author | lcp |
Thu, 30 Mar 1995 13:44:34 +0200 | |
changeset 982 | 4fe0b642b7d5 |
parent 747 | bdc066781063 |
child 1010 | a7693f30065d |
permissions | -rw-r--r-- |
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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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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 : 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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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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fun rep_claset (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) = |
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{safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper}; |
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fun getwrapper (CS{wrapper,...}) = wrapper; |
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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. Sorts the list of pairs by |
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the number of new subgoals generated. *) |
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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,wrapper} = |
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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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wrapper = wrapper, |
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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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end; |
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(*** Manipulation of clasets ***) |
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val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[], wrapper=I}; |
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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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(** Adding new (un)safe introduction or elimination rules **) |
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fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addSIs ths = |
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make_cs {safeIs=ths@safeIs, |
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safeEs=safeEs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper}; |
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fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addSEs ths = |
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make_cs {safeEs=ths@safeEs, |
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safeIs=safeIs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper}; |
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|
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fun cs addSDs ths = cs addSEs (map make_elim ths); |
|
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||
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fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addIs ths = |
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make_cs {hazIs=ths@hazIs, |
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safeIs=safeIs, safeEs=safeEs, hazEs=hazEs, wrapper=wrapper}; |
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fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addEs ths = |
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make_cs {hazEs=ths@hazEs, |
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safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, wrapper=wrapper}; |
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fun cs addDs ths = cs addEs (map make_elim ths); |
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(** Setting or modifying the wrapper tactical **) |
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|
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(*Set a new wrapper*) |
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fun (CS{safeIs,safeEs,hazIs,hazEs,...}) setwrapper wrapper = |
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make_cs {wrapper=wrapper, |
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safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs}; |
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|
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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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|
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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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|
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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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|
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|
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|
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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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|
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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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|
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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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|
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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 | 249 |
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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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|
0 | 253 |
(*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 | 257 |
|
258 |
(*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 | 263 |
|
264 |
(*** The following tactics all fail unless they solve one goal ***) |
|
265 |
||
266 |
(*Dumb but fast*) |
|
267 |
fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1)); |
|
268 |
||
269 |
(*Slower but smarter than fast_tac*) |
|
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fun best_tac cs = |
|
271 |
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1)); |
|
272 |
||
273 |
fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1)); |
|
274 |
||
275 |
fun slow_best_tac cs = |
|
276 |
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1)); |
|
277 |
||
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|
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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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283 |
|
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(*Non-deterministic! Could always expand the first unsafe connective. |
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285 |
That's hard to implement and did not perform better in experiments, due to |
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286 |
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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|
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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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292 |
SELECT_GOAL |
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293 |
(REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE |
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294 |
(DEPTH_SOLVE (depth_tac cs m 1), |
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295 |
inst0_step_tac cs 1 APPEND |
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296 |
COND (K(m=0)) no_tac |
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297 |
((instp_step_tac cs 1 APPEND dup_step_tac cs 1) |
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THEN DEPTH_SOLVE (depth_tac cs (m-1) 1)))) |
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299 |
i); |
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300 |
|
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301 |
(*Iterative deepening tactical. Allows us to "deepen" any search tactic*) |
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302 |
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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304 |
else (writeln ("Depth = " ^ string_of_int m); |
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|
305 |
tacf m i ORELSE DEEPEN tacf (m+2) i)); |
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|
306 |
|
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|
307 |
fun safe_depth_tac cs m = |
681
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SUBGOAL |
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309 |
(fn (prem,i) => |
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310 |
let val deti = |
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311 |
(*No Vars in the goal? No need to backtrack between goals.*) |
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|
312 |
case term_vars prem of |
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[] => DETERM |
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314 |
| _::_ => I |
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|
315 |
in SELECT_GOAL (TRY (safe_tac cs) THEN |
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316 |
DEPTH_SOLVE (deti (depth_tac cs m 1))) i |
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317 |
end); |
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318 |
|
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319 |
fun deepen_tac cs = DEEPEN (safe_depth_tac cs); |
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320 |
|
0 | 321 |
end; |
322 |
end; |