(* Title: FOLP/classical
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Like Provers/classical but modified because match_tac is unsuitable for
proof objects.
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.
*)
signature CLASSICAL_DATA =
sig
val mp: thm (* [| P-->Q; P |] ==> Q *)
val not_elim: thm (* [| ~P; P |] ==> R *)
val swap: thm (* ~P ==> (~Q ==> P) ==> Q *)
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;
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 print_cs: claset -> unit
val rep_cs: claset ->
{safeIs: thm list, safeEs: thm list, hazIs: thm list, hazEs: thm list,
safe0_brls:(bool*thm)list, safep_brls: (bool*thm)list,
haz_brls: (bool*thm)list}
val best_tac : claset -> int -> tactic
val contr_tac : int -> tactic
val fast_tac : claset -> int -> tactic
val inst_step_tac : 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 step_tac : claset -> int -> tactic
val swapify : thm list -> thm list
val swap_res_tac : thm list -> int -> tactic
val uniq_mp_tac: 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 = etac not_elim THEN' assume_tac;
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
fun mp_tac i = eresolve_tac ([not_elim,imp_elim]) i THEN assume_tac i;
(*Like mp_tac but instantiates no variables*)
fun uniq_mp_tac i = ematch_tac ([not_elim,imp_elim]) i THEN uniq_assume_tac i;
(*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 tacf rl = rtac rl ORELSE' etac (rl RSN (2,swap))
in assume_tac ORELSE' contr_tac ORELSE' FIRST' (map tacf rls)
end;
(*** Classical rule sets ***)
datatype claset =
CS of {safeIs: thm list,
safeEs: thm list,
hazIs: thm list,
hazEs: thm list,
(*the following are computed from the above*)
safe0_brls: (bool*thm)list,
safep_brls: (bool*thm)list,
haz_brls: (bool*thm)list};
fun rep_cs (CS x) = x;
(*For use with biresolve_tac. Combines intrs with swap to catch negated
assumptions. Also pairs elims with true. *)
fun joinrules (intrs,elims) =
map (pair true) (elims @ swapify intrs) @ map (pair false) intrs;
(*Note that allE precedes exI in haz_brls*)
fun make_cs {safeIs,safeEs,hazIs,hazEs} =
let val (safe0_brls, safep_brls) = (*0 subgoals vs 1 or more*)
List.partition (curry (op =) 0 o subgoals_of_brl)
(sort (make_ord lessb) (joinrules(safeIs, safeEs)))
in CS{safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs,
safe0_brls=safe0_brls, safep_brls=safep_brls,
haz_brls = sort (make_ord lessb) (joinrules(hazIs, hazEs))}
end;
(*** Manipulation of clasets ***)
val empty_cs = make_cs{safeIs=[], safeEs=[], hazIs=[], hazEs=[]};
fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) =
writeln (cat_lines
(["Introduction rules"] @ map Display.string_of_thm_without_context hazIs @
["Safe introduction rules"] @ map Display.string_of_thm_without_context safeIs @
["Elimination rules"] @ map Display.string_of_thm_without_context hazEs @
["Safe elimination rules"] @ map Display.string_of_thm_without_context safeEs));
fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSIs ths =
make_cs {safeIs=ths@safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=hazEs};
fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addSEs ths =
make_cs {safeIs=safeIs, safeEs=ths@safeEs, hazIs=hazIs, hazEs=hazEs};
fun cs addSDs ths = cs addSEs (map make_elim ths);
fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addIs ths =
make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=ths@hazIs, hazEs=hazEs};
fun (CS{safeIs,safeEs,hazIs,hazEs,...}) addEs ths =
make_cs {safeIs=safeIs, safeEs=safeEs, hazIs=hazIs, hazEs=ths@hazEs};
fun cs addDs ths = cs addEs (map make_elim ths);
(*** Simple tactics for theorem proving ***)
(*Attack subgoals using safe inferences*)
fun safe_step_tac (CS{safe0_brls,safep_brls,...}) =
FIRST' [uniq_assume_tac,
uniq_mp_tac,
biresolve_tac safe0_brls,
FIRST' hyp_subst_tacs,
biresolve_tac safep_brls] ;
(*Repeatedly attack subgoals using safe inferences*)
fun safe_tac cs = DETERM (REPEAT_FIRST (safe_step_tac cs));
(*These steps could instantiate variables and are therefore unsafe.*)
val inst_step_tac = assume_tac APPEND' contr_tac;
(*Single step for the prover. FAILS unless it makes progress. *)
fun step_tac (cs as (CS{haz_brls,...})) i =
FIRST [safe_tac cs,
inst_step_tac i,
biresolve_tac haz_brls 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));
(*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 as (CS{haz_brls,...})) i =
safe_tac cs ORELSE (assume_tac i APPEND biresolve_tac haz_brls i);
end;
end;