Term.logicS);

    val all = Term.all aT;

    val x = Free ("x", aT);

    val phi = Free ("phi", propT);

    val psi = Free ("psi", aT --> propT);

    val cx = cert x;

    val cphi = cert phi;

    val lhs = cert (Logic.mk_implies (phi, all $ Abs ("x", aT, psi $ Bound 0)));

    val rhs = cert (all $ Abs ("x", aT, Logic.mk_implies (phi, psi $ Bound 0)));

  in

    Thm.equal_intr

      (Thm.implies_elim (Thm.assume lhs) (Thm.assume cphi)

        |> Thm.forall_elim cx

        |> Thm.implies_intr cphi

        |> Thm.forall_intr cx

        |> Thm.implies_intr lhs)

      (Thm.implies_elim

          (Thm.assume rhs |> Thm.forall_elim cx) (Thm.assume cphi)

        |> Thm.forall_intr cx

        |> Thm.implies_intr cphi

        |> Thm.implies_intr rhs)

    |> store_standard_thm "norm_hhf_eq"

  end;

(*** Instantiate theorem th, reading instantiations under signature sg ****)

(*Version that normalizes the result: Thm.instantiate no longer does that*)

fun instantiate instpair th = Thm.instantiate instpair th  COMP   asm_rl;

fun read_instantiate_sg sg sinsts th =

    let val ts = types_sorts th;

        val used = add_term_tvarnames(#prop(rep_thm th),[]);

    in  instantiate (read_insts sg ts ts used sinsts) th  end;

(*Instantiate theorem th, reading instantiations under theory of th*)

fun read_instantiate sinsts th =

    read_instantiate_sg (#sign (rep_thm th)) sinsts th;

(*Left-to-right replacements: tpairs = [...,(vi,ti),...].

  Instantiates distinct Vars by terms, inferring type instantiations. *)

local

  fun add_types ((ct,cu), (sign,tye,maxidx)) =

    let val {sign=signt, t=t, T= T, maxidx=maxt,...} = rep_cterm ct

        and {sign=signu, t=u, T= U, maxidx=maxu,...} = rep_cterm cu;

        val maxi = Int.max(maxidx, Int.max(maxt, maxu));

        val sign' = Sign.merge(sign, Sign.merge(signt, signu))

        val (tye',maxi') = Type.unify (#tsig(Sign.rep_sg sign')) maxi tye (T,U)

          handle Type.TUNIFY => raise TYPE("Ill-typed instantiation", [T,U], [t,u])

    in  (sign', tye', maxi')  end;

in

fun cterm_instantiate ctpairs0 th =

  let val (sign,tye,_) = foldr add_types (ctpairs0, (#sign(rep_thm th), Vartab.empty, 0))

      fun instT(ct,cu) = let val inst = subst_TVars_Vartab tye

                         in (cterm_fun inst ct, cterm_fun inst cu) end

      fun ctyp2 (ix,T) = (ix, ctyp_of sign T)

  in  instantiate (map ctyp2 (Vartab.dest tye), map instT ctpairs0) th  end

  handle TERM _ =>

           raise THM("cterm_instantiate: incompatible signatures",0,[th])

       | TYPE (msg, _, _) => raise THM(msg, 0, [th])

end;

(** Derived rules mainly for METAHYPS **)

(*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)

fun equal_abs_elim ca eqth =

  let val {sign=signa, t=a, ...} = rep_cterm ca

      and combth = combination eqth (reflexive ca)

      val {sign,prop,...} = rep_thm eqth

      val (abst,absu) = Logic.dest_equals prop

      val cterm = cterm_of (Sign.merge (sign,signa))

  in  transitive (symmetric (beta_conversion false (cterm (abst$a))))

           (transitive combth (beta_conversion false (cterm (absu$a))))

  end

  handle THM _ => raise THM("equal_abs_elim", 0, [eqth]);

(*Calling equal_abs_elim with multiple terms*)

fun equal_abs_elim_list cts th = foldr (uncurry equal_abs_elim) (rev cts, th);

local

  val alpha = TVar(("'a",0), [])     (*  type ?'a::{}  *)

  fun err th = raise THM("flexpair_inst: ", 0, [th])

  fun flexpair_inst def th =

    let val {prop = Const _ $ t $ u,  sign,...} = rep_thm th

        val cterm = cterm_of sign

        fun cvar a = cterm(Var((a,0),alpha))

        val def' = cterm_instantiate [(cvar"t", cterm t), (cvar"u", cterm u)]

                   def

    in  equal_elim def' th

    end

    handle THM _ => err th | Bind => err th

in

val flexpair_intr = flexpair_inst (symmetric ProtoPure.flexpair_def)

and flexpair_elim = flexpair_inst ProtoPure.flexpair_def

end;

(*Version for flexflex pairs -- this supports lifting.*)

fun flexpair_abs_elim_list cts =

    flexpair_intr o equal_abs_elim_list cts o flexpair_elim;

(*** GOAL (PROP A) <==> PROP A ***)

local

  val A = read_prop "PROP A";

  val G = read_prop "GOAL (PROP A)";

  val (G_def, _) = freeze_thaw ProtoPure.Goal_def;

in

  val triv_goal = store_thm "triv_goal"

    (tag_rule internal_tag (standard (Thm.equal_elim (Thm.symmetric G_def) (Thm.assume A))));

  val rev_triv_goal = store_thm "rev_triv_goal"

    (tag_rule internal_tag (standard (Thm.equal_elim G_def (Thm.assume G))));

end;

val mk_cgoal = Thm.capply (Thm.cterm_of proto_sign Logic.goal_const);

fun assume_goal ct = Thm.assume (mk_cgoal ct) RS rev_triv_goal;

(** variations on instantiate **)

(*shorthand for instantiating just one variable in the current theory*)

fun inst x t = read_instantiate_sg (sign_of (the_context())) [(x,t)];

(* collect vars *)

val add_tvarsT = foldl_atyps (fn (vs, TVar v) => v ins vs | (vs, _) => vs);

val add_tvars = foldl_types add_tvarsT;

val add_vars = foldl_aterms (fn (vs, Var v) => v ins vs | (vs, _) => vs);

fun tvars_of_terms ts = rev (foldl add_tvars ([], ts));

fun vars_of_terms ts = rev (foldl add_vars ([], ts));

fun tvars_of thm = tvars_of_terms [#prop (Thm.rep_thm thm)];

fun vars_of thm = vars_of_terms [#prop (Thm.rep_thm thm)];

(* instantiate by left-to-right occurrence of variables *)

fun instantiate' cTs cts thm =

  let

    fun err msg =

      raise TYPE ("instantiate': " ^ msg,

        mapfilter (apsome Thm.typ_of) cTs,

        mapfilter (apsome Thm.term_of) cts);

    fun inst_of (v, ct) =

      (Thm.cterm_of (#sign (Thm.rep_cterm ct)) (Var v), ct)

        handle TYPE (msg, _, _) => err msg;

    fun zip_vars _ [] = []

      | zip_vars (_ :: vs) (None :: opt_ts) = zip_vars vs opt_ts

      | zip_vars (v :: vs) (Some t :: opt_ts) = (v, t) :: zip_vars vs opt_ts

      | zip_vars [] _ = err "more instantiations than variables in thm";

    (*instantiate types first!*)

    val thm' =

      if forall is_none cTs then thm

      else Thm.instantiate (zip_vars (map fst (tvars_of thm)) cTs, []) thm;

    in

      if forall is_none cts then thm'

      else Thm.instantiate ([], map inst_of (zip_vars (vars_of thm') cts)) thm'

    end;

(* unvarify(T) *)

(*assume thm in standard form, i.e. no frees, 0 var indexes*)

fun unvarifyT thm =

  let

    val cT = Thm.ctyp_of (Thm.sign_of_thm thm);

    val tfrees = map (fn ((x, _), S) => Some (cT (TFree (x, S)))) (tvars_of thm);

  in instantiate' tfrees [] thm end;

fun unvarify raw_thm =

  let

    val thm = unvarifyT raw_thm;

    val ct = Thm.cterm_of (Thm.sign_of_thm thm);

    val frees = map (fn ((x, _), T) => Some (ct (Free (x, T)))) (vars_of thm);

  in instantiate' [] frees thm end;

(* tvars_intr_list *)

fun tfrees_of thm =

  let val {hyps, prop, ...} = Thm.rep_thm thm

  in foldr Term.add_term_tfree_names (prop :: hyps, []) end;

fun tvars_intr_list tfrees thm =

  Thm.varifyT' (tfrees_of thm \\ tfrees) thm;

(* increment var indexes *)

fun incr_indexes_wrt is cTs cts thms =

  let

    val maxidx =

      foldl Int.max (~1, is @

        map (maxidx_of_typ o #T o Thm.rep_ctyp) cTs @

        map (#maxidx o Thm.rep_cterm) cts @

        map (#maxidx o Thm.rep_thm) thms);

  in Thm.incr_indexes (maxidx + 1) end;

(* freeze_all *)

(*freeze all (T)Vars; assumes thm in standard form*)

fun freeze_all_TVars thm =

  (case tvars_of thm of

    [] => thm

  | tvars =>

      let val cert = Thm.ctyp_of (Thm.sign_of_thm thm)

      in instantiate' (map (fn ((x, _), S) => Some (cert (TFree (x, S)))) tvars) [] thm end);

fun freeze_all_Vars thm =

  (case vars_of thm of

    [] => thm

  | vars =>

      let val cert = Thm.cterm_of (Thm.sign_of_thm thm)

      in instantiate' [] (map (fn ((x, _), T) => Some (cert (Free (x, T)))) vars) thm end);

val freeze_all = freeze_all_Vars o freeze_all_TVars;

(* mk_triv_goal *)

(*make an initial proof state, "PROP A ==> (PROP A)" *)

fun mk_triv_goal ct = instantiate' [] [Some ct] triv_goal;

end;

structure BasicDrule: BASIC_DRULE = Drule;

open BasicDrule;