(*  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 and theories

 *)

 infix 0 RS RSN RL RLN MRS MRL COMP;

 signature DRULE =

   sig

   structure Thm : THM

   local open Thm  in

   val asm_rl: thm

   val assume_ax: theory -> string -> thm

   val COMP: thm * thm -> thm

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

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

   val cut_rl: thm

   val equal_abs_elim: cterm  -> thm -> thm

   val equal_abs_elim_list: cterm list -> thm -> thm

   val eq_thm: thm * thm -> bool

   val eq_thm_sg: thm * thm -> bool

   val flexpair_abs_elim_list: cterm list -> thm -> thm

   val forall_intr_list: cterm list -> thm -> thm

   val forall_intr_frees: 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 implies_elim_list: thm -> thm list -> thm

   val implies_intr_list: cterm list -> thm -> thm

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

   val MRS: thm list * thm -> thm

   val pprint_cterm: cterm -> pprint_args -> unit

   val pprint_ctyp: ctyp -> pprint_args -> unit

   val pprint_theory: theory -> pprint_args -> unit

   val pprint_thm: thm -> pprint_args -> unit

   val pretty_thm: thm -> Sign.Syntax.Pretty.T

   val print_cterm: cterm -> unit

   val print_ctyp: ctyp -> unit

   val print_goals: int -> thm -> unit

   val print_goals_ref: (int -> thm -> unit) ref

   val print_theory: theory -> unit

   val print_thm: thm -> unit

   val prth: thm -> thm

   val prthq: thm Sequence.seq -> thm Sequence.seq

   val prths: thm list -> thm list

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

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

   val read_insts:

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

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

                   -> (string*string)list

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

   val reflexive_thm: thm

   val revcut_rl: thm

   val rewrite_goal_rule: bool*bool -> (meta_simpset -> thm -> thm option)

         -> meta_simpset -> int -> thm -> thm

   val rewrite_goals_rule: thm list -> thm -> thm

   val rewrite_rule: thm list -> thm -> thm

   val RS: thm * thm -> thm

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

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

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

   val show_hyps: bool ref

   val size_of_thm: thm -> int

   val standard: thm -> thm

   val string_of_cterm: cterm -> string

   val string_of_ctyp: ctyp -> string

   val string_of_thm: thm -> string

   val symmetric_thm: thm

   val transitive_thm: thm

   val triv_forall_equality: thm

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

   val zero_var_indexes: thm -> thm

   end

   end;

 functor DruleFun (structure Logic: LOGIC and Thm: THM): DRULE =

 struct

 structure Thm = Thm;

 structure Sign = Thm.Sign;

 structure Type = Sign.Type;

 structure Pretty = Sign.Syntax.Pretty

 local open Thm

 in

 (**** More derived rules and operations on theorems ****)

 (** reading of instantiations **)

 fun indexname cs = case Syntax.scan_varname cs of (v,[]) => v

         | _ => error("Lexical error in variable name " ^ quote (implode cs));

 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) insts =

 let val {tsig,...} = Sign.rep_sg sign

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

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

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

                 | cs => split(l,tvs,(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 Type.typ_instance(tsig,T,TVar(ixn,S)) then (ixn,T)

            else inst_failure ixn

         end

     val tye = map readT tvs;

     fun add_cterm ((cts,tye), (ixn,st)) =

         let val T = case rtypes ixn of

                       Some T => typ_subst_TVars tye T

                     | None => absent ixn;

             val (ct,tye2) = read_def_cterm (sign,types,sorts) (st,T);

             val cv = cterm_of sign (Var(ixn,typ_subst_TVars tye2 T))

         in ((cv,ct)::cts,tye2 @ tye) end

     val (cterms,tye') = foldl add_cterm (([],tye), vs);

 in (map (fn (ixn,T) => (ixn,ctyp_of sign T)) tye', cterms) end;

 (*** Printing of theories, theorems, etc. ***)

 (*If false, hypotheses are printed as dots*)

 val show_hyps = ref true;

 fun pretty_thm th =

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

     val hsymbs = if null hyps then []

                  else if !show_hyps then

                       [Pretty.brk 2,

                        Pretty.lst("[","]") (map (Sign.pretty_term sign) hyps)]

                  else Pretty.str" [" :: map (fn _ => Pretty.str".") hyps @

                       [Pretty.str"]"];

 in Pretty.blk(0, Sign.pretty_term sign prop :: hsymbs) end;

 val string_of_thm = Pretty.string_of o pretty_thm;

 val pprint_thm = Pretty.pprint o Pretty.quote o pretty_thm;

 (** Top-level commands for printing theorems **)

 val print_thm = writeln o string_of_thm;

 fun prth th = (print_thm th; th);

 (*Print and return a sequence of theorems, separated by blank lines. *)

 fun prthq thseq =

   (Sequence.prints (fn _ => print_thm) 100000 thseq; thseq);

 (*Print and return a list of theorems, separated by blank lines. *)

 fun prths ths = (print_list_ln print_thm ths; ths);

 (* other printing commands *)

 fun pprint_ctyp cT =

   let val {sign, T} = rep_ctyp cT in Sign.pprint_typ sign T end;

 fun string_of_ctyp cT =

   let val {sign, T} = rep_ctyp cT in Sign.string_of_typ sign T end;

 val print_ctyp = writeln o string_of_ctyp;

 fun pprint_cterm ct =

   let val {sign, t, ...} = rep_cterm ct in Sign.pprint_term sign t end;

 fun string_of_cterm ct =

   let val {sign, t, ...} = rep_cterm ct in Sign.string_of_term sign t end;

 val print_cterm = writeln o string_of_cterm;

 (* print theory *)

 val pprint_theory = Sign.pprint_sg o sign_of;

 fun print_theory thy =

   let

     fun prt_thm (name, thm) = Pretty.block

       [Pretty.str (name ^ ":"), Pretty.brk 1, Pretty.quote (pretty_thm thm)];

     val sg = sign_of thy;

     val axioms =        (* FIXME should rather fix axioms_of *)

       sort (fn ((x, _), (y, _)) => x <= y)

         (gen_distinct eq_fst (axioms_of thy));

   in

     Sign.print_sg sg;

     Pretty.writeln (Pretty.big_list "axioms:" (map prt_thm axioms))

   end;

 (** Print thm A1,...,An/B in "goal style" -- premises as numbered subgoals **)

 fun prettyprints es = writeln(Pretty.string_of(Pretty.blk(0,es)));

 fun print_goals maxgoals th : unit =

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

     fun printgoals (_, []) = ()

       | printgoals (n, A::As) =

         let val prettyn = Pretty.str(" " ^ string_of_int n ^ ". ");

             val prettyA = Sign.pretty_term sign A

         in prettyprints[prettyn,prettyA];

            printgoals (n+1,As)

         end;

     fun prettypair(t,u) =

         Pretty.blk(0, [Sign.pretty_term sign t, Pretty.str" =?=", Pretty.brk 1,

                        Sign.pretty_term sign u]);

     fun printff [] = ()

       | printff tpairs =

          writeln("\nFlex-flex pairs:\n" ^

                  Pretty.string_of(Pretty.lst("","") (map prettypair tpairs)))

     val (tpairs,As,B) = Logic.strip_horn(prop);

     val ngoals = length As

 in

    writeln (Sign.string_of_term sign B);

    if ngoals=0 then writeln"No subgoals!"

    else if ngoals>maxgoals

         then (printgoals (1, take(maxgoals,As));

               writeln("A total of " ^ string_of_int ngoals ^ " subgoals..."))

         else printgoals (1, As);

    printff tpairs

 end;

 (*"hook" for user interfaces: allows print_goals to be replaced*)

 val print_goals_ref = ref print_goals;

 (*** 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;

 (** Standardization of rules **)

 (*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 atless (term_frees prop)))

          th

     end;

 (*Replace outermost quantified variable by Var of given index.

     Could clash with Vars already present.*)

 fun forall_elim_var i th =

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

     in case prop of

           Const("all",_) $ Abs(a,T,_) =>

               forall_elim (cterm_of sign (Var((a,i), T)))  th

         | _ => raise THM("forall_elim_var", i, [th])

     end;

 (*Repeat forall_elim_var until all outer quantifiers are removed*)

 fun forall_elim_vars i th =

     forall_elim_vars i (forall_elim_var i 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 = 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 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 standard th =

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

     in  varifyT (zero_var_indexes (forall_elim_vars(maxidx+1)

                          (forall_intr_frees(implies_intr_hyps th))))

     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 = 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 Sequence.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 = Sequence.list_of_s (resolve thb) handle THM _ => []

   in  flat (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;

 (*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) =

     Sequence.list_of_s (bicompose false (false,tha,0) i 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]);

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

 fun read_instantiate_sg sg sinsts th =

     let val ts = types_sorts th;

     in  instantiate (read_insts sg ts ts 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)) =

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

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

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

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

           handle Type.TUNIFY => raise TYPE("add_types", [T,U], [t,u])

     in  (sign', tye')  end;

 in

 fun cterm_instantiate ctpairs0 th =

   let val (sign,tye) = foldr add_types (ctpairs0, (#sign(rep_thm th),[]))

       val tsig = #tsig(Sign.rep_sg sign);

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

                          in (cterm_fun inst ct, cterm_fun inst cu) end

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

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

   handle TERM _ =>

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

        | TYPE _ => raise THM("cterm_instantiate: types", 0, [th])

 end;

 (** theorem equality test is exported and used by BEST_FIRST **)

 (*equality of theorems uses equality of signatures and

   the a-convertible test for terms*)

 fun eq_thm (th1,th2) =

     let val {sign=sg1, hyps=hyps1, prop=prop1, ...} = rep_thm th1

         and {sign=sg2, hyps=hyps2, prop=prop2, ...} = rep_thm th2

     in  Sign.eq_sg (sg1,sg2) andalso

         aconvs(hyps1,hyps2) andalso

         prop1 aconv prop2

     end;

 (*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;

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

 val reflexive_thm =

   let val cx = cterm_of Sign.pure (Var(("x",0),TVar(("'a",0),["logic"])))

   in Thm.reflexive cx end;

 val symmetric_thm =

   let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)

   in standard(Thm.implies_intr_hyps(Thm.symmetric(Thm.assume xy))) end;

 val transitive_thm =

   let val xy = read_cterm Sign.pure ("x::'a::logic == y",propT)

       val yz = read_cterm Sign.pure ("y::'a::logic == z",propT)

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

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

 (** Below, a "conversion" has type cterm -> thm **)

 val refl_cimplies = reflexive (cterm_of Sign.pure implies);

 (*In [A1,...,An]==>B, rewrite the selected A's only -- for rewrite_goals_tac*)

 (*Do not rewrite flex-flex pairs*)

 fun goals_conv pred cv =

   let fun gconv i ct =

         let val (A,B) = Thm.dest_cimplies ct

             val (thA,j) = case term_of A of

                   Const("=?=",_)$_$_ => (reflexive A, i)

                 | _ => (if pred i then cv A else reflexive A, i+1)

         in  combination (combination refl_cimplies thA) (gconv j B) end

         handle TERM _ => reflexive ct

   in gconv 1 end;

 (*Use a conversion to transform a theorem*)

 fun fconv_rule cv th = equal_elim (cv (cprop_of th)) th;

 (*rewriting conversion*)

 fun rew_conv mode prover mss = rewrite_cterm mode mss prover;

 (*Rewrite a theorem*)

 fun rewrite_rule thms =

   fconv_rule (rew_conv (true,false) (K(K None)) (Thm.mss_of thms));

 (*Rewrite the subgoals of a proof state (represented by a theorem) *)

 fun rewrite_goals_rule thms =

   fconv_rule (goals_conv (K true) (rew_conv (true,false) (K(K None))

              (Thm.mss_of thms)));

 (*Rewrite the subgoal of a proof state (represented by a theorem) *)

 fun rewrite_goal_rule mode prover mss i thm =

   if 0 < i  andalso  i <= nprems_of thm

   then fconv_rule (goals_conv (fn j => j=i) (rew_conv mode prover mss)) thm

   else raise THM("rewrite_goal_rule",i,[thm]);

 (** 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 (cterm (abst$a))))

            (transitive combth (beta_conversion (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

   open Logic

   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 flexpair_def)

 and flexpair_elim = flexpair_inst 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;

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

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

 val asm_rl = trivial(read_cterm Sign.pure ("PROP ?psi",propT));

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

 val cut_rl = trivial(read_cterm Sign.pure

         ("PROP ?psi ==> PROP ?theta", propT));

 (*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_cterm Sign.pure ("PROP V", propT)

       and VW = read_cterm Sign.pure ("PROP V ==> PROP W", propT);

   in  standard (implies_intr V

                 (implies_intr VW

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

   end;

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

 val triv_forall_equality =

   let val V  = read_cterm Sign.pure ("PROP V", propT)

       and QV = read_cterm Sign.pure ("!!x::'a. PROP V", propT)

       and x  = read_cterm Sign.pure ("x", TFree("'a",["logic"]));

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

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

   end;

 end

 end;