(* Title: FOL/int-prover ID: $Id: intprover.ML,v 1.11 2006/11/26 22:43:54 wenzelm Exp $ 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 Intuitionistic Logic. J. Symbolic Logic 57(3), 1992, pages 795-807. The approach was developed independently by Roy Dyckhoff and L C Paulson. *) signature INT_PROVER = sig val best_tac: int -> tactic val best_dup_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 step_dup_tac: int -> tactic val haz_brls: (bool * thm) list val haz_dup_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, thm "FalseE"), (false, thm "TrueI"), (false, thm "refl"), (false, thm "impI"), (false, thm "notI"), (false, thm "allI"), (true, thm "conjE"), (true, thm "exE"), (false, thm "conjI"), (true, thm "conj_impE"), (true, thm "disj_impE"), (true, thm "disjE"), (false, thm "iffI"), (true, thm "iffE"), (true, thm "not_to_imp") ]; val haz_brls = [ (false, thm "disjI1"), (false, thm "disjI2"), (false, thm "exI"), (true, thm "allE"), (true, thm "not_impE"), (true, thm "imp_impE"), (true, thm "iff_impE"), (true, thm "all_impE"), (true, thm "ex_impE"), (true, thm "impE") ]; val haz_dup_brls = [ (false, thm "disjI1"), (false, thm "disjI2"), (false, thm "exI"), (true, thm "all_dupE"), (true, thm "not_impE"), (true, thm "imp_impE"), (true, thm "iff_impE"), (true, thm "all_impE"), (true, thm "ex_impE"), (true, thm "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 -- matching, not resolution*) val safe_step_tac = FIRST' [eq_assume_tac, eq_mp_tac, bimatch_tac safe0_brls, hyp_subst_tac, bimatch_tac safep_brls] ; (*Repeatedly attack subgoals using safe inferences -- it's deterministic!*) val safe_tac = REPEAT_DETERM_FIRST safe_step_tac; (*These steps could instantiate variables and are therefore unsafe.*) val inst_step_tac = assume_tac APPEND' mp_tac APPEND' biresolve_tac (safe0_brls @ safep_brls); (*One safe or unsafe step. *) fun step_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_brls i]; fun step_dup_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_dup_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)); (*Uses all_dupE: allows multiple use of universal assumptions. VERY slow.*) val best_dup_tac = SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_dup_tac 1)); end;