(* Title: FOLP/classical ID: $Id$ 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"Introduction rules"; Display.prths hazIs; writeln"Safe introduction rules"; Display.prths safeIs; writeln"Elimination rules"; Display.prths hazEs; writeln"Safe elimination rules"; Display.prths 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;