author  berghofe 
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child 1800  3d9d2ef0cd3b 
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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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 merge_cs : claset * claset > 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 fast_tac : claset > int > tactic 
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val slow_tac : claset > int > tactic 
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val weight_ASTAR : int ref 
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val astar_tac : claset > int > tactic 
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val slow_astar_tac : claset > int > tactic 
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val best_tac : claset > int > tactic 
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val slow_best_tac : claset > 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 contr_tac : 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 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 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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val claset : claset ref 

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val AddDs : thm list > unit 

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val AddEs : thm list > unit 

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val AddIs : thm list > unit 

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val AddSDs : thm list > unit 

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val AddSEs : thm list > unit 

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val AddSIs : thm list > unit 

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val Fast_tac : 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 = (*cannot use bind_thm within a structure!*) 
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store_thm ("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 = 
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store_thm ("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  169 

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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  180 

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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, 
1231  205 
then rules added most recently (preferring the head of the list).*) 
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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  257 

258 
fun cs addSDs ths = cs addSEs (map make_elim ths); 

259 

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(** Hazardous (unsafe) rules **) 
0  262 

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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; 
0  296 

297 
fun cs addDs ths = cs addEs (map make_elim ths); 

298 

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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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1711  331 
(*Merge works by adding all new rules of the 2nd claset into the 1st claset. 
332 
Merging the term nets may look more efficient, but the rather delicate 

333 
treatment of priority might get muddled up.*) 

334 
fun merge_cs 

335 
(cs as CS{safeIs, safeEs, hazIs, hazEs, wrapper, ...}, 

336 
CS{safeIs=safeIs2, safeEs=safeEs2, hazIs=hazIs2, hazEs=hazEs2,...}) = 

337 
let val safeIs' = gen_rems eq_thm (safeIs2,safeIs) 

338 
val safeEs' = gen_rems eq_thm (safeEs2,safeEs) 

339 
val hazIs' = gen_rems eq_thm (hazIs2,hazIs) 

340 
val hazEs' = gen_rems eq_thm (hazEs2,hazEs) 

341 
in cs addSIs safeIs' 

342 
addSEs safeEs' 

343 
addIs hazIs' 

344 
addEs hazEs' 

345 
end; 

346 

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0  348 
(*** Simple tactics for theorem proving ***) 
349 

350 
(*Attack subgoals using safe inferences  matching, not resolution*) 

351 
fun safe_step_tac (CS{safe0_netpair,safep_netpair,...}) = 

352 
FIRST' [eq_assume_tac, 

353 
eq_mp_tac, 

354 
bimatch_from_nets_tac safe0_netpair, 

355 
FIRST' hyp_subst_tacs, 

356 
bimatch_from_nets_tac safep_netpair] ; 

357 

358 
(*Repeatedly attack subgoals using safe inferences  it's deterministic!*) 

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359 
fun safe_tac cs = REPEAT_DETERM_FIRST (safe_step_tac cs); 
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360 

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361 
(*But these unsafe steps at least solve a subgoal!*) 
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362 
fun inst0_step_tac (CS{safe0_netpair,safep_netpair,...}) = 
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363 
assume_tac APPEND' 
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364 
contr_tac APPEND' 
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365 
biresolve_from_nets_tac safe0_netpair; 
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366 

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367 
(*These are much worse since they could generate more and more subgoals*) 
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368 
fun instp_step_tac (CS{safep_netpair,...}) = 
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369 
biresolve_from_nets_tac safep_netpair; 
0  370 

371 
(*These steps could instantiate variables and are therefore unsafe.*) 

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372 
fun inst_step_tac cs = inst0_step_tac cs APPEND' instp_step_tac cs; 
0  373 

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374 
fun haz_step_tac (CS{haz_netpair,...}) = 
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375 
biresolve_from_nets_tac haz_netpair; 
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376 

0  377 
(*Single step for the prover. FAILS unless it makes progress. *) 
681
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378 
fun step_tac cs i = 
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379 
getwrapper cs 
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380 
(FIRST [safe_tac cs, inst_step_tac cs i, haz_step_tac cs i]); 
0  381 

382 
(*Using a "safe" rule to instantiate variables is unsafe. This tactic 

383 
allows backtracking from "safe" rules to "unsafe" rules here.*) 

681
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384 
fun slow_step_tac cs i = 
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385 
getwrapper cs 
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386 
(safe_tac cs ORELSE (inst_step_tac cs i APPEND haz_step_tac cs i)); 
0  387 

388 
(*** The following tactics all fail unless they solve one goal ***) 

389 

390 
(*Dumb but fast*) 

391 
fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1)); 

392 

393 
(*Slower but smarter than fast_tac*) 

394 
fun best_tac cs = 

395 
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1)); 

396 

397 
fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1)); 

398 

399 
fun slow_best_tac cs = 

400 
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1)); 

401 

681
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402 

1587
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403 
(**ASTAR with weight weight_ASTAR, by Norbert Voelker*) 
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404 
val weight_ASTAR = ref 5; 
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Now provides astar versions (thanks to Norbert Voelker)
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405 

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406 
fun astar_tac cs = 
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407 
SELECT_GOAL ( ASTAR (has_fewer_prems 1 
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408 
, fn level =>(fn thm =>size_of_thm thm + !weight_ASTAR *level)) 
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Now provides astar versions (thanks to Norbert Voelker)
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409 
(step_tac cs 1)); 
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410 

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411 
fun slow_astar_tac cs = 
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412 
SELECT_GOAL ( ASTAR (has_fewer_prems 1 
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413 
, fn level =>(fn thm =>size_of_thm thm + !weight_ASTAR *level)) 
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Now provides astar versions (thanks to Norbert Voelker)
paulson
parents:
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414 
(slow_step_tac cs 1)); 
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Now provides astar versions (thanks to Norbert Voelker)
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415 

982
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416 
(*** Complete tactic, loosely based upon LeanTaP. This tactic is the outcome 
747
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diff
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417 
of much experimentation! Changing APPEND to ORELSE below would prove 
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418 
easy theorems faster, but loses completeness  and many of the harder 
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419 
theorems such as 43. ***) 
681
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parents:
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420 

747
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421 
(*Nondeterministic! Could always expand the first unsafe connective. 
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422 
That's hard to implement and did not perform better in experiments, due to 
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parents:
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423 
greater search depth required.*) 
681
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parents:
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424 
fun dup_step_tac (cs as (CS{dup_netpair,...})) = 
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425 
biresolve_from_nets_tac dup_netpair; 
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426 

747
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427 
(*Searching to depth m.*) 
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428 
fun depth_tac cs m i = STATE(fn state => 
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429 
SELECT_GOAL 
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430 
(REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE 
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431 
(DEPTH_SOLVE (depth_tac cs m 1), 
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432 
inst0_step_tac cs 1 APPEND 
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433 
COND (K(m=0)) no_tac 
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434 
((instp_step_tac cs 1 APPEND dup_step_tac cs 1) 
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435 
THEN DEPTH_SOLVE (depth_tac cs (m1) 1)))) 
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436 
i); 
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437 

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parents:
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438 
(*Iterative deepening tactical. Allows us to "deepen" any search tactic*) 
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diff
changeset

439 
fun DEEPEN tacf m i = STATE(fn state => 
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440 
if has_fewer_prems i state then no_tac 
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441 
else (writeln ("Depth = " ^ string_of_int m); 
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changeset

442 
tacf m i ORELSE DEEPEN tacf (m+2) i)); 
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changeset

443 

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changeset

444 
fun safe_depth_tac cs m = 
681
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parents:
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diff
changeset

445 
SUBGOAL 
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446 
(fn (prem,i) => 
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447 
let val deti = 
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changeset

448 
(*No Vars in the goal? No need to backtrack between goals.*) 
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Provers/classical: now takes theorem "classical" as argument, proves "swap"
lcp
parents:
469
diff
changeset

449 
case term_vars prem of 
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lcp
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diff
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450 
[] => DETERM 
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lcp
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diff
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451 
 _::_ => I 
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lcp
parents:
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diff
changeset

452 
in SELECT_GOAL (TRY (safe_tac cs) THEN 
747
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453 
DEPTH_SOLVE (deti (depth_tac cs m 1))) i 
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changeset

454 
end); 
681
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Provers/classical: now takes theorem "classical" as argument, proves "swap"
lcp
parents:
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diff
changeset

455 

747
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changeset

456 
fun deepen_tac cs = DEEPEN (safe_depth_tac cs); 
681
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Provers/classical: now takes theorem "classical" as argument, proves "swap"
lcp
parents:
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diff
changeset

457 

1724  458 
val claset = ref empty_cs; 
459 

460 
fun AddDs ts = (claset := !claset addDs ts); 

461 

462 
fun AddEs ts = (claset := !claset addEs ts); 

463 

464 
fun AddIs ts = (claset := !claset addIs ts); 

465 

466 
fun AddSDs ts = (claset := !claset addSDs ts); 

467 

468 
fun AddSEs ts = (claset := !claset addSEs ts); 

469 

470 
fun AddSIs ts = (claset := !claset addSIs ts); 

471 

472 
fun Fast_tac i = fast_tac (!claset) i; 

473 

0  474 
end; 
475 
end; 