--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/LK/lk.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,237 @@
+(* Title: LK/lk.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1992 University of Cambridge
+
+Tactics and lemmas for lk.thy (thanks also to Philippe de Groote)
+*)
+
+open LK;
+
+(*Higher precedence than := facilitates use of references*)
+infix 4 add_safes add_unsafes;
+
+signature LK_RESOLVE =
+ sig
+ datatype pack = Pack of thm list * thm list
+ val add_safes: pack * thm list -> pack
+ val add_unsafes: pack * thm list -> pack
+ val allL_thin: thm
+ val best_tac: pack -> int -> tactic
+ val could_res: term * term -> bool
+ val could_resolve_seq: term * term -> bool
+ val cutL_tac: string -> int -> tactic
+ val cutR_tac: string -> int -> tactic
+ val conL: thm
+ val conR: thm
+ val empty_pack: pack
+ val exR_thin: thm
+ val fast_tac: pack -> int -> tactic
+ val filseq_resolve_tac: thm list -> int -> int -> tactic
+ val forms_of_seq: term -> term list
+ val has_prems: int -> thm -> bool
+ val iffL: thm
+ val iffR: thm
+ val less: thm * thm -> bool
+ val LK_dup_pack: pack
+ val LK_pack: pack
+ val pc_tac: pack -> int -> tactic
+ val prop_pack: pack
+ val repeat_goal_tac: pack -> int -> tactic
+ val reresolve_tac: thm list -> int -> tactic
+ val RESOLVE_THEN: thm list -> (int -> tactic) -> int -> tactic
+ val safe_goal_tac: pack -> int -> tactic
+ val step_tac: pack -> int -> tactic
+ val symL: thm
+ val TrueR: thm
+ end;
+
+
+structure LK_Resolve : LK_RESOLVE =
+struct
+
+(*Cut and thin, replacing the right-side formula*)
+fun cutR_tac (sP: string) i =
+ res_inst_tac [ ("P",sP) ] cut i THEN rtac thinR i;
+
+(*Cut and thin, replacing the left-side formula*)
+fun cutL_tac (sP: string) i =
+ res_inst_tac [ ("P",sP) ] cut i THEN rtac thinL (i+1);
+
+
+(** If-and-only-if rules **)
+val iffR = prove_goalw LK.thy [iff_def]
+ "[| $H,P |- $E,Q,$F; $H,Q |- $E,P,$F |] ==> $H |- $E, P <-> Q, $F"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[conjR,impR]) 1)) ]);
+
+val iffL = prove_goalw LK.thy [iff_def]
+ "[| $H,$G |- $E,P,Q; $H,Q,P,$G |- $E |] ==> $H, P <-> Q, $G |- $E"
+ (fn prems=> [ (REPEAT (resolve_tac (prems@[conjL,impL,basic]) 1)) ]);
+
+val TrueR = prove_goalw LK.thy [True_def]
+ "$H |- $E, True, $F"
+ (fn _=> [ rtac impR 1, rtac basic 1 ]);
+
+
+(** Weakened quantifier rules. Incomplete, they let the search terminate.**)
+
+val allL_thin = prove_goal LK.thy
+ "$H, P(x), $G |- $E ==> $H, ALL x.P(x), $G |- $E"
+ (fn prems=> [ (rtac allL 1), (rtac thinL 1), (resolve_tac prems 1) ]);
+
+val exR_thin = prove_goal LK.thy
+ "$H |- $E, P(x), $F ==> $H |- $E, EX x.P(x), $F"
+ (fn prems=> [ (rtac exR 1), (rtac thinR 1), (resolve_tac prems 1) ]);
+
+(* Symmetry of equality in hypotheses *)
+val symL = prove_goal LK.thy
+ "$H, $G, B = A |- $E ==> $H, A = B, $G |- $E"
+ (fn prems=>
+ [ (rtac cut 1),
+ (rtac thinL 2),
+ (resolve_tac prems 2),
+ (resolve_tac [basic RS sym] 1) ]);
+
+
+(**** Theorem Packs ****)
+
+datatype pack = Pack of thm list * thm list;
+
+(*A theorem pack has the form (safe rules, unsafe rules)
+ An unsafe rule is incomplete or introduces variables in subgoals,
+ and is tried only when the safe rules are not applicable. *)
+
+fun less (rl1,rl2) = (nprems_of rl1) < (nprems_of rl2);
+
+val empty_pack = Pack([],[]);
+
+fun (Pack(safes,unsafes)) add_safes ths =
+ Pack(sort less (ths@safes), unsafes);
+
+fun (Pack(safes,unsafes)) add_unsafes ths =
+ Pack(safes, sort less (ths@unsafes));
+
+(*The rules of LK*)
+val prop_pack = empty_pack add_safes
+ [basic, refl, conjL, conjR, disjL, disjR, impL, impR,
+ notL, notR, iffL, iffR];
+
+val LK_pack = prop_pack add_safes [allR, exL]
+ add_unsafes [allL_thin, exR_thin];
+
+val LK_dup_pack = prop_pack add_safes [allR, exL]
+ add_unsafes [allL, exR];
+
+
+
+(*Returns the list of all formulas in the sequent*)
+fun forms_of_seq (Const("Seqof",_) $ P $ u) = P :: forms_of_seq u
+ | forms_of_seq (H $ u) = forms_of_seq u
+ | forms_of_seq _ = [];
+
+(*Tests whether two sequences (left or right sides) could be resolved.
+ seqp is a premise (subgoal), seqc is a conclusion of an object-rule.
+ Assumes each formula in seqc is surrounded by sequence variables
+ -- checks that each concl formula looks like some subgoal formula.
+ It SHOULD check order as well, using recursion rather than forall/exists*)
+fun could_res (seqp,seqc) =
+ forall (fn Qc => exists (fn Qp => could_unify (Qp,Qc))
+ (forms_of_seq seqp))
+ (forms_of_seq seqc);
+
+(*Tests whether two sequents G|-H could be resolved, comparing each side.*)
+fun could_resolve_seq (prem,conc) =
+ case (prem,conc) of
+ (_ $ Abs(_,_,leftp) $ Abs(_,_,rightp),
+ _ $ Abs(_,_,leftc) $ Abs(_,_,rightc)) =>
+ could_res (leftp,leftc) andalso could_res (rightp,rightc)
+ | _ => false;
+
+
+(*Like filt_resolve_tac, using could_resolve_seq
+ Much faster than resolve_tac when there are many rules.
+ Resolve subgoal i using the rules, unless more than maxr are compatible. *)
+fun filseq_resolve_tac rules maxr = SUBGOAL(fn (prem,i) =>
+ let val rls = filter_thms could_resolve_seq (maxr+1, prem, rules)
+ in if length rls > maxr then no_tac else resolve_tac rls i
+ end);
+
+
+(*Predicate: does the rule have n premises? *)
+fun has_prems n rule = (nprems_of rule = n);
+
+(*Continuation-style tactical for resolution.
+ The list of rules is partitioned into 0, 1, 2 premises.
+ The resulting tactic, gtac, tries to resolve with rules.
+ If successful, it recursively applies nextac to the new subgoals only.
+ Else fails. (Treatment of goals due to Ph. de Groote)
+ Bind (RESOLVE_THEN rules) to a variable: it preprocesses the rules. *)
+
+(*Takes rule lists separated in to 0, 1, 2, >2 premises.
+ The abstraction over state prevents needless divergence in recursion.
+ The 9999 should be a parameter, to delay treatment of flexible goals. *)
+fun RESOLVE_THEN rules =
+ let val [rls0,rls1,rls2] = partition_list has_prems 0 2 rules;
+ fun tac nextac i = STATE (fn state =>
+ filseq_resolve_tac rls0 9999 i
+ ORELSE
+ (DETERM(filseq_resolve_tac rls1 9999 i) THEN TRY(nextac i))
+ ORELSE
+ (DETERM(filseq_resolve_tac rls2 9999 i) THEN TRY(nextac(i+1))
+ THEN TRY(nextac i)) )
+ in tac end;
+
+
+(*repeated resolution applied to the designated goal*)
+fun reresolve_tac rules =
+ let val restac = RESOLVE_THEN rules; (*preprocessing done now*)
+ fun gtac i = restac gtac i
+ in gtac end;
+
+(*tries the safe rules repeatedly before the unsafe rules. *)
+fun repeat_goal_tac (Pack(safes,unsafes)) =
+ let val restac = RESOLVE_THEN safes
+ and lastrestac = RESOLVE_THEN unsafes;
+ fun gtac i = restac gtac i ORELSE lastrestac gtac i
+ in gtac end;
+
+
+(*Tries safe rules only*)
+fun safe_goal_tac (Pack(safes,unsafes)) = reresolve_tac safes;
+
+(*Tries a safe rule or else a unsafe rule. Single-step for tracing. *)
+fun step_tac (thm_pack as Pack(safes,unsafes)) =
+ safe_goal_tac thm_pack ORELSE'
+ filseq_resolve_tac unsafes 9999;
+
+
+(* Tactic for reducing a goal, using Predicate Calculus rules.
+ A decision procedure for Propositional Calculus, it is incomplete
+ for Predicate-Calculus because of allL_thin and exR_thin.
+ Fails if it can do nothing. *)
+fun pc_tac thm_pack = SELECT_GOAL (DEPTH_SOLVE (repeat_goal_tac thm_pack 1));
+
+(*The following two tactics are analogous to those provided by
+ Provers/classical. In fact, pc_tac is usually FASTER than fast_tac!*)
+fun fast_tac thm_pack =
+ SELECT_GOAL (DEPTH_SOLVE (step_tac thm_pack 1));
+
+fun best_tac thm_pack =
+ SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, size_of_thm)
+ (step_tac thm_pack 1));
+
+(** Contraction. Useful since some rules are not complete. **)
+
+val conR = prove_goal LK.thy
+ "$H |- $E, P, $F, P ==> $H |- $E, P, $F"
+ (fn prems=>
+ [ (rtac cut 1), (REPEAT (resolve_tac (prems@[basic]) 1)) ]);
+
+val conL = prove_goal LK.thy
+ "$H, P, $G, P |- $E ==> $H, P, $G |- $E"
+ (fn prems=>
+ [ (rtac cut 1), (REPEAT (resolve_tac (prems@[basic]) 1)) ]);
+
+end;
+
+open LK_Resolve;