(* Title: Provers/classical
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Theorem prover for classical reasoning, including predicate calculus, set
theory, etc.
Rules must be classified as intr, elim, safe, hazardous.
A rule is unsafe unless it can be applied blindly without harmful results.
For a rule to be safe, its premises and conclusion should be logically
equivalent. There should be no variables in the premises that are not in
the conclusion.
*)
infix 1 THEN_MAYBE;
signature CLASSICAL_DATA =
sig
val mp : thm (* [| P-->Q; P |] ==> Q *)
val not_elim : thm (* [| ~P; P |] ==> R *)
val classical : thm (* (~P ==> P) ==> P *)
val sizef : thm -> int (* size function for BEST_FIRST *)
val hyp_subst_tacs: (int -> tactic) list
end;
(*Higher precedence than := facilitates use of references*)
infix 4 addSIs addSEs addSDs addIs addEs addDs
setwrapper compwrapper addbefore addafter;
signature CLASSICAL =
sig
type claset
val empty_cs : claset
val addDs : claset * thm list -> claset
val addEs : claset * thm list -> claset
val addIs : claset * thm list -> claset
val addSDs : claset * thm list -> claset
val addSEs : claset * thm list -> claset
val addSIs : claset * thm list -> claset
val setwrapper : claset * (tactic->tactic) -> claset
val compwrapper : claset * (tactic->tactic) -> claset
val addbefore : claset * tactic -> claset
val addafter : claset * tactic -> claset
val print_cs : claset -> unit
val rep_claset : claset -> {safeIs: thm list, safeEs: thm list,
hazIs: thm list, hazEs: thm list,
wrapper: tactic -> tactic}
val getwrapper : claset -> tactic -> tactic
val THEN_MAYBE : tactic * tactic -> tactic
val best_tac : claset -> int -> tactic
val contr_tac : int -> tactic
val depth_tac : claset -> int -> int -> tactic
val deepen_tac : claset -> int -> int -> tactic
val dup_elim : thm -> thm
val dup_intr : thm -> thm
val dup_step_tac : claset -> int -> tactic
val eq_mp_tac : int -> tactic
val fast_tac : claset -> int -> tactic
val haz_step_tac : claset -> int -> tactic
val joinrules : thm list * thm list -> (bool * thm) list
val mp_tac : int -> tactic
val safe_tac : claset -> tactic
val safe_step_tac : claset -> int -> tactic
val slow_step_tac : claset -> int -> tactic
val slow_best_tac : claset -> int -> tactic
val slow_tac : claset -> int -> tactic
val step_tac : claset -> int -> tactic
val swap : thm (* ~P ==> (~Q ==> P) ==> Q *)
val swapify : thm list -> thm list
val swap_res_tac : thm list -> int -> tactic
val inst_step_tac : claset -> int -> tactic
val inst0_step_tac : claset -> int -> tactic
val instp_step_tac : claset -> int -> tactic
end;
functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL =
struct
local open Data in
(** Useful tactics for classical reasoning **)
val imp_elim = make_elim mp;
(*Solve goal that assumes both P and ~P. *)
val contr_tac = eresolve_tac [not_elim] THEN' assume_tac;
(*Finds P-->Q and P in the assumptions, replaces implication by Q.
Could do the same thing for P<->Q and P... *)
fun mp_tac i = eresolve_tac [not_elim, imp_elim] i THEN assume_tac i;
(*Like mp_tac but instantiates no variables*)
fun eq_mp_tac i = ematch_tac [not_elim, imp_elim] i THEN eq_assume_tac i;
val swap = rule_by_tactic (etac thin_rl 1) (not_elim RS classical);
(*Creates rules to eliminate ~A, from rules to introduce A*)
fun swapify intrs = intrs RLN (2, [swap]);
(*Uses introduction rules in the normal way, or on negated assumptions,
trying rules in order. *)
fun swap_res_tac rls =
let fun addrl (rl,brls) = (false, rl) :: (true, rl RSN (2,swap)) :: brls
in assume_tac ORELSE'
contr_tac ORELSE'
biresolve_tac (foldr addrl (rls,[]))
end;
(*Duplication of hazardous rules, for complete provers*)
fun dup_intr th = standard (th RS classical);
fun dup_elim th = th RSN (2, revcut_rl) |> assumption 2 |> Sequence.hd |>
rule_by_tactic (TRYALL (etac revcut_rl));
(*** Classical rule sets ***)
type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;
datatype claset =
CS of {safeIs : thm list, (*safe introduction rules*)
safeEs : thm list, (*safe elimination rules*)
hazIs : thm list, (*unsafe introduction rules*)
hazEs : thm list, (*unsafe elimination rules*)
wrapper : tactic->tactic, (*for transforming step_tac*)
safe0_netpair : netpair, (*nets for trivial cases*)
safep_netpair : netpair, (*nets for >0 subgoals*)
haz_netpair : netpair, (*nets for unsafe rules*)
dup_netpair : netpair}; (*nets for duplication*)
fun rep_claset (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) =
{safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper};
fun getwrapper (CS{wrapper,...}) = wrapper;
(*For use with biresolve_tac. Combines intr rules with swap to handle negated
assumptions. Pairs elim rules with true. Sorts the list of pairs by
the number of new subgoals generated. *)
fun joinrules (intrs,elims) =
sort lessb
(map (pair true) (elims @ swapify intrs) @
map (pair false) intrs);
val build = build_netpair(Net.empty,Net.empty);
(*Make a claset from the four kinds of rules*)
fun make_cs {safeIs,safeEs,hazIs,hazEs,wrapper} =
let val (safe0_brls, safep_brls) = (*0 subgoals vs 1 or more*)
take_prefix (fn brl => subgoals_of_brl brl=0)
(joinrules(safeIs, safeEs))
in CS{safeIs = safeIs,
safeEs = safeEs,
hazIs = hazIs,
hazEs = hazEs,
wrapper = wrapper,
safe0_netpair = build safe0_brls,
safep_netpair = build safep_brls,
haz_netpair = build (joinrules(hazIs, hazEs)),
dup_netpair = build (joinrules(map dup_intr hazIs,
map dup_elim hazEs))}
end;
(*** Manipulation of clasets ***)
val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[], wrapper=I};
fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) =
(writeln"Introduction rules"; prths hazIs;
writeln"Safe introduction rules"; prths safeIs;
writeln"Elimination rules"; prths hazEs;
writeln"Safe elimination rules"; prths safeEs;
());
(** Adding new (un)safe introduction or elimination rules
In case of overlap, new rules are tried BEFORE old ones
**)
fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addSIs ths =
make_cs {safeIs=ths@safeIs,
safeEs=safeEs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper};
fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addSEs ths =
make_cs {safeEs=ths@safeEs,
safeIs=safeIs, hazIs=hazIs, hazEs=hazEs, wrapper=wrapper};
fun cs addSDs ths = cs addSEs (map make_elim ths);
fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addIs ths =
make_cs {hazIs=ths@hazIs,
safeIs=safeIs, safeEs=safeEs, hazEs=hazEs, wrapper=wrapper};
fun (CS{safeIs,safeEs,hazIs,hazEs,wrapper,...}) addEs ths =
make_cs {hazEs=ths@hazEs,
safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, wrapper=wrapper};
fun cs addDs ths = cs addEs (map make_elim ths);
(** Setting or modifying the wrapper tactical **)
(*Set a new wrapper*)
fun (CS{safeIs,safeEs,hazIs,hazEs,...}) setwrapper wrapper =
make_cs {wrapper=wrapper,
safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs};
(*Compose a tactical with the existing wrapper*)
fun cs compwrapper wrapper' = cs setwrapper (wrapper' o getwrapper cs);
(*Execute tac1, but only execute tac2 if there are at least as many subgoals
as before. This ensures that tac2 is only applied to an outcome of tac1.*)
fun tac1 THEN_MAYBE tac2 =
STATE (fn state =>
tac1 THEN
COND (has_fewer_prems (nprems_of state)) all_tac tac2);
(*Cause a tactic to be executed before/after the step tactic*)
fun cs addbefore tac2 = cs compwrapper (fn tac1 => tac2 THEN_MAYBE tac1);
fun cs addafter tac2 = cs compwrapper (fn tac1 => tac1 THEN_MAYBE tac2);
(*** Simple tactics for theorem proving ***)
(*Attack subgoals using safe inferences -- matching, not resolution*)
fun safe_step_tac (CS{safe0_netpair,safep_netpair,...}) =
FIRST' [eq_assume_tac,
eq_mp_tac,
bimatch_from_nets_tac safe0_netpair,
FIRST' hyp_subst_tacs,
bimatch_from_nets_tac safep_netpair] ;
(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
fun safe_tac cs = REPEAT_DETERM_FIRST (safe_step_tac cs);
(*But these unsafe steps at least solve a subgoal!*)
fun inst0_step_tac (CS{safe0_netpair,safep_netpair,...}) =
assume_tac APPEND'
contr_tac APPEND'
biresolve_from_nets_tac safe0_netpair;
(*These are much worse since they could generate more and more subgoals*)
fun instp_step_tac (CS{safep_netpair,...}) =
biresolve_from_nets_tac safep_netpair;
(*These steps could instantiate variables and are therefore unsafe.*)
fun inst_step_tac cs = inst0_step_tac cs APPEND' instp_step_tac cs;
fun haz_step_tac (CS{haz_netpair,...}) =
biresolve_from_nets_tac haz_netpair;
(*Single step for the prover. FAILS unless it makes progress. *)
fun step_tac cs i =
getwrapper cs
(FIRST [safe_tac cs, inst_step_tac cs i, haz_step_tac cs i]);
(*Using a "safe" rule to instantiate variables is unsafe. This tactic
allows backtracking from "safe" rules to "unsafe" rules here.*)
fun slow_step_tac cs i =
getwrapper cs
(safe_tac cs ORELSE (inst_step_tac cs i APPEND haz_step_tac cs i));
(*** The following tactics all fail unless they solve one goal ***)
(*Dumb but fast*)
fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1));
(*Slower but smarter than fast_tac*)
fun best_tac cs =
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1));
fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1));
fun slow_best_tac cs =
SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1));
(*** Complete tactic, loosely based upon LeanTaP. This tactic is the outcome
of much experimentation! Changing APPEND to ORELSE below would prove
easy theorems faster, but loses completeness -- and many of the harder
theorems such as 43. ***)
(*Non-deterministic! Could always expand the first unsafe connective.
That's hard to implement and did not perform better in experiments, due to
greater search depth required.*)
fun dup_step_tac (cs as (CS{dup_netpair,...})) =
biresolve_from_nets_tac dup_netpair;
(*Searching to depth m.*)
fun depth_tac cs m i = STATE(fn state =>
SELECT_GOAL
(REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE
(DEPTH_SOLVE (depth_tac cs m 1),
inst0_step_tac cs 1 APPEND
COND (K(m=0)) no_tac
((instp_step_tac cs 1 APPEND dup_step_tac cs 1)
THEN DEPTH_SOLVE (depth_tac cs (m-1) 1))))
i);
(*Iterative deepening tactical. Allows us to "deepen" any search tactic*)
fun DEEPEN tacf m i = STATE(fn state =>
if has_fewer_prems i state then no_tac
else (writeln ("Depth = " ^ string_of_int m);
tacf m i ORELSE DEEPEN tacf (m+2) i));
fun safe_depth_tac cs m =
SUBGOAL
(fn (prem,i) =>
let val deti =
(*No Vars in the goal? No need to backtrack between goals.*)
case term_vars prem of
[] => DETERM
| _::_ => I
in SELECT_GOAL (TRY (safe_tac cs) THEN
DEPTH_SOLVE (deti (depth_tac cs m 1))) i
end);
fun deepen_tac cs = DEEPEN (safe_depth_tac cs);
end;
end;