src/Pure/drule.ML
author lcp
Tue Jan 18 13:46:08 1994 +0100 (1994-01-18)
changeset 229 4002c4cd450c
parent 214 ed6a3e2b1a33
child 252 7532f95d7f44
permissions -rw-r--r--
Pure: MAJOR CHANGE. Moved ML types ctyp and cterm and their associated
functions from sign.ML to thm.ML or drule.ML. This allows the "prop" field
of a theorem to be regarded as a cterm -- avoids expensive calls to
cterm_of.
     1 (*  Title: 	Pure/drule.ML
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Derived rules and other operations on theorems and theories
     7 *)
     8 
     9 infix 0 RS RSN RL RLN MRS MRL COMP;
    10 
    11 signature DRULE =
    12   sig
    13   structure Thm : THM
    14   local open Thm  in
    15   val asm_rl: thm
    16   val assume_ax: theory -> string -> thm
    17   val cterm_fun: (term -> term) -> (cterm -> cterm)
    18   val COMP: thm * thm -> thm
    19   val compose: thm * int * thm -> thm list
    20   val cterm_instantiate: (cterm*cterm)list -> thm -> thm
    21   val cut_rl: thm
    22   val equal_abs_elim: cterm  -> thm -> thm
    23   val equal_abs_elim_list: cterm list -> thm -> thm
    24   val eq_sg: Sign.sg * Sign.sg -> bool
    25   val eq_thm: thm * thm -> bool
    26   val eq_thm_sg: thm * thm -> bool
    27   val flexpair_abs_elim_list: cterm list -> thm -> thm
    28   val forall_intr_list: cterm list -> thm -> thm
    29   val forall_intr_frees: thm -> thm
    30   val forall_elim_list: cterm list -> thm -> thm
    31   val forall_elim_var: int -> thm -> thm
    32   val forall_elim_vars: int -> thm -> thm
    33   val implies_elim_list: thm -> thm list -> thm
    34   val implies_intr_list: cterm list -> thm -> thm
    35   val MRL: thm list list * thm list -> thm list
    36   val MRS: thm list * thm -> thm
    37   val pprint_cterm: cterm -> pprint_args -> unit
    38   val pprint_ctyp: ctyp -> pprint_args -> unit
    39   val pprint_sg: Sign.sg -> pprint_args -> unit
    40   val pprint_theory: theory -> pprint_args -> unit
    41   val pprint_thm: thm -> pprint_args -> unit
    42   val pretty_thm: thm -> Sign.Syntax.Pretty.T
    43   val print_cterm: cterm -> unit
    44   val print_ctyp: ctyp -> unit
    45   val print_goals: int -> thm -> unit
    46   val print_goals_ref: (int -> thm -> unit) ref
    47   val print_sg: Sign.sg -> unit
    48   val print_theory: theory -> unit
    49   val print_thm: thm -> unit
    50   val prth: thm -> thm
    51   val prthq: thm Sequence.seq -> thm Sequence.seq
    52   val prths: thm list -> thm list
    53   val read_ctyp: Sign.sg -> string -> ctyp
    54   val read_instantiate: (string*string)list -> thm -> thm
    55   val read_instantiate_sg: Sign.sg -> (string*string)list -> thm -> thm
    56   val read_insts: 
    57           Sign.sg -> (indexname -> typ option) * (indexname -> sort option)
    58                   -> (indexname -> typ option) * (indexname -> sort option)
    59                   -> (string*string)list
    60                   -> (indexname*ctyp)list * (cterm*cterm)list
    61   val reflexive_thm: thm
    62   val revcut_rl: thm
    63   val rewrite_goal_rule: bool*bool -> (meta_simpset -> thm -> thm option)
    64         -> meta_simpset -> int -> thm -> thm
    65   val rewrite_goals_rule: thm list -> thm -> thm
    66   val rewrite_rule: thm list -> thm -> thm
    67   val RS: thm * thm -> thm
    68   val RSN: thm * (int * thm) -> thm
    69   val RL: thm list * thm list -> thm list
    70   val RLN: thm list * (int * thm list) -> thm list
    71   val show_hyps: bool ref
    72   val size_of_thm: thm -> int
    73   val standard: thm -> thm
    74   val string_of_cterm: cterm -> string
    75   val string_of_ctyp: ctyp -> string
    76   val string_of_thm: thm -> string
    77   val symmetric_thm: thm
    78   val transitive_thm: thm
    79   val triv_forall_equality: thm
    80   val types_sorts: thm -> (indexname-> typ option) * (indexname-> sort option)
    81   val zero_var_indexes: thm -> thm
    82   end
    83   end;
    84 
    85 functor DruleFun (structure Logic: LOGIC and Thm: THM)(* : DRULE *) = (* FIXME *)
    86 struct
    87 structure Thm = Thm;
    88 structure Sign = Thm.Sign;
    89 structure Type = Sign.Type;
    90 structure Pretty = Sign.Syntax.Pretty
    91 local open Thm
    92 in
    93 
    94 (**** More derived rules and operations on theorems ****)
    95 
    96 fun cterm_fun f ct =
    97  let val {sign,t,...} = rep_cterm ct in cterm_of sign (f t) end;
    98 
    99 fun read_ctyp sign = ctyp_of sign o Sign.read_typ(sign, K None);
   100 
   101 
   102 (** reading of instantiations **)
   103 
   104 fun indexname cs = case Syntax.scan_varname cs of (v,[]) => v
   105         | _ => error("Lexical error in variable name " ^ quote (implode cs));
   106 
   107 fun absent ixn =
   108   error("No such variable in term: " ^ Syntax.string_of_vname ixn);
   109 
   110 fun inst_failure ixn =
   111   error("Instantiation of " ^ Syntax.string_of_vname ixn ^ " fails");
   112 
   113 fun read_insts sign (rtypes,rsorts) (types,sorts) insts =
   114 let val {tsig,...} = Sign.rep_sg sign
   115     fun split([],tvs,vs) = (tvs,vs)
   116       | split((sv,st)::l,tvs,vs) = (case explode sv of
   117                   "'"::cs => split(l,(indexname cs,st)::tvs,vs)
   118                 | cs => split(l,tvs,(indexname cs,st)::vs));
   119     val (tvs,vs) = split(insts,[],[]);
   120     fun readT((a,i),st) =
   121         let val ixn = ("'" ^ a,i);
   122             val S = case rsorts ixn of Some S => S | None => absent ixn;
   123             val T = Sign.read_typ (sign,sorts) st;
   124         in if Type.typ_instance(tsig,T,TVar(ixn,S)) then (ixn,T)
   125            else inst_failure ixn
   126         end
   127     val tye = map readT tvs;
   128     fun add_cterm ((cts,tye), (ixn,st)) =
   129         let val T = case rtypes ixn of
   130                       Some T => typ_subst_TVars tye T
   131                     | None => absent ixn;
   132             val (ct,tye2) = read_def_cterm (sign,types,sorts) (st,T);
   133             val cv = cterm_of sign (Var(ixn,typ_subst_TVars tye2 T))
   134         in ((cv,ct)::cts,tye2 @ tye) end
   135     val (cterms,tye') = foldl add_cterm (([],tye), vs);
   136 in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) tye', cterms) end;
   137 
   138 
   139 (*** Printing of theorems ***)
   140 
   141 (*If false, hypotheses are printed as dots*)
   142 val show_hyps = ref true;
   143 
   144 fun pretty_thm th =
   145 let val {sign, hyps, prop,...} = rep_thm th
   146     val hsymbs = if null hyps then []
   147 		 else if !show_hyps then
   148 		      [Pretty.brk 2,
   149 		       Pretty.lst("[","]") (map (Sign.pretty_term sign) hyps)]
   150 		 else Pretty.str" [" :: map (fn _ => Pretty.str".") hyps @
   151 		      [Pretty.str"]"];
   152 in Pretty.blk(0, Sign.pretty_term sign prop :: hsymbs) end;
   153 
   154 val string_of_thm = Pretty.string_of o pretty_thm;
   155 
   156 val pprint_thm = Pretty.pprint o Pretty.quote o pretty_thm;
   157 
   158 
   159 (** Top-level commands for printing theorems **)
   160 val print_thm = writeln o string_of_thm;
   161 
   162 fun prth th = (print_thm th; th);
   163 
   164 (*Print and return a sequence of theorems, separated by blank lines. *)
   165 fun prthq thseq =
   166     (Sequence.prints (fn _ => print_thm) 100000 thseq;
   167      thseq);
   168 
   169 (*Print and return a list of theorems, separated by blank lines. *)
   170 fun prths ths = (print_list_ln print_thm ths; ths);
   171 
   172 (*Other printing commands*)
   173 fun pprint_ctyp cT = 
   174  let val {sign,T} = rep_ctyp cT in  Sign.pprint_typ sign T  end;
   175 
   176 fun string_of_ctyp cT = 
   177  let val {sign,T} = rep_ctyp cT in  Sign.string_of_typ sign T  end;
   178 
   179 val print_ctyp = writeln o string_of_ctyp;
   180 
   181 fun pprint_cterm ct = 
   182  let val {sign,t,...} = rep_cterm ct in  Sign.pprint_term sign t  end;
   183 
   184 fun string_of_cterm ct = 
   185  let val {sign,t,...} = rep_cterm ct in  Sign.string_of_term sign t  end;
   186 
   187 val print_cterm = writeln o string_of_cterm;
   188 
   189 fun pretty_sg sg = 
   190   Pretty.lst ("{", "}") (map (Pretty.str o !) (#stamps (Sign.rep_sg sg)));
   191 
   192 val pprint_sg = Pretty.pprint o pretty_sg;
   193 
   194 val pprint_theory = pprint_sg o sign_of;
   195 
   196 val print_sg = writeln o Pretty.string_of o pretty_sg;
   197 val print_theory = print_sg o sign_of;
   198 
   199 
   200 (** Print thm A1,...,An/B in "goal style" -- premises as numbered subgoals **)
   201 
   202 fun prettyprints es = writeln(Pretty.string_of(Pretty.blk(0,es)));
   203 
   204 fun print_goals maxgoals th : unit =
   205 let val {sign, hyps, prop,...} = rep_thm th;
   206     fun printgoals (_, []) = ()
   207       | printgoals (n, A::As) =
   208 	let val prettyn = Pretty.str(" " ^ string_of_int n ^ ". ");
   209 	    val prettyA = Sign.pretty_term sign A
   210 	in prettyprints[prettyn,prettyA]; 
   211            printgoals (n+1,As) 
   212         end;
   213     fun prettypair(t,u) =
   214         Pretty.blk(0, [Sign.pretty_term sign t, Pretty.str" =?=", Pretty.brk 1,
   215 		       Sign.pretty_term sign u]);
   216     fun printff [] = ()
   217       | printff tpairs =
   218 	 writeln("\nFlex-flex pairs:\n" ^
   219 		 Pretty.string_of(Pretty.lst("","") (map prettypair tpairs)))
   220     val (tpairs,As,B) = Logic.strip_horn(prop);
   221     val ngoals = length As
   222 in 
   223    writeln (Sign.string_of_term sign B);
   224    if ngoals=0 then writeln"No subgoals!"
   225    else if ngoals>maxgoals 
   226         then (printgoals (1, take(maxgoals,As));
   227 	      writeln("A total of " ^ string_of_int ngoals ^ " subgoals..."))
   228         else printgoals (1, As);
   229    printff tpairs
   230 end;
   231 
   232 (*"hook" for user interfaces: allows print_goals to be replaced*)
   233 val print_goals_ref = ref print_goals;
   234 
   235 (*** Find the type (sort) associated with a (T)Var or (T)Free in a term 
   236      Used for establishing default types (of variables) and sorts (of
   237      type variables) when reading another term.
   238      Index -1 indicates that a (T)Free rather than a (T)Var is wanted.
   239 ***)
   240 
   241 fun types_sorts thm =
   242     let val {prop,hyps,...} = rep_thm thm;
   243 	val big = list_comb(prop,hyps); (* bogus term! *)
   244 	val vars = map dest_Var (term_vars big);
   245 	val frees = map dest_Free (term_frees big);
   246 	val tvars = term_tvars big;
   247 	val tfrees = term_tfrees big;
   248 	fun typ(a,i) = if i<0 then assoc(frees,a) else assoc(vars,(a,i));
   249 	fun sort(a,i) = if i<0 then assoc(tfrees,a) else assoc(tvars,(a,i));
   250     in (typ,sort) end;
   251 
   252 (** Standardization of rules **)
   253 
   254 (*Generalization over a list of variables, IGNORING bad ones*)
   255 fun forall_intr_list [] th = th
   256   | forall_intr_list (y::ys) th =
   257 	let val gth = forall_intr_list ys th
   258 	in  forall_intr y gth   handle THM _ =>  gth  end;
   259 
   260 (*Generalization over all suitable Free variables*)
   261 fun forall_intr_frees th =
   262     let val {prop,sign,...} = rep_thm th
   263     in  forall_intr_list
   264          (map (cterm_of sign) (sort atless (term_frees prop))) 
   265          th
   266     end;
   267 
   268 (*Replace outermost quantified variable by Var of given index.
   269     Could clash with Vars already present.*)
   270 fun forall_elim_var i th = 
   271     let val {prop,sign,...} = rep_thm th
   272     in case prop of
   273 	  Const("all",_) $ Abs(a,T,_) =>
   274 	      forall_elim (cterm_of sign (Var((a,i), T)))  th
   275 	| _ => raise THM("forall_elim_var", i, [th])
   276     end;
   277 
   278 (*Repeat forall_elim_var until all outer quantifiers are removed*)
   279 fun forall_elim_vars i th = 
   280     forall_elim_vars i (forall_elim_var i th)
   281 	handle THM _ => th;
   282 
   283 (*Specialization over a list of cterms*)
   284 fun forall_elim_list cts th = foldr (uncurry forall_elim) (rev cts, th);
   285 
   286 (* maps [A1,...,An], B   to   [| A1;...;An |] ==> B  *)
   287 fun implies_intr_list cAs th = foldr (uncurry implies_intr) (cAs,th);
   288 
   289 (* maps [| A1;...;An |] ==> B and [A1,...,An]   to   B *)
   290 fun implies_elim_list impth ths = foldl (uncurry implies_elim) (impth,ths);
   291 
   292 (*Reset Var indexes to zero, renaming to preserve distinctness*)
   293 fun zero_var_indexes th = 
   294     let val {prop,sign,...} = rep_thm th;
   295         val vars = term_vars prop
   296         val bs = foldl add_new_id ([], map (fn Var((a,_),_)=>a) vars)
   297 	val inrs = add_term_tvars(prop,[]);
   298 	val nms' = rev(foldl add_new_id ([], map (#1 o #1) inrs));
   299 	val tye = map (fn ((v,rs),a) => (v, TVar((a,0),rs))) (inrs ~~ nms')
   300 	val ctye = map (fn (v,T) => (v,ctyp_of sign T)) tye;
   301 	fun varpairs([],[]) = []
   302 	  | varpairs((var as Var(v,T)) :: vars, b::bs) =
   303 		let val T' = typ_subst_TVars tye T
   304 		in (cterm_of sign (Var(v,T')),
   305 		    cterm_of sign (Var((b,0),T'))) :: varpairs(vars,bs)
   306 		end
   307 	  | varpairs _ = raise TERM("varpairs", []);
   308     in instantiate (ctye, varpairs(vars,rev bs)) th end;
   309 
   310 
   311 (*Standard form of object-rule: no hypotheses, Frees, or outer quantifiers;
   312     all generality expressed by Vars having index 0.*)
   313 fun standard th =
   314     let val {maxidx,...} = rep_thm th
   315     in  varifyT (zero_var_indexes (forall_elim_vars(maxidx+1) 
   316                          (forall_intr_frees(implies_intr_hyps th))))
   317     end;
   318 
   319 (*Assume a new formula, read following the same conventions as axioms. 
   320   Generalizes over Free variables,
   321   creates the assumption, and then strips quantifiers.
   322   Example is [| ALL x:?A. ?P(x) |] ==> [| ?P(?a) |]
   323 	     [ !(A,P,a)[| ALL x:A. P(x) |] ==> [| P(a) |] ]    *)
   324 fun assume_ax thy sP =
   325     let val sign = sign_of thy
   326 	val prop = Logic.close_form (term_of (read_cterm sign
   327 			 (sP, propT)))
   328     in forall_elim_vars 0 (assume (cterm_of sign prop))  end;
   329 
   330 (*Resolution: exactly one resolvent must be produced.*) 
   331 fun tha RSN (i,thb) =
   332   case Sequence.chop (2, biresolution false [(false,tha)] i thb) of
   333       ([th],_) => th
   334     | ([],_)   => raise THM("RSN: no unifiers", i, [tha,thb])
   335     |      _   => raise THM("RSN: multiple unifiers", i, [tha,thb]);
   336 
   337 (*resolution: P==>Q, Q==>R gives P==>R. *)
   338 fun tha RS thb = tha RSN (1,thb);
   339 
   340 (*For joining lists of rules*)
   341 fun thas RLN (i,thbs) = 
   342   let val resolve = biresolution false (map (pair false) thas) i
   343       fun resb thb = Sequence.list_of_s (resolve thb) handle THM _ => []
   344   in  flat (map resb thbs)  end;
   345 
   346 fun thas RL thbs = thas RLN (1,thbs);
   347 
   348 (*Resolve a list of rules against bottom_rl from right to left;
   349   makes proof trees*)
   350 fun rls MRS bottom_rl = 
   351   let fun rs_aux i [] = bottom_rl
   352 	| rs_aux i (rl::rls) = rl RSN (i, rs_aux (i+1) rls)
   353   in  rs_aux 1 rls  end;
   354 
   355 (*As above, but for rule lists*)
   356 fun rlss MRL bottom_rls = 
   357   let fun rs_aux i [] = bottom_rls
   358 	| rs_aux i (rls::rlss) = rls RLN (i, rs_aux (i+1) rlss)
   359   in  rs_aux 1 rlss  end;
   360 
   361 (*compose Q and [...,Qi,Q(i+1),...]==>R to [...,Q(i+1),...]==>R 
   362   with no lifting or renaming!  Q may contain ==> or meta-quants
   363   ALWAYS deletes premise i *)
   364 fun compose(tha,i,thb) = 
   365     Sequence.list_of_s (bicompose false (false,tha,0) i thb);
   366 
   367 (*compose Q and [Q1,Q2,...,Qk]==>R to [Q2,...,Qk]==>R getting unique result*)
   368 fun tha COMP thb =
   369     case compose(tha,1,thb) of
   370         [th] => th  
   371       | _ =>   raise THM("COMP", 1, [tha,thb]);
   372 
   373 (*Instantiate theorem th, reading instantiations under signature sg*)
   374 fun read_instantiate_sg sg sinsts th =
   375     let val ts = types_sorts th;
   376     in  instantiate (read_insts sg ts ts sinsts) th  end;
   377 
   378 (*Instantiate theorem th, reading instantiations under theory of th*)
   379 fun read_instantiate sinsts th =
   380     read_instantiate_sg (#sign (rep_thm th)) sinsts th;
   381 
   382 
   383 (*Left-to-right replacements: tpairs = [...,(vi,ti),...].
   384   Instantiates distinct Vars by terms, inferring type instantiations. *)
   385 local
   386   fun add_types ((ct,cu), (sign,tye)) =
   387     let val {sign=signt, t=t, T= T, ...} = rep_cterm ct
   388         and {sign=signu, t=u, T= U, ...} = rep_cterm cu
   389         val sign' = Sign.merge(sign, Sign.merge(signt, signu))
   390 	val tye' = Type.unify (#tsig(Sign.rep_sg sign')) ((T,U), tye)
   391 	  handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])
   392     in  (sign', tye')  end;
   393 in
   394 fun cterm_instantiate ctpairs0 th = 
   395   let val (sign,tye) = foldr add_types (ctpairs0, (#sign(rep_thm th),[]))
   396       val tsig = #tsig(Sign.rep_sg sign);
   397       fun instT(ct,cu) = let val inst = subst_TVars tye
   398 			 in (cterm_fun inst ct, cterm_fun inst cu) end
   399       fun ctyp2 (ix,T) = (ix, ctyp_of sign T)
   400   in  instantiate (map ctyp2 tye, map instT ctpairs0) th  end
   401   handle TERM _ => 
   402            raise THM("cterm_instantiate: incompatible signatures",0,[th])
   403        | TYPE _ => raise THM("cterm_instantiate: types", 0, [th])
   404 end;
   405 
   406 
   407 (** theorem equality test is exported and used by BEST_FIRST **)
   408 
   409 (*equality of signatures means exact identity -- by ref equality*)
   410 fun eq_sg (sg1,sg2) = (#stamps(Sign.rep_sg sg1) = #stamps(Sign.rep_sg sg2));
   411 
   412 (*equality of theorems uses equality of signatures and 
   413   the a-convertible test for terms*)
   414 fun eq_thm (th1,th2) = 
   415     let val {sign=sg1, hyps=hyps1, prop=prop1, ...} = rep_thm th1
   416 	and {sign=sg2, hyps=hyps2, prop=prop2, ...} = rep_thm th2
   417     in  eq_sg (sg1,sg2) andalso 
   418         aconvs(hyps1,hyps2) andalso 
   419         prop1 aconv prop2  
   420     end;
   421 
   422 (*Do the two theorems have the same signature?*)
   423 fun eq_thm_sg (th1,th2) = eq_sg(#sign(rep_thm th1), #sign(rep_thm th2));
   424 
   425 (*Useful "distance" function for BEST_FIRST*)
   426 val size_of_thm = size_of_term o #prop o rep_thm;
   427 
   428 
   429 (*** Meta-Rewriting Rules ***)
   430 
   431 
   432 val reflexive_thm =
   433   let val cx = cterm_of Sign.pure (Var(("x",0),TVar(("'a",0),["logic"])))
   434   in Thm.reflexive cx end;
   435 
   436 val symmetric_thm =
   437   let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)
   438   in standard(Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;
   439 
   440 val transitive_thm =
   441   let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)
   442       val yz = read_cterm Sign.pure ("y::'a::logic == z",propT)
   443       val xythm = Thm.assume xy and yzthm = Thm.assume yz
   444   in standard(Thm.implies_intr yz (Thm.transitive xythm yzthm)) end;
   445 
   446 (** Below, a "conversion" has type cterm -> thm **)
   447 
   448 val refl_cimplies = reflexive (cterm_of Sign.pure implies);
   449 
   450 (*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)
   451 (*Do not rewrite flex-flex pairs*)
   452 fun goals_conv pred cv = 
   453   let fun gconv i ct =
   454         let val (A,B) = Thm.dest_cimplies ct
   455             val (thA,j) = case term_of A of
   456                   Const("=?=",_)$_$_ => (reflexive A, i)
   457                 | _ => (if pred i then cv A else reflexive A, i+1)
   458 	in  combination (combination refl_cimplies thA) (gconv j B) end
   459         handle TERM _ => reflexive ct
   460   in gconv 1 end;
   461 
   462 (*Use a conversion to transform a theorem*)
   463 fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th;
   464 
   465 (*rewriting conversion*)
   466 fun rew_conv mode prover mss = rewrite_cterm mode mss prover;
   467 
   468 (*Rewrite a theorem*)
   469 fun rewrite_rule thms =
   470   fconv_rule (rew_conv (true,false) (K(K None)) (Thm.mss_of thms));
   471 
   472 (*Rewrite the subgoals of a proof state (represented by a theorem) *)
   473 fun rewrite_goals_rule thms =
   474   fconv_rule (goals_conv (K true) (rew_conv (true,false) (K(K None))
   475              (Thm.mss_of thms)));
   476 
   477 (*Rewrite the subgoal of a proof state (represented by a theorem) *)
   478 fun rewrite_goal_rule mode prover mss i thm =
   479   if 0 < i  andalso  i <= nprems_of thm
   480   then fconv_rule (goals_conv (fn j => j=i) (rew_conv mode prover mss)) thm
   481   else raise THM("rewrite_goal_rule",i,[thm]);
   482 
   483 
   484 (** Derived rules mainly for METAHYPS **)
   485 
   486 (*Given the term "a", takes (%x.t)==(%x.u) to t[a/x]==u[a/x]*)
   487 fun equal_abs_elim ca eqth =
   488   let val {sign=signa, t=a, ...} = rep_cterm ca
   489       and combth = combination eqth (reflexive ca)
   490       val {sign,prop,...} = rep_thm eqth
   491       val (abst,absu) = Logic.dest_equals prop
   492       val cterm = cterm_of (Sign.merge (sign,signa))
   493   in  transitive (symmetric (beta_conversion (cterm (abst$a))))
   494            (transitive combth (beta_conversion (cterm (absu$a))))
   495   end
   496   handle THM _ => raise THM("equal_abs_elim", 0, [eqth]);
   497 
   498 (*Calling equal_abs_elim with multiple terms*)
   499 fun equal_abs_elim_list cts th = foldr (uncurry equal_abs_elim) (rev cts, th);
   500 
   501 local
   502   open Logic
   503   val alpha = TVar(("'a",0), [])     (*  type ?'a::{}  *)
   504   fun err th = raise THM("flexpair_inst: ", 0, [th])
   505   fun flexpair_inst def th =
   506     let val {prop = Const _ $ t $ u,  sign,...} = rep_thm th
   507 	val cterm = cterm_of sign
   508 	fun cvar a = cterm(Var((a,0),alpha))
   509 	val def' = cterm_instantiate [(cvar"t", cterm t), (cvar"u", cterm u)] 
   510 		   def
   511     in  equal_elim def' th
   512     end
   513     handle THM _ => err th | bind => err th
   514 in
   515 val flexpair_intr = flexpair_inst (symmetric flexpair_def)
   516 and flexpair_elim = flexpair_inst flexpair_def
   517 end;
   518 
   519 (*Version for flexflex pairs -- this supports lifting.*)
   520 fun flexpair_abs_elim_list cts = 
   521     flexpair_intr o equal_abs_elim_list cts o flexpair_elim;
   522 
   523 
   524 (*** Some useful meta-theorems ***)
   525 
   526 (*The rule V/V, obtains assumption solving for eresolve_tac*)
   527 val asm_rl = trivial(read_cterm Sign.pure ("PROP ?psi",propT));
   528 
   529 (*Meta-level cut rule: [| V==>W; V |] ==> W *)
   530 val cut_rl = trivial(read_cterm Sign.pure 
   531 	("PROP ?psi ==> PROP ?theta", propT));
   532 
   533 (*Generalized elim rule for one conclusion; cut_rl with reversed premises: 
   534      [| PROP V;  PROP V ==> PROP W |] ==> PROP W *)
   535 val revcut_rl =
   536   let val V = read_cterm Sign.pure ("PROP V", propT)
   537       and VW = read_cterm Sign.pure ("PROP V ==> PROP W", propT);
   538   in  standard (implies_intr V 
   539 		(implies_intr VW
   540 		 (implies_elim (assume VW) (assume V))))
   541   end;
   542 
   543 (* (!!x. PROP ?V) == PROP ?V       Allows removal of redundant parameters*)
   544 val triv_forall_equality =
   545   let val V  = read_cterm Sign.pure ("PROP V", propT)
   546       and QV = read_cterm Sign.pure ("!!x::'a. PROP V", propT)
   547       and x  = read_cterm Sign.pure ("x", TFree("'a",["logic"]));
   548   in  standard (equal_intr (implies_intr QV (forall_elim x (assume QV)))
   549 		           (implies_intr V  (forall_intr x (assume V))))
   550   end;
   551 
   552 end
   553 end;