src/FOL/intprover.ML
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(*  Title:      FOL/intprover.ML
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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A naive prover for intuitionistic logic
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BEWARE OF NAME CLASHES WITH CLASSICAL TACTICS -- use IntPr.fast_tac ...
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Completeness (for propositional logic) is proved in 
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Roy Dyckhoff.
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Contraction-Free Sequent Calculi for Intuitionistic Logic.
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J. Symbolic Logic  57(3), 1992, pages 795-807.
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The approach was developed independently by Roy Dyckhoff and L C Paulson.
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*)
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signature INT_PROVER = 
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  sig
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  val best_tac: int -> tactic
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  val best_dup_tac: int -> tactic
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  val fast_tac: int -> tactic
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  val inst_step_tac: int -> tactic
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  val safe_step_tac: int -> tactic
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  val safe_brls: (bool * thm) list
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  val safe_tac: tactic
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  val step_tac: int -> tactic
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  val step_dup_tac: int -> tactic
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  val haz_brls: (bool * thm) list
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  val haz_dup_brls: (bool * thm) list
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  end;
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structure IntPr : INT_PROVER   = 
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struct
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(*Negation is treated as a primitive symbol, with rules notI (introduction),
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  not_to_imp (converts the assumption ~P to P-->False), and not_impE
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  (handles double negations).  Could instead rewrite by not_def as the first
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  step of an intuitionistic proof.
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*)
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val safe_brls = sort (make_ord lessb)
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    [ (true, @{thm FalseE}), (false, @{thm TrueI}), (false, @{thm refl}),
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      (false, @{thm impI}), (false, @{thm notI}), (false, @{thm allI}),
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      (true, @{thm conjE}), (true, @{thm exE}),
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      (false, @{thm conjI}), (true, @{thm conj_impE}),
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      (true, @{thm disj_impE}), (true, @{thm disjE}), 
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      (false, @{thm iffI}), (true, @{thm iffE}), (true, @{thm not_to_imp}) ];
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val haz_brls =
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    [ (false, @{thm disjI1}), (false, @{thm disjI2}), (false, @{thm exI}), 
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      (true, @{thm allE}), (true, @{thm not_impE}), (true, @{thm imp_impE}), (true, @{thm iff_impE}),
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      (true, @{thm all_impE}), (true, @{thm ex_impE}), (true, @{thm impE}) ];
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val haz_dup_brls =
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    [ (false, @{thm disjI1}), (false, @{thm disjI2}), (false, @{thm exI}),
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      (true, @{thm all_dupE}), (true, @{thm not_impE}), (true, @{thm imp_impE}), (true, @{thm iff_impE}),
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      (true, @{thm all_impE}), (true, @{thm ex_impE}), (true, @{thm impE}) ];
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(*0 subgoals vs 1 or more: the p in safep is for positive*)
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val (safe0_brls, safep_brls) =
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    List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls;
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(*Attack subgoals using safe inferences -- matching, not resolution*)
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val safe_step_tac = FIRST' [eq_assume_tac,
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                            eq_mp_tac,
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                            bimatch_tac safe0_brls,
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                            hyp_subst_tac,
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                            bimatch_tac safep_brls] ;
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(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
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val safe_tac = REPEAT_DETERM_FIRST safe_step_tac;
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(*These steps could instantiate variables and are therefore unsafe.*)
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val inst_step_tac =
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  assume_tac APPEND' mp_tac APPEND' 
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  biresolve_tac (safe0_brls @ safep_brls);
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(*One safe or unsafe step. *)
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fun step_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_brls i];
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fun step_dup_tac i = FIRST [safe_tac, inst_step_tac i, biresolve_tac haz_dup_brls i];
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(*Dumb but fast*)
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val fast_tac = SELECT_GOAL (DEPTH_SOLVE (step_tac 1));
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(*Slower but smarter than fast_tac*)
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val best_tac = 
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  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_tac 1));
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(*Uses all_dupE: allows multiple use of universal assumptions.  VERY slow.*)
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val best_dup_tac = 
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  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm) (step_dup_tac 1));
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end;
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