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(* Title: FOL/intprover


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ID: $Id$


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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 Int.fast_tac ...


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Completeness (for propositional logic) is proved in


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Roy Dyckhoff.


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ContractionFree Sequent Calculi for Intuitionistic Logic.


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J. Symbolic Logic (in press)


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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 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 haz_brls: (bool * thm) list


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end;


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structure Int : 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 lessb


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[ (true,FalseE), (false,TrueI), (false,refl),


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(false,impI), (false,notI), (false,allI),


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(true,conjE), (true,exE),


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(false,conjI), (true,conj_impE),


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(true,disj_impE), (true,ex_impE),


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(true,disjE), (false,iffI), (true,iffE), (true,not_to_imp) ];


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val haz_brls =


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[ (false,disjI1), (false,disjI2), (false,exI),


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(true,allE), (true,not_impE), (true,imp_impE), (true,iff_impE),


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(true,all_impE), (true,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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partition (apl(0,op=) 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 = DETERM (REPEAT_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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(*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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end;


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