(* Title: FOLP/intprover.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, @{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}) ]; (*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;