src/Sequents/prover.ML
author paulson
Wed, 09 Oct 1996 13:32:33 +0200
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New unified treatment of sequent calculi by Sara Kalvala combines the old LK and Modal with the new ILL (Int. Linear Logic)
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(*  Title:      LK/LK.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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(**** Theorem Packs ****)
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(* based largely on LK *)
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datatype pack = Pack of thm list * thm list;
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(*A theorem pack has the form  (safe rules, unsafe rules)
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  An unsafe rule is incomplete or introduces variables in subgoals,
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  and is tried only when the safe rules are not applicable.  *)
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fun less (rl1,rl2) = (nprems_of rl1) < (nprems_of rl2);
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val empty_pack = Pack([],[]);
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infix 4 add_safes add_unsafes;
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fun (Pack(safes,unsafes)) add_safes ths   = 
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    Pack(sort less (ths@safes), unsafes);
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fun (Pack(safes,unsafes)) add_unsafes ths = 
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    Pack(safes, sort less (ths@unsafes));
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(*Returns the list of all formulas in the sequent*)
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fun forms_of_seq (Const("SeqO'",_) $ P $ u) = P :: forms_of_seq u
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  | forms_of_seq (H $ u) = forms_of_seq u
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  | forms_of_seq _ = [];
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(*Tests whether two sequences (left or right sides) could be resolved.
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  seqp is a premise (subgoal), seqc is a conclusion of an object-rule.
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  Assumes each formula in seqc is surrounded by sequence variables
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  -- checks that each concl formula looks like some subgoal formula.
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  It SHOULD check order as well, using recursion rather than forall/exists*)
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fun could_res (seqp,seqc) =
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      forall (fn Qc => exists (fn Qp => could_unify (Qp,Qc)) 
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                              (forms_of_seq seqp))
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             (forms_of_seq seqc);
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(*Tests whether two sequents or pairs of sequents could be resolved*)
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fun could_resolve_seq (prem,conc) =
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  case (prem,conc) of
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      (_ $ Abs(_,_,leftp) $ Abs(_,_,rightp),
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       _ $ Abs(_,_,leftc) $ Abs(_,_,rightc)) =>
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	  could_res (leftp,leftc) andalso could_res (rightp,rightc)
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    | (_ $ Abs(_,_,leftp) $ rightp,
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       _ $ Abs(_,_,leftc) $ rightc) =>
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	  could_res (leftp,leftc)  andalso  could_unify (rightp,rightc)
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    | _ => false;
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(*Like filt_resolve_tac, using could_resolve_seq
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  Much faster than resolve_tac when there are many rules.
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  Resolve subgoal i using the rules, unless more than maxr are compatible. *)
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fun filseq_resolve_tac rules maxr = SUBGOAL(fn (prem,i) =>
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  let val rls = filter_thms could_resolve_seq (maxr+1, prem, rules)
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  in  if length rls > maxr  then  no_tac
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	  else (*((rtac derelict 1 THEN rtac impl 1
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		 THEN (rtac identity 2 ORELSE rtac ll_mp 2)
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		 THEN rtac context1 1)
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		 ORELSE *) resolve_tac rls i
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  end);
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(*Predicate: does the rule have n premises? *)
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fun has_prems n rule =  (nprems_of rule = n);
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(*Continuation-style tactical for resolution.
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  The list of rules is partitioned into 0, 1, 2 premises.
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  The resulting tactic, gtac, tries to resolve with rules.
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  If successful, it recursively applies nextac to the new subgoals only.
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  Else fails.  (Treatment of goals due to Ph. de Groote) 
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  Bind (RESOLVE_THEN rules) to a variable: it preprocesses the rules. *)
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(*Takes rule lists separated in to 0, 1, 2, >2 premises.
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  The abstraction over state prevents needless divergence in recursion.
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  The 9999 should be a parameter, to delay treatment of flexible goals. *)
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fun RESOLVE_THEN rules =
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  let val [rls0,rls1,rls2] = partition_list has_prems 0 2 rules;
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      fun tac nextac i = STATE (fn state =>  
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	  filseq_resolve_tac rls0 9999 i 
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	  ORELSE
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	  (DETERM(filseq_resolve_tac rls1 9999 i) THEN  TRY(nextac i))
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	  ORELSE
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	  (DETERM(filseq_resolve_tac rls2 9999 i) THEN  TRY(nextac(i+1))
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					THEN  TRY(nextac i)) )
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  in  tac  end;
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(*repeated resolution applied to the designated goal*)
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fun reresolve_tac rules = 
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  let val restac = RESOLVE_THEN rules;  (*preprocessing done now*)
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      fun gtac i = restac gtac i
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  in  gtac  end; 
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(*tries the safe rules repeatedly before the unsafe rules. *)
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fun repeat_goal_tac (Pack(safes,unsafes)) = 
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  let val restac  =    RESOLVE_THEN safes
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      and lastrestac = RESOLVE_THEN unsafes;
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      fun gtac i = restac gtac i  ORELSE  (print_tac THEN lastrestac gtac i)
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  in  gtac  end; 
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(*Tries safe rules only*)
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fun safe_goal_tac (Pack(safes,unsafes)) = reresolve_tac safes;
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(*Tries a safe rule or else a unsafe rule.  Single-step for tracing. *)
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fun step_tac (thm_pack as Pack(safes,unsafes)) =
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    safe_goal_tac thm_pack  ORELSE'
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    filseq_resolve_tac unsafes 9999;
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(* Tactic for reducing a goal, using Predicate Calculus rules.
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   A decision procedure for Propositional Calculus, it is incomplete
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   for Predicate-Calculus because of allL_thin and exR_thin.  
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   Fails if it can do nothing.      *)
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fun pc_tac thm_pack = SELECT_GOAL (DEPTH_SOLVE (repeat_goal_tac thm_pack 1));
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(*The following two tactics are analogous to those provided by 
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  Provers/classical.  In fact, pc_tac is usually FASTER than fast_tac!*)
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fun fast_tac thm_pack =
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  SELECT_GOAL (DEPTH_SOLVE (step_tac thm_pack 1));
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fun best_tac thm_pack  = 
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  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) 
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	       (step_tac thm_pack 1));
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signature MODAL_PROVER_RULE =
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sig
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    val rewrite_rls      : thm list
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    val safe_rls         : thm list
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    val unsafe_rls       : thm list
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    val bound_rls        : thm list
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    val aside_rls        : thm list
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end;
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signature MODAL_PROVER = 
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sig
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    val rule_tac   : thm list -> int ->tactic
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    val step_tac   : int -> tactic
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    val solven_tac : int -> int -> tactic
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    val solve_tac  : int -> tactic
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end;
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functor Modal_ProverFun (Modal_Rule: MODAL_PROVER_RULE) : MODAL_PROVER = 
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struct
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local open Modal_Rule
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in 
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(*Returns the list of all formulas in the sequent*)
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fun forms_of_seq (Const("SeqO",_) $ P $ u) = P :: forms_of_seq u
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  | forms_of_seq (H $ u) = forms_of_seq u
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  | forms_of_seq _ = [];
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(*Tests whether two sequences (left or right sides) could be resolved.
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  seqp is a premise (subgoal), seqc is a conclusion of an object-rule.
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  Assumes each formula in seqc is surrounded by sequence variables
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  -- checks that each concl formula looks like some subgoal formula.*)
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fun could_res (seqp,seqc) =
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      forall (fn Qc => exists (fn Qp => could_unify (Qp,Qc)) 
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                              (forms_of_seq seqp))
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             (forms_of_seq seqc);
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(*Tests whether two sequents G|-H could be resolved, comparing each side.*)
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fun could_resolve_seq (prem,conc) =
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  case (prem,conc) of
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      (_ $ Abs(_,_,leftp) $ Abs(_,_,rightp),
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       _ $ Abs(_,_,leftc) $ Abs(_,_,rightc)) =>
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          could_res (leftp,leftc)  andalso  could_res (rightp,rightc)
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    | _ => false;
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(*Like filt_resolve_tac, using could_resolve_seq
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  Much faster than resolve_tac when there are many rules.
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  Resolve subgoal i using the rules, unless more than maxr are compatible. *)
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fun filseq_resolve_tac rules maxr = SUBGOAL(fn (prem,i) =>
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  let val rls = filter_thms could_resolve_seq (maxr+1, prem, rules)
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  in  if length rls > maxr  then  no_tac  else resolve_tac rls i
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  end);
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fun fresolve_tac rls n = filseq_resolve_tac rls 999 n;
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(* NB No back tracking possible with aside rules *)
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fun aside_tac n = DETERM(REPEAT (filt_resolve_tac aside_rls 999 n));
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fun rule_tac rls n = fresolve_tac rls n THEN aside_tac n;
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val fres_safe_tac = fresolve_tac safe_rls;
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val fres_unsafe_tac = fresolve_tac unsafe_rls THEN' aside_tac;
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val fres_bound_tac = fresolve_tac bound_rls;
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fun UPTOGOAL n tf = let fun tac i = if i<n then all_tac
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                                    else tf(i) THEN tac(i-1)
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                    in STATE(fn state=> tac(nprems_of state)) end;
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(* Depth first search bounded by d *)
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fun solven_tac d n = STATE (fn state =>
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        if d<0 then no_tac
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        else if (nprems_of state = 0) then all_tac 
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        else (DETERM(fres_safe_tac n) THEN UPTOGOAL n (solven_tac d)) ORELSE
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                 ((fres_unsafe_tac n  THEN UPTOGOAL n (solven_tac d)) APPEND
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                   (fres_bound_tac n  THEN UPTOGOAL n (solven_tac (d-1)))));
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fun solve_tac d = rewrite_goals_tac rewrite_rls THEN solven_tac d 1;
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fun step_tac n = STATE (fn state =>  
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      if (nprems_of state = 0) then all_tac 
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      else (DETERM(fres_safe_tac n)) ORELSE 
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           (fres_unsafe_tac n APPEND fres_bound_tac n));
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end;
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end;