(*  Title:      FOLP/int-prover.ML

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

 A naive prover for intuitionistic logic

 BEWARE OF NAME CLASHES WITH CLASSICAL TACTICS -- use IntPr.fast_tac ...

 Completeness (for propositional logic) is proved in 

 Roy Dyckhoff.

 Contraction-Free Sequent Calculi for IntPruitionistic Logic.

 J. Symbolic Logic (in press)

 *)

 signature INT_PROVER = 

   sig

   val best_tac: int -> tactic

   val fast_tac: int -> tactic

   val inst_step_tac: int -> tactic

   val safe_step_tac: int -> tactic

   val safe_brls: (bool * thm) list

   val safe_tac: tactic

   val step_tac: int -> tactic

   val haz_brls: (bool * thm) list

   end;

 structure IntPr : INT_PROVER   = 

 struct

 (*Negation is treated as a primitive symbol, with rules notI (introduction),

   not_to_imp (converts the assumption ~P to P-->False), and not_impE

   (handles double negations).  Could instead rewrite by not_def as the first

   step of an intuitionistic proof.

 *)

 val safe_brls = sort (make_ord lessb)

     [ (true,FalseE), (false,TrueI), (false,refl),

       (false,impI), (false,notI), (false,allI),

       (true,conjE), (true,exE),

       (false,conjI), (true,conj_impE),

       (true,disj_impE), (true,disjE), 

       (false,iffI), (true,iffE), (true,not_to_imp) ];

 val haz_brls =

     [ (false,disjI1), (false,disjI2), (false,exI), 

       (true,allE), (true,not_impE), (true,imp_impE), (true,iff_impE),

       (true,all_impE), (true,ex_impE), (true,impE) ];

 (*0 subgoals vs 1 or more: the p in safep is for positive*)

 val (safe0_brls, safep_brls) =

     partition (apl(0,op=) o subgoals_of_brl) safe_brls;

 (*Attack subgoals using safe inferences*)

 val safe_step_tac = FIRST' [uniq_assume_tac,

                             int_uniq_mp_tac,

                             biresolve_tac safe0_brls,

                             hyp_subst_tac,

                             biresolve_tac safep_brls] ;

 (*Repeatedly attack subgoals using safe inferences*)

 val safe_tac = DETERM (REPEAT_FIRST safe_step_tac);

 (*These steps could instantiate variables and are therefore unsafe.*)

 val inst_step_tac = assume_tac APPEND' mp_tac;

 (*One safe or unsafe step. *)

 fun step_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_brls i];

 (*Dumb but fast*)

 val fast_tac = SELECT_GOAL (DEPTH_SOLVE (step_tac 1));

 (*Slower but smarter than fast_tac*)

 val best_tac = 

   SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_tac 1));

 end;