author  lcp 
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(* Title: Provers/classical 
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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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Theorem prover for classical reasoning, including predicate calculus, set 

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theory, etc. 

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Rules must be classified as intr, elim, safe, hazardous. 

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A rule is unsafe unless it can be applied blindly without harmful results. 

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For a rule to be safe, its premises and conclusion should be logically 

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equivalent. There should be no variables in the premises that are not in 

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the conclusion. 

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*) 

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infix 1 THEN_MAYBE; 
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signature CLASSICAL_DATA = 
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sig 

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val mp : thm (* [ P>Q; P ] ==> Q *) 
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val not_elim : thm (* [ ~P; P ] ==> R *) 
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val classical : thm (* (~P ==> P) ==> P *) 
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val sizef : thm > int (* size function for BEST_FIRST *) 
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val hyp_subst_tacs: (int > tactic) list 
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end; 

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(*Higher precedence than := facilitates use of references*) 

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infix 4 addSIs addSEs addSDs addIs addEs addDs 
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setwrapper compwrapper addbefore addafter; 
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signature CLASSICAL = 

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sig 

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type claset 

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type netpair 
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val empty_cs : claset 
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val addDs : claset * thm list > claset 
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val addEs : claset * thm list > claset 
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val addIs : claset * thm list > claset 
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val addSDs : claset * thm list > claset 
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val addSEs : claset * thm list > claset 
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val addSIs : claset * thm list > claset 
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val setwrapper : claset * (tactic>tactic) > claset 
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val compwrapper : claset * (tactic>tactic) > claset 
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val addbefore : claset * tactic > claset 
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val addafter : claset * tactic > claset 
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val print_cs : claset > unit 
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val rep_claset : 
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claset > {safeIs: thm list, safeEs: thm list, 
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hazIs: thm list, hazEs: thm list, 
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wrapper: tactic > tactic, 
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safe0_netpair: netpair, safep_netpair: netpair, 
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haz_netpair: netpair, dup_netpair: netpair} 
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val getwrapper : claset > tactic > tactic 
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val THEN_MAYBE : tactic * tactic > tactic 
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val best_tac : claset > int > tactic 
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val contr_tac : int > tactic 
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val depth_tac : claset > int > int > tactic 
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val deepen_tac : claset > int > int > tactic 
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val dup_elim : thm > thm 
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val dup_intr : thm > thm 
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val dup_step_tac : claset > int > tactic 
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val eq_mp_tac : int > tactic 
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val fast_tac : claset > int > tactic 
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val haz_step_tac : claset > int > tactic 
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val joinrules : thm list * thm list > (bool * thm) list 
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val mp_tac : int > tactic 
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val safe_tac : claset > tactic 
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val safe_step_tac : claset > int > tactic 
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val slow_step_tac : claset > int > tactic 
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val slow_best_tac : claset > int > tactic 
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val slow_tac : claset > int > tactic 
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val step_tac : claset > int > tactic 
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val swap : thm (* ~P ==> (~Q ==> P) ==> Q *) 
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val swapify : thm list > thm list 
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val swap_res_tac : thm list > int > tactic 
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val inst_step_tac : claset > int > tactic 
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val inst0_step_tac : claset > int > tactic 
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val instp_step_tac : claset > int > tactic 
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end; 
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functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL = 

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struct 

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local open Data in 

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(** Useful tactics for classical reasoning **) 

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val imp_elim = make_elim mp; 

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(*Solve goal that assumes both P and ~P. *) 

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val contr_tac = eresolve_tac [not_elim] THEN' assume_tac; 

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(*Finds P>Q and P in the assumptions, replaces implication by Q. 
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Could do the same thing for P<>Q and P... *) 
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fun mp_tac i = eresolve_tac [not_elim, imp_elim] i THEN assume_tac i; 
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(*Like mp_tac but instantiates no variables*) 

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fun eq_mp_tac i = ematch_tac [not_elim, imp_elim] i THEN eq_assume_tac i; 
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val swap = rule_by_tactic (etac thin_rl 1) (not_elim RS classical); 
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(*Creates rules to eliminate ~A, from rules to introduce A*) 

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fun swapify intrs = intrs RLN (2, [swap]); 

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(*Uses introduction rules in the normal way, or on negated assumptions, 

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trying rules in order. *) 

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fun swap_res_tac rls = 

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let fun addrl (rl,brls) = (false, rl) :: (true, rl RSN (2,swap)) :: brls 
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in assume_tac ORELSE' 

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contr_tac ORELSE' 

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biresolve_tac (foldr addrl (rls,[])) 

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end; 
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(*Duplication of hazardous rules, for complete provers*) 
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fun dup_intr th = standard (th RS classical); 
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fun dup_elim th = th RSN (2, revcut_rl) > assumption 2 > Sequence.hd > 
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rule_by_tactic (TRYALL (etac revcut_rl)); 
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(*** Classical rule sets ***) 
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type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net; 

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datatype claset = 

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CS of {safeIs : thm list, (*safe introduction rules*) 
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safeEs : thm list, (*safe elimination rules*) 
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hazIs : thm list, (*unsafe introduction rules*) 
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hazEs : thm list, (*unsafe elimination rules*) 
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wrapper : tactic>tactic, (*for transforming step_tac*) 
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safe0_netpair : netpair, (*nets for trivial cases*) 
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safep_netpair : netpair, (*nets for >0 subgoals*) 
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haz_netpair : netpair, (*nets for unsafe rules*) 
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dup_netpair : netpair}; (*nets for duplication*) 
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(*Desired invariants are 
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safe0_netpair = build safe0_brls, 
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safep_netpair = build safep_brls, 
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haz_netpair = build (joinrules(hazIs, hazEs)), 
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dup_netpair = build (joinrules(map dup_intr hazIs, 
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map dup_elim hazEs))} 
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where build = build_netpair(Net.empty,Net.empty), 
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safe0_brls contains all brules that solve the subgoal, and 
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safep_brls contains all brules that generate 1 or more new subgoals. 
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Nets must be built incrementally, to save space and time. 
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*) 
0  153 

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val empty_cs = 
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CS{safeIs = [], 
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safeEs = [], 
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hazIs = [], 
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hazEs = [], 
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wrapper = I, 
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safe0_netpair = (Net.empty,Net.empty), 
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safep_netpair = (Net.empty,Net.empty), 
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haz_netpair = (Net.empty,Net.empty), 
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dup_netpair = (Net.empty,Net.empty)}; 
0  164 

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fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) = 

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(writeln"Introduction rules"; prths hazIs; 
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writeln"Safe introduction rules"; prths safeIs; 
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writeln"Elimination rules"; prths hazEs; 
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writeln"Safe elimination rules"; prths safeEs; 
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()); 
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fun rep_claset (CS args) = args; 
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fun getwrapper (CS{wrapper,...}) = wrapper; 
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(** Adding (un)safe introduction or elimination rules. 
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In case of overlap, new rules are tried BEFORE old ones!! 
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**) 
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(*For use with biresolve_tac. Combines intr rules with swap to handle negated 
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assumptions. Pairs elim rules with true. *) 
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fun joinrules (intrs,elims) = 
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(map (pair true) (elims @ swapify intrs) @ 
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map (pair false) intrs); 
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(*Priority: prefer rules with fewest subgoals, 
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then rules added most recently.*) 
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fun tag_brls k [] = [] 
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 tag_brls k (brl::brls) = 
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(1000000*subgoals_of_brl brl + k, brl) :: 
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tag_brls (k+1) brls; 
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fun insert_tagged_list kbrls np = foldr insert_tagged_brl (kbrls, np); 
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(*Insert into netpair that already has nI intr rules and nE elim rules. 
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Count the intr rules double (to account for swapify). Negate to give the 
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new insertions the lowest priority.*) 
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fun insert (nI,nE) = insert_tagged_list o (tag_brls (~(2*nI+nE))) o joinrules; 
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(** Safe rules **) 
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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addSIs ths = 
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let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*) 
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take_prefix (fn rl => nprems_of rl=0) ths 
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val nI = length safeIs + length ths 
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and nE = length safeEs 
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in CS{safeIs = ths@safeIs, 
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safe0_netpair = insert (nI,nE) (safe0_rls, []) safe0_netpair, 
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safep_netpair = insert (nI,nE) (safep_rls, []) safep_netpair, 
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safeEs = safeEs, 
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hazIs = hazIs, 
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hazEs = hazEs, 
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wrapper = wrapper, 
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haz_netpair = haz_netpair, 
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dup_netpair = dup_netpair} 
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end; 
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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addSEs ths = 
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let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*) 
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take_prefix (fn rl => nprems_of rl=1) ths 
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val nI = length safeIs 
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and nE = length safeEs + length ths 
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in 
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CS{safeEs = ths@safeEs, 
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safe0_netpair = insert (nI,nE) ([], safe0_rls) safe0_netpair, 
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safep_netpair = insert (nI,nE) ([], safep_rls) safep_netpair, 
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safeIs = safeIs, 
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hazIs = hazIs, 
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hazEs = hazEs, 
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wrapper = wrapper, 
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haz_netpair = haz_netpair, 
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dup_netpair = dup_netpair} 
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end; 
0  241 

242 
fun cs addSDs ths = cs addSEs (map make_elim ths); 

243 

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(** Hazardous (unsafe) rules **) 
0  246 

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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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addIs ths = 
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let val nI = length hazIs + length ths 
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and nE = length hazEs 
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in 
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CS{hazIs = ths@hazIs, 
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haz_netpair = insert (nI,nE) (ths, []) haz_netpair, 
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dup_netpair = insert (nI,nE) (map dup_intr ths, []) dup_netpair, 
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safeIs = safeIs, 
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safeEs = safeEs, 
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hazEs = hazEs, 
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wrapper = wrapper, 
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safe0_netpair = safe0_netpair, 
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safep_netpair = safep_netpair} 
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end; 
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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addEs ths = 
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let val nI = length hazIs 
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and nE = length hazEs + length ths 
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in 
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CS{hazEs = ths@hazEs, 
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haz_netpair = insert (nI,nE) ([], ths) haz_netpair, 
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dup_netpair = insert (nI,nE) ([], map dup_elim ths) dup_netpair, 
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safeIs = safeIs, 
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safeEs = safeEs, 
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hazIs = hazIs, 
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wrapper = wrapper, 
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safe0_netpair = safe0_netpair, 
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safep_netpair = safep_netpair} 
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end; 
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281 
fun cs addDs ths = cs addEs (map make_elim ths); 

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(** Setting or modifying the wrapper tactical **) 
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(*Set a new wrapper*) 
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fun (CS{safeIs, safeEs, hazIs, hazEs, 
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safe0_netpair, safep_netpair, haz_netpair, dup_netpair, ...}) 
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setwrapper new_wrapper = 
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CS{wrapper = new_wrapper, 
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safeIs = safeIs, 
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safeEs = safeEs, 
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hazIs = hazIs, 
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hazEs = hazEs, 
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safe0_netpair = safe0_netpair, 
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safep_netpair = safep_netpair, 
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haz_netpair = haz_netpair, 
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dup_netpair = dup_netpair}; 
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(*Compose a tactical with the existing wrapper*) 
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fun cs compwrapper wrapper' = cs setwrapper (wrapper' o getwrapper cs); 
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(*Execute tac1, but only execute tac2 if there are at least as many subgoals 
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as before. This ensures that tac2 is only applied to an outcome of tac1.*) 
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fun tac1 THEN_MAYBE tac2 = 
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STATE (fn state => 
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tac1 THEN 
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COND (has_fewer_prems (nprems_of state)) all_tac tac2); 
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(*Cause a tactic to be executed before/after the step tactic*) 
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fun cs addbefore tac2 = cs compwrapper (fn tac1 => tac2 THEN_MAYBE tac1); 
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fun cs addafter tac2 = cs compwrapper (fn tac1 => tac1 THEN_MAYBE tac2); 
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0  316 
(*** Simple tactics for theorem proving ***) 
317 

318 
(*Attack subgoals using safe inferences  matching, not resolution*) 

319 
fun safe_step_tac (CS{safe0_netpair,safep_netpair,...}) = 

320 
FIRST' [eq_assume_tac, 

321 
eq_mp_tac, 

322 
bimatch_from_nets_tac safe0_netpair, 

323 
FIRST' hyp_subst_tacs, 

324 
bimatch_from_nets_tac safep_netpair] ; 

325 

326 
(*Repeatedly attack subgoals using safe inferences  it's deterministic!*) 

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fun safe_tac cs = REPEAT_DETERM_FIRST (safe_step_tac cs); 
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(*But these unsafe steps at least solve a subgoal!*) 
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fun inst0_step_tac (CS{safe0_netpair,safep_netpair,...}) = 
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assume_tac APPEND' 
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contr_tac APPEND' 
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biresolve_from_nets_tac safe0_netpair; 
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(*These are much worse since they could generate more and more subgoals*) 
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fun instp_step_tac (CS{safep_netpair,...}) = 
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337 
biresolve_from_nets_tac safep_netpair; 
0  338 

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(*These steps could instantiate variables and are therefore unsafe.*) 

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fun inst_step_tac cs = inst0_step_tac cs APPEND' instp_step_tac cs; 
0  341 

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fun haz_step_tac (CS{haz_netpair,...}) = 
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biresolve_from_nets_tac haz_netpair; 
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344 

0  345 
(*Single step for the prover. FAILS unless it makes progress. *) 
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fun step_tac cs i = 
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getwrapper cs 
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(FIRST [safe_tac cs, inst_step_tac cs i, haz_step_tac cs i]); 
0  349 

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(*Using a "safe" rule to instantiate variables is unsafe. This tactic 

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allows backtracking from "safe" rules to "unsafe" rules here.*) 

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fun slow_step_tac cs i = 
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getwrapper cs 
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(safe_tac cs ORELSE (inst_step_tac cs i APPEND haz_step_tac cs i)); 
0  355 

356 
(*** The following tactics all fail unless they solve one goal ***) 

357 

358 
(*Dumb but fast*) 

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fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1)); 

360 

361 
(*Slower but smarter than fast_tac*) 

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fun best_tac cs = 

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SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1)); 

364 

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fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1)); 

366 

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fun slow_best_tac cs = 

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SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1)); 

369 

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(*** Complete tactic, loosely based upon LeanTaP. This tactic is the outcome 
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of much experimentation! Changing APPEND to ORELSE below would prove 
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373 
easy theorems faster, but loses completeness  and many of the harder 
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374 
theorems such as 43. ***) 
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375 

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(*Nondeterministic! Could always expand the first unsafe connective. 
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377 
That's hard to implement and did not perform better in experiments, due to 
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378 
greater search depth required.*) 
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fun dup_step_tac (cs as (CS{dup_netpair,...})) = 
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biresolve_from_nets_tac dup_netpair; 
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381 

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(*Searching to depth m.*) 
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fun depth_tac cs m i = STATE(fn state => 
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384 
SELECT_GOAL 
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(REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE 
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386 
(DEPTH_SOLVE (depth_tac cs m 1), 
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387 
inst0_step_tac cs 1 APPEND 
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388 
COND (K(m=0)) no_tac 
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389 
((instp_step_tac cs 1 APPEND dup_step_tac cs 1) 
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THEN DEPTH_SOLVE (depth_tac cs (m1) 1)))) 
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391 
i); 
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392 

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(*Iterative deepening tactical. Allows us to "deepen" any search tactic*) 
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394 
fun DEEPEN tacf m i = STATE(fn state => 
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if has_fewer_prems i state then no_tac 
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396 
else (writeln ("Depth = " ^ string_of_int m); 
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397 
tacf m i ORELSE DEEPEN tacf (m+2) i)); 
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398 

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399 
fun safe_depth_tac cs m = 
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400 
SUBGOAL 
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401 
(fn (prem,i) => 
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402 
let val deti = 
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403 
(*No Vars in the goal? No need to backtrack between goals.*) 
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404 
case term_vars prem of 
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405 
[] => DETERM 
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406 
 _::_ => I 
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407 
in SELECT_GOAL (TRY (safe_tac cs) THEN 
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408 
DEPTH_SOLVE (deti (depth_tac cs m 1))) i 
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409 
end); 
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410 

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411 
fun deepen_tac cs = DEEPEN (safe_depth_tac cs); 
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412 

0  413 
end; 
414 
end; 