(*  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 gen_all           : 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 standard'         : thm -> thm

  val rotate_prems      : int -> thm -> thm

  val rearrange_prems   : int list -> 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 eq_thm_sg         : thm * thm -> bool

  val eq_thm_prop	: thm * thm -> bool

  val weak_eq_thm       : 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 extensional       : 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 equal_elim_rule1  : 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 strip_comb: cterm -> cterm * cterm list

  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 get_kind: thm -> string

  val kind: string -> 'a attribute

  val theoremK: string

  val lemmaK: string

  val corollaryK: string

  val internalK: string

  val kind_internal: 'a attribute

  val has_internal: tag list -> bool

  val impose_hyps: cterm list -> thm -> thm

  val close_derivation: thm -> thm

  val local_standard: thm -> thm

  val compose_single: thm * int * thm -> thm

  val add_rule: thm -> thm list -> thm list

  val del_rule: thm -> thm list -> thm list

  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 is_norm_hhf: term -> bool

  val norm_hhf: Sign.sg -> term -> term

  val triv_goal: thm

  val rev_triv_goal: thm

  val implies_intr_goals: cterm list -> thm -> thm

  val freeze_all: thm -> thm

  val mk_triv_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 * (string * indexname) list

  val remdups_rl: thm

  val conj_intr: thm -> thm -> thm

  val conj_intr_list: thm list -> thm

  val conj_elim: thm -> thm * thm

  val conj_elim_list: thm -> thm list

  val conj_elim_precise: int -> thm -> thm list

  val conj_intr_thm: thm

  val abs_def: 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) = Thm.dest_comb ct

            in  (#2 (Thm.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) = Thm.dest_comb ct

            in  (#2 (Thm.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 (Thm.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) = Thm.capply (Thm.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));

(*cterm version of strip_comb: maps  f(t1,...,tn)  to  (f, [t1,...,tn]) *)

fun strip_comb ct = 

  let

    fun stripc (p as (ct, cts)) =

      let val (ct1, ct2) = Thm.dest_comb ct

      in stripc (ct1, ct2 :: cts) end handle CTERM _ => p

  in stripc (ct, []) end;

(** 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;

(* theorem kinds *)

val theoremK = "theorem";

val lemmaK = "lemma";

val corollaryK = "corollary";

val internalK = "internal";

fun get_kind thm =

  (case Library.assoc (#2 (Thm.get_name_tags thm), "kind") of

    Some (k :: _) => k

  | _ => "unknown");

fun kind_rule k = tag_rule ("kind", [k]) o untag_rule "kind";

fun kind k x = if k = "" then x else rule_attribute (K (kind_rule k)) x;

fun kind_internal x = kind internalK x;

fun has_internal tags = exists (equal internalK o fst) 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 gen_all thm =

  let

    val {sign, prop, maxidx, ...} = Thm.rep_thm thm;

    fun elim (th, (x, T)) = Thm.forall_elim (Thm.cterm_of sign (Var ((x, maxidx + 1), T))) th;

    val vs = Term.strip_all_vars prop;

  in foldl elim (thm, Term.variantlist (map #1 vs, []) ~~ map #2 vs) end;

(*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);

(* maps |- B to A1,...,An |- B *)

fun impose_hyps chyps th =

  let val chyps' = gen_rems (op aconv o apfst Thm.term_of) (chyps, #hyps (Thm.rep_thm th))

  in implies_elim_list (implies_intr_list chyps' th) (map Thm.assume chyps') end;

(*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

  end;

val standard = close_derivation o standard';

fun local_standard th =

  th |> strip_shyps |> zero_var_indexes

  |> Thm.compress |> close_derivation;

(*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;

(* permute prems, where the i-th position in the argument list (counting from 0)

   gives the position within the original thm to be transferred to position i.

   Any remaining trailing positions are left unchanged. *)

val rearrange_prems = let

  fun rearr new []      thm = thm

  |   rearr new (p::ps) thm = rearr (new+1)

     (map (fn q => if new<=q andalso q<p then q+1 else q) ps)

     (permute_prems (new+1) (new-p) (permute_prems new (p-new) thm))

  in rearr 0 end;

(*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 **)

val eq_thm_sg = Sign.eq_sg o pairself Thm.sign_of_thm;

val eq_thm_prop = op aconv o pairself Thm.prop_of;

(*Useful "distance" function for BEST_FIRST*)

val size_of_thm = size_of_term o prop_of;

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

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

fun add_rules rs rules = rs @ del_rules rs rules;

val del_rule = del_rules o single;

val add_rule = add_rules o single;

fun merge_rules (rules1, rules2) = gen_merge_lists' eq_thm_prop 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;

val weak_eq_thm = Thm.eq_thm o pairself (forall_intr_vars o freezeT);

(*** 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);

fun store_thm_open name thm = hd (PureThy.smart_store_thms_open (name, [thm]));

fun store_standard_thm_open name thm = store_thm_open name (standard' thm);

val reflexive_thm =

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

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

val symmetric_thm =

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

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

fun symmetric_fun thm = thm RS symmetric_thm;

fun extensional eq =

  let val eq' =

    abstract_rule "x" (snd (Thm.dest_comb (fst (dest_equals (cprop_of eq))))) eq

  in equal_elim (eta_conversion (cprop_of eq')) eq' end;

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_open "imp_cong" (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_open "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;

fun abs_def thm =

  let

    val (_, cvs) = strip_comb (fst (dest_equals (cprop_of thm)));

    val thm' = foldr (fn (ct, thm) => Thm.abstract_rule

      (case term_of ct of Var ((a, _), _) => a | Free (a, _) => a | _ => "x")

        ct thm) (cvs, thm)

  in transitive

    (symmetric (eta_conversion (fst (dest_equals (cprop_of thm'))))) thm'

  end;

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

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

val asm_rl = store_standard_thm_open "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_open "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_open "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_open "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_open "triv_forall_equality"

      (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_open "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_open "equal_intr_rule"

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

  end;

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

val equal_elim_rule1 =

  let val eq = read_prop "PROP phi == PROP psi"

      and P = read_prop "PROP phi"

  in store_standard_thm_open "equal_elim_rule1"

    (Thm.equal_elim (assume eq) (assume P) |> implies_intr_list [eq, P])

  end;

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

val remdups_rl =

  let val P = read_prop "PROP phi" and Q = read_prop "PROP psi";

  in store_standard_thm_open "remdups_rl" (implies_intr_list [P, P, Q] (Thm.assume Q)) end;

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

  Rewrite rule for HHF normalization.*)

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_open "norm_hhf_eq"

  end;

fun is_norm_hhf tm =

  let

    fun is_norm (Const ("==>", _) $ _ $ (Const ("all", _) $ _)) = false

      | is_norm (t $ u) = is_norm t andalso is_norm u

      | is_norm (Abs (_, _, t)) = is_norm t

      | is_norm _ = true;

  in is_norm (Pattern.beta_eta_contract tm) end;

fun norm_hhf sg t =

  if is_norm_hhf t then t

  else Pattern.rewrite_term (Sign.tsig_of sg) [Logic.dest_equals (prop_of norm_hhf_eq)] [] t;

(*** 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_of 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')) (tye, maxi) (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 cert = Thm.cterm_of proto_sign;

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

  val G = Logic.mk_goal A;

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

in

  val triv_goal = store_thm "triv_goal" (kind_rule internalK (standard

      (Thm.equal_elim (Thm.symmetric G_def) (Thm.assume (cert A)))));

  val rev_triv_goal = store_thm "rev_triv_goal" (kind_rule internalK (standard

      (Thm.equal_elim G_def (Thm.assume (cert 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;

fun implies_intr_goals cprops thm =

  implies_elim_list (implies_intr_list cprops thm) (map assume_goal cprops)

  |> implies_intr_list (map mk_cgoal cprops);

(** 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 in left-to-right order *)

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

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

fun tvars_of thm = tvars_of_terms [prop_of thm];

fun vars_of thm = vars_of_terms [prop_of 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;

(** meta-level conjunction **)

local

  val A = read_prop "PROP A";

  val B = read_prop "PROP B";

  val C = read_prop "PROP C";

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

  val proj1 =

    forall_intr_list [A, B] (implies_intr_list [A, B] (Thm.assume A))

    |> forall_elim_vars 0;

  val proj2 =

    forall_intr_list [A, B] (implies_intr_list [A, B] (Thm.assume B))

    |> forall_elim_vars 0;

  val conj_intr_rule =

    forall_intr_list [A, B] (implies_intr_list [A, B]

      (Thm.forall_intr C (Thm.implies_intr ABC

        (implies_elim_list (Thm.assume ABC) [Thm.assume A, Thm.assume B]))))

    |> forall_elim_vars 0;

  val incr = incr_indexes_wrt [] [] [];

in

fun conj_intr tha thb = thb COMP (tha COMP incr [tha, thb] conj_intr_rule);

fun conj_intr_list [] = asm_rl

  | conj_intr_list ths = foldr1 (uncurry conj_intr) ths;

fun conj_elim th =

  let val th' = forall_elim_var (#maxidx (Thm.rep_thm th) + 1) th

  in (incr [th'] proj1 COMP th', incr [th'] proj2 COMP th') end;

fun conj_elim_list th =

  let val (th1, th2) = conj_elim th

  in conj_elim_list th1 @ conj_elim_list th2 end handle THM _ => [th];

fun conj_elim_precise 0 _ = []

  | conj_elim_precise 1 th = [th]

  | conj_elim_precise n th =

      let val (th1, th2) = conj_elim th

      in th1 :: conj_elim_precise (n - 1) th2 end;

val conj_intr_thm = store_standard_thm_open "conjunctionI"

  (implies_intr_list [A, B] (conj_intr (Thm.assume A) (Thm.assume B)));

end;

end;

structure BasicDrule: BASIC_DRULE = Drule;

open BasicDrule;