author  lcp 
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permissions  rwrr 
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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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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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signature CLASSICAL = 

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sig 

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type claset 

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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 print_cs : claset > unit 
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val rep_claset : claset > {safeIs: thm list, safeEs: thm list, 
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hazIs: thm list, hazEs: thm list} 
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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, 

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safeEs : thm list, 

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hazIs : thm list, 

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hazEs : thm list, 

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safe0_netpair : netpair, 

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safep_netpair : netpair, 

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haz_netpair : netpair, 
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dup_netpair : netpair}; 
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fun rep_claset (CS{safeIs,safeEs,hazIs,hazEs,...}) = 

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{safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs}; 

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(*For use with biresolve_tac. Combines intrs with swap to catch negated 

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assumptions; pairs elims with true; sorts. *) 

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fun joinrules (intrs,elims) = 

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sort lessb 

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(map (pair true) (elims @ swapify intrs) @ 

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map (pair false) intrs); 

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val build = build_netpair(Net.empty,Net.empty); 
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(*Make a claset from the four kinds of rules*) 
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fun make_cs {safeIs,safeEs,hazIs,hazEs} = 

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let val (safe0_brls, safep_brls) = (*0 subgoals vs 1 or more*) 

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take_prefix (fn brl => subgoals_of_brl brl=0) 

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(joinrules(safeIs, safeEs)) 

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in CS{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 = 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))} 
0  149 
end; 
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(*** Manipulation of clasets ***) 

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val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[]}; 

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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 (CS{safeIs,safeEs,hazIs,hazEs,...}) addSIs ths = 

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make_cs {safeIs=ths@safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs}; 

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

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make_cs {safeIs=safeIs, safeEs=ths@safeEs, hazIs=hazIs, hazEs=hazEs}; 

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

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

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make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=ths@hazIs, hazEs=hazEs}; 

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

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make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=ths@hazEs}; 

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fun cs addDs ths = cs addEs (map make_elim ths); 

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(*** Simple tactics for theorem proving ***) 

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

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

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FIRST' [eq_assume_tac, 

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eq_mp_tac, 

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bimatch_from_nets_tac safe0_netpair, 

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FIRST' hyp_subst_tacs, 

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bimatch_from_nets_tac safep_netpair] ; 

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(*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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biresolve_from_nets_tac safep_netpair; 
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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  203 

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fun haz_step_tac (cs as (CS{haz_netpair,...})) = 
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biresolve_from_nets_tac haz_netpair; 
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(*Single step for the prover. FAILS unless it makes progress. *) 
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fun step_tac cs i = 
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FIRST [safe_tac cs, inst_step_tac cs i, haz_step_tac cs i]; 
0  210 

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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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safe_tac cs ORELSE (inst_step_tac cs i APPEND haz_step_tac cs i); 
0  215 

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

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(*Dumb but fast*) 

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

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(*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)); 

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

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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)); 

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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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easy theorems faster, but loses completeness  and many of the harder 
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theorems such as 43. ***) 
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(*Nondeterministic! Could always expand the first unsafe connective. 
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That's hard to implement and did not perform better in experiments, due to 
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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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(*Searching to depth m.*) 
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fun depth_tac cs m i = STATE(fn state => 
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SELECT_GOAL 
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(REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE 
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(DEPTH_SOLVE (depth_tac cs m 1), 
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inst0_step_tac cs 1 APPEND 
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COND (K(m=0)) no_tac 
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((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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i); 
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(*Iterative deepening tactical. Allows us to "deepen" any search tactic*) 
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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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else (writeln ("Depth = " ^ string_of_int m); 
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tacf m i ORELSE DEEPEN tacf (m+2) i)); 
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fun safe_depth_tac cs m = 
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SUBGOAL 
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(fn (prem,i) => 
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let val deti = 
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(*No Vars in the goal? No need to backtrack between goals.*) 
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case term_vars prem of 
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[] => DETERM 
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 _::_ => I 
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in SELECT_GOAL (TRY (safe_tac cs) THEN 
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DEPTH_SOLVE (deti (depth_tac cs m 1))) i 
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end); 
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fun deepen_tac cs = DEEPEN (safe_depth_tac cs); 
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0  273 
end; 
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end; 