1(* Title: FOL/intprover.ML 2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 3 Copyright 1992 University of Cambridge 4 5A naive prover for intuitionistic logic 6 7BEWARE OF NAME CLASHES WITH CLASSICAL TACTICS -- use IntPr.fast_tac ... 8 9Completeness (for propositional logic) is proved in 10 11Roy Dyckhoff. 12Contraction-Free Sequent Calculi for Intuitionistic Logic. 13J. Symbolic Logic 57(3), 1992, pages 795-807. 14 15The approach was developed independently by Roy Dyckhoff and L C Paulson. 16*) 17 18signature INT_PROVER = 19sig 20 val best_tac: Proof.context -> int -> tactic 21 val best_dup_tac: Proof.context -> int -> tactic 22 val fast_tac: Proof.context -> int -> tactic 23 val inst_step_tac: Proof.context -> int -> tactic 24 val safe_step_tac: Proof.context -> int -> tactic 25 val safe_brls: (bool * thm) list 26 val safe_tac: Proof.context -> tactic 27 val step_tac: Proof.context -> int -> tactic 28 val step_dup_tac: Proof.context -> int -> tactic 29 val haz_brls: (bool * thm) list 30 val haz_dup_brls: (bool * thm) list 31end; 32 33 34structure IntPr : INT_PROVER = 35struct 36 37(*Negation is treated as a primitive symbol, with rules notI (introduction), 38 not_to_imp (converts the assumption ~P to P-->False), and not_impE 39 (handles double negations). Could instead rewrite by not_def as the first 40 step of an intuitionistic proof. 41*) 42val safe_brls = sort (make_ord lessb) 43 [ (true, @{thm FalseE}), (false, @{thm TrueI}), (false, @{thm refl}), 44 (false, @{thm impI}), (false, @{thm notI}), (false, @{thm allI}), 45 (true, @{thm conjE}), (true, @{thm exE}), 46 (false, @{thm conjI}), (true, @{thm conj_impE}), 47 (true, @{thm disj_impE}), (true, @{thm disjE}), 48 (false, @{thm iffI}), (true, @{thm iffE}), (true, @{thm not_to_imp}) ]; 49 50val haz_brls = 51 [ (false, @{thm disjI1}), (false, @{thm disjI2}), (false, @{thm exI}), 52 (true, @{thm allE}), (true, @{thm not_impE}), (true, @{thm imp_impE}), (true, @{thm iff_impE}), 53 (true, @{thm all_impE}), (true, @{thm ex_impE}), (true, @{thm impE}) ]; 54 55val haz_dup_brls = 56 [ (false, @{thm disjI1}), (false, @{thm disjI2}), (false, @{thm exI}), 57 (true, @{thm all_dupE}), (true, @{thm not_impE}), (true, @{thm imp_impE}), (true, @{thm iff_impE}), 58 (true, @{thm all_impE}), (true, @{thm ex_impE}), (true, @{thm impE}) ]; 59 60(*0 subgoals vs 1 or more: the p in safep is for positive*) 61val (safe0_brls, safep_brls) = 62 List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls; 63 64(*Attack subgoals using safe inferences -- matching, not resolution*) 65fun safe_step_tac ctxt = 66 FIRST' [ 67 eq_assume_tac, 68 eq_mp_tac ctxt, 69 bimatch_tac ctxt safe0_brls, 70 hyp_subst_tac ctxt, 71 bimatch_tac ctxt safep_brls]; 72 73(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*) 74fun safe_tac ctxt = REPEAT_DETERM_FIRST (safe_step_tac ctxt); 75 76(*These steps could instantiate variables and are therefore unsafe.*) 77fun inst_step_tac ctxt = 78 assume_tac ctxt APPEND' mp_tac ctxt APPEND' 79 biresolve_tac ctxt (safe0_brls @ safep_brls); 80 81(*One safe or unsafe step. *) 82fun step_tac ctxt i = 83 FIRST [safe_tac ctxt, inst_step_tac ctxt i, biresolve_tac ctxt haz_brls i]; 84 85fun step_dup_tac ctxt i = 86 FIRST [safe_tac ctxt, inst_step_tac ctxt i, biresolve_tac ctxt haz_dup_brls i]; 87 88(*Dumb but fast*) 89fun fast_tac ctxt = SELECT_GOAL (DEPTH_SOLVE (step_tac ctxt 1)); 90 91(*Slower but smarter than fast_tac*) 92fun best_tac ctxt = 93 SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_tac ctxt 1)); 94 95(*Uses all_dupE: allows multiple use of universal assumptions. VERY slow.*) 96fun best_dup_tac ctxt = 97 SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_dup_tac ctxt 1)); 98 99 100end; 101 102