(* Title: FOL/int-prover
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 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;