src/LK/LK.ML

+(*  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;