(* 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) =
List.partition (curry (op =) 0 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;