src/Provers/classical.ML
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(*  Title: 	Provers/classical
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    ID:         $Id$
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    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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Theorem prover for classical reasoning, including predicate calculus, set
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theory, etc.
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Rules must be classified as intr, elim, safe, hazardous.
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A rule is unsafe unless it can be applied blindly without harmful results.
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For a rule to be safe, its premises and conclusion should be logically
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equivalent.  There should be no variables in the premises that are not in
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the conclusion.
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*)
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infix 1 THEN_MAYBE;
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signature CLASSICAL_DATA =
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  sig
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  val mp	: thm    	(* [| P-->Q;  P |] ==> Q *)
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  val not_elim	: thm		(* [| ~P;  P |] ==> R *)
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  val classical	: thm		(* (~P ==> P) ==> P *)
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  val sizef 	: thm -> int	(* size function for BEST_FIRST *)
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  val hyp_subst_tacs: (int -> tactic) list
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  end;
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(*Higher precedence than := facilitates use of references*)
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infix 4 addSIs addSEs addSDs addIs addEs addDs 
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        setwrapper compwrapper addbefore addafter;
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signature CLASSICAL =
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  sig
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  type claset
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  type netpair
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  val empty_cs		: claset
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  val addDs 		: claset * thm list -> claset
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  val addEs 		: claset * thm list -> claset
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  val addIs 		: claset * thm list -> claset
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  val addSDs		: claset * thm list -> claset
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  val addSEs		: claset * thm list -> claset
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  val addSIs		: claset * thm list -> claset
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  val setwrapper 	: claset * (tactic->tactic) -> claset
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  val compwrapper 	: claset * (tactic->tactic) -> claset
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  val addbefore 	: claset * tactic -> claset
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  val addafter 		: claset * tactic -> claset
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  val print_cs		: claset -> unit
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  val rep_claset	: 
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      claset -> {safeIs: thm list, safeEs: thm list, 
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		 hazIs: thm list, hazEs: thm list,
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		 wrapper: tactic -> tactic,
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		 safe0_netpair: netpair, safep_netpair: netpair,
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		 haz_netpair: netpair, dup_netpair: netpair}
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  val getwrapper	: claset -> tactic -> tactic
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  val THEN_MAYBE	: tactic * tactic -> tactic
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  val best_tac 		: claset -> int -> tactic
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  val contr_tac 	: int -> tactic
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  val depth_tac		: claset -> int -> int -> tactic
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  val deepen_tac	: claset -> int -> int -> tactic
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  val dup_elim		: thm -> thm
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  val dup_intr		: thm -> thm
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  val dup_step_tac	: claset -> int -> tactic
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  val eq_mp_tac		: int -> tactic
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  val fast_tac 		: claset -> int -> tactic
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  val haz_step_tac 	: claset -> int -> tactic
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  val joinrules 	: thm list * thm list -> (bool * thm) list
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  val mp_tac		: int -> tactic
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  val safe_tac 		: claset -> tactic
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  val safe_step_tac 	: claset -> int -> tactic
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  val slow_step_tac 	: claset -> int -> tactic
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  val slow_best_tac 	: claset -> int -> tactic
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  val slow_tac 		: claset -> int -> tactic
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  val step_tac 		: claset -> int -> tactic
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  val swap		: thm                 (* ~P ==> (~Q ==> P) ==> Q *)
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  val swapify 		: thm list -> thm list
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  val swap_res_tac 	: thm list -> int -> tactic
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  val inst_step_tac 	: claset -> int -> tactic
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  val inst0_step_tac 	: claset -> int -> tactic
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  val instp_step_tac 	: claset -> int -> tactic
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  end;
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functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL = 
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struct
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local open Data in
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(** Useful tactics for classical reasoning **)
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val imp_elim = (*cannot use bind_thm within a structure!*)
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  store_thm ("imp_elim", make_elim mp);
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(*Solve goal that assumes both P and ~P. *)
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val contr_tac = eresolve_tac [not_elim]  THEN'  assume_tac;
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(*Finds P-->Q and P in the assumptions, replaces implication by Q.
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  Could do the same thing for P<->Q and P... *)
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fun mp_tac i = eresolve_tac [not_elim, imp_elim] i  THEN  assume_tac i;
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(*Like mp_tac but instantiates no variables*)
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fun eq_mp_tac i = ematch_tac [not_elim, imp_elim] i  THEN  eq_assume_tac i;
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val swap =
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  store_thm ("swap", rule_by_tactic (etac thin_rl 1) (not_elim RS classical));
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(*Creates rules to eliminate ~A, from rules to introduce A*)
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fun swapify intrs = intrs RLN (2, [swap]);
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(*Uses introduction rules in the normal way, or on negated assumptions,
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  trying rules in order. *)
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fun swap_res_tac rls = 
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    let fun addrl (rl,brls) = (false, rl) :: (true, rl RSN (2,swap)) :: brls
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    in  assume_tac 	ORELSE' 
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	contr_tac 	ORELSE' 
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        biresolve_tac (foldr addrl (rls,[]))
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    end;
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(*Duplication of hazardous rules, for complete provers*)
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fun dup_intr th = standard (th RS classical);
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fun dup_elim th = th RSN (2, revcut_rl) |> assumption 2 |> Sequence.hd |> 
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                  rule_by_tactic (TRYALL (etac revcut_rl));
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(*** Classical rule sets ***)
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type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;
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datatype claset =
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  CS of {safeIs		: thm list,		(*safe introduction rules*)
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	 safeEs		: thm list,		(*safe elimination rules*)
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	 hazIs		: thm list,		(*unsafe introduction rules*)
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	 hazEs		: thm list,		(*unsafe elimination rules*)
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	 wrapper	: tactic->tactic,	(*for transforming step_tac*)
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	 safe0_netpair	: netpair,		(*nets for trivial cases*)
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	 safep_netpair	: netpair,		(*nets for >0 subgoals*)
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	 haz_netpair  	: netpair,		(*nets for unsafe rules*)
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	 dup_netpair	: netpair};		(*nets for duplication*)
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(*Desired invariants are
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	safe0_netpair = build safe0_brls,
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	safep_netpair = build safep_brls,
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	haz_netpair = build (joinrules(hazIs, hazEs)),
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	dup_netpair = build (joinrules(map dup_intr hazIs, 
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				       map dup_elim hazEs))}
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where build = build_netpair(Net.empty,Net.empty), 
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      safe0_brls contains all brules that solve the subgoal, and
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      safep_brls contains all brules that generate 1 or more new subgoals.
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Nets must be built incrementally, to save space and time.
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*)
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val empty_cs = 
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  CS{safeIs	= [],
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     safeEs	= [],
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     hazIs	= [],
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     hazEs	= [],
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     wrapper 	= I,
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     safe0_netpair = (Net.empty,Net.empty),
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     safep_netpair = (Net.empty,Net.empty),
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     haz_netpair   = (Net.empty,Net.empty),
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     dup_netpair   = (Net.empty,Net.empty)};
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fun print_cs (CS{safeIs,safeEs,hazIs,hazEs,...}) =
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 (writeln"Introduction rules";  	prths hazIs;
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  writeln"Safe introduction rules";  	prths safeIs;
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  writeln"Elimination rules";  		prths hazEs;
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  writeln"Safe elimination rules";  	prths safeEs;
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  ());
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fun rep_claset (CS args) = args;
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fun getwrapper (CS{wrapper,...}) = wrapper;
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(** Adding (un)safe introduction or elimination rules.
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    In case of overlap, new rules are tried BEFORE old ones!!
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**)
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(*For use with biresolve_tac.  Combines intr rules with swap to handle negated
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  assumptions.  Pairs elim rules with true. *)
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fun joinrules (intrs,elims) =  
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    (map (pair true) (elims @ swapify intrs)  @
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     map (pair false) intrs);
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(*Priority: prefer rules with fewest subgoals, 
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  then rules added most recently (preferring the head of the list).*)
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fun tag_brls k [] = []
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  | tag_brls k (brl::brls) =
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      (1000000*subgoals_of_brl brl + k, brl) :: 
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      tag_brls (k+1) brls;
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fun insert_tagged_list kbrls np = foldr insert_tagged_brl (kbrls, np);
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(*Insert into netpair that already has nI intr rules and nE elim rules.
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  Count the intr rules double (to account for swapify).  Negate to give the
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  new insertions the lowest priority.*)
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fun insert (nI,nE) = insert_tagged_list o (tag_brls (~(2*nI+nE))) o joinrules;
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(** Safe rules **)
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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	safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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    addSIs  ths  =
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  let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
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          take_prefix (fn rl => nprems_of rl=0) ths
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      val nI = length safeIs + length ths
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      and nE = length safeEs
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  in CS{safeIs	= ths@safeIs,
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        safe0_netpair = insert (nI,nE) (safe0_rls, []) safe0_netpair,
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	safep_netpair = insert (nI,nE) (safep_rls, []) safep_netpair,
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	safeEs	= safeEs,
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	hazIs	= hazIs,
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	hazEs	= hazEs,
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	wrapper = wrapper,
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	haz_netpair = haz_netpair,
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	dup_netpair = dup_netpair}
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  end;
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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	safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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    addSEs  ths  =
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  let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
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          take_prefix (fn rl => nprems_of rl=1) ths
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      val nI = length safeIs
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      and nE = length safeEs + length ths
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  in 
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     CS{safeEs	= ths@safeEs,
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        safe0_netpair = insert (nI,nE) ([], safe0_rls) safe0_netpair,
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	safep_netpair = insert (nI,nE) ([], safep_rls) safep_netpair,
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	safeIs	= safeIs,
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	hazIs	= hazIs,
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	hazEs	= hazEs,
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	wrapper = wrapper,
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	haz_netpair = haz_netpair,
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	dup_netpair = dup_netpair}
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  end;
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fun cs addSDs ths = cs addSEs (map make_elim ths);
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(** Hazardous (unsafe) rules **)
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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	safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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    addIs  ths  =
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  let val nI = length hazIs + length ths
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      and nE = length hazEs
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  in 
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     CS{hazIs	= ths@hazIs,
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	haz_netpair = insert (nI,nE) (ths, []) haz_netpair,
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	dup_netpair = insert (nI,nE) (map dup_intr ths, []) dup_netpair,
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	safeIs 	= safeIs, 
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	safeEs	= safeEs,
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	hazEs	= hazEs,
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	wrapper 	= wrapper,
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	safe0_netpair = safe0_netpair,
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	safep_netpair = safep_netpair}
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  end;
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fun (CS{safeIs, safeEs, hazIs, hazEs, wrapper, 
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	safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) 
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    addEs  ths  =
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  let val nI = length hazIs 
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      and nE = length hazEs + length ths
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  in 
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     CS{hazEs	= ths@hazEs,
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	haz_netpair = insert (nI,nE) ([], ths) haz_netpair,
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	dup_netpair = insert (nI,nE) ([], map dup_elim ths) dup_netpair,
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	safeIs	= safeIs, 
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	safeEs	= safeEs,
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	hazIs	= hazIs,
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	wrapper	= wrapper,
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	safe0_netpair = safe0_netpair,
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	safep_netpair = safep_netpair}
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  end;
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fun cs addDs ths = cs addEs (map make_elim ths);
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(** Setting or modifying the wrapper tactical **)
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(*Set a new wrapper*)
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fun (CS{safeIs, safeEs, hazIs, hazEs, 
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	safe0_netpair, safep_netpair, haz_netpair, dup_netpair, ...}) 
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    setwrapper new_wrapper  =
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  CS{wrapper 	= new_wrapper,
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     safeIs	= safeIs,
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     safeEs	= safeEs,
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     hazIs	= hazIs,
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     hazEs	= hazEs,
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     safe0_netpair = safe0_netpair,
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     safep_netpair = safep_netpair,
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     haz_netpair = haz_netpair,
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     dup_netpair = dup_netpair};
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(*Compose a tactical with the existing wrapper*)
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fun cs compwrapper wrapper' = cs setwrapper (wrapper' o getwrapper cs);
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(*Execute tac1, but only execute tac2 if there are at least as many subgoals
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  as before.  This ensures that tac2 is only applied to an outcome of tac1.*)
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fun tac1 THEN_MAYBE tac2 = 
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  STATE (fn state =>
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	 tac1  THEN  
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	 COND (has_fewer_prems (nprems_of state)) all_tac tac2);
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(*Cause a tactic to be executed before/after the step tactic*)
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fun cs addbefore tac2 = cs compwrapper (fn tac1 => tac2 THEN_MAYBE tac1);
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fun cs addafter tac2  = cs compwrapper (fn tac1 => tac1 THEN_MAYBE tac2);
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(*** Simple tactics for theorem proving ***)
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(*Attack subgoals using safe inferences -- matching, not resolution*)
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fun safe_step_tac (CS{safe0_netpair,safep_netpair,...}) = 
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  FIRST' [eq_assume_tac,
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	  eq_mp_tac,
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	  bimatch_from_nets_tac safe0_netpair,
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	  FIRST' hyp_subst_tacs,
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	  bimatch_from_nets_tac safep_netpair] ;
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(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
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fun safe_tac cs = REPEAT_DETERM_FIRST (safe_step_tac cs);
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(*But these unsafe steps at least solve a subgoal!*)
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fun inst0_step_tac (CS{safe0_netpair,safep_netpair,...}) =
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  assume_tac 			  APPEND' 
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  contr_tac 			  APPEND' 
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  biresolve_from_nets_tac safe0_netpair;
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(*These are much worse since they could generate more and more subgoals*)
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fun instp_step_tac (CS{safep_netpair,...}) =
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  biresolve_from_nets_tac safep_netpair;
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(*These steps could instantiate variables and are therefore unsafe.*)
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fun inst_step_tac cs = inst0_step_tac cs APPEND' instp_step_tac cs;
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fun haz_step_tac (CS{haz_netpair,...}) = 
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  biresolve_from_nets_tac haz_netpair;
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(*Single step for the prover.  FAILS unless it makes progress. *)
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fun step_tac cs i = 
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  getwrapper cs 
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    (FIRST [safe_tac cs, inst_step_tac cs i, haz_step_tac cs i]);
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(*Using a "safe" rule to instantiate variables is unsafe.  This tactic
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  allows backtracking from "safe" rules to "unsafe" rules here.*)
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fun slow_step_tac cs i = 
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  getwrapper cs 
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    (safe_tac cs ORELSE (inst_step_tac cs i APPEND haz_step_tac cs i));
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(*** The following tactics all fail unless they solve one goal ***)
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(*Dumb but fast*)
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fun fast_tac cs = SELECT_GOAL (DEPTH_SOLVE (step_tac cs 1));
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(*Slower but smarter than fast_tac*)
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fun best_tac cs = 
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  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (step_tac cs 1));
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fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1));
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fun slow_best_tac cs = 
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  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1));
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(*** Complete tactic, loosely based upon LeanTaP.  This tactic is the outcome
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  of much experimentation!  Changing APPEND to ORELSE below would prove
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  easy theorems faster, but loses completeness -- and many of the harder
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  theorems such as 43. ***)
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(*Non-deterministic!  Could always expand the first unsafe connective.
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  That's hard to implement and did not perform better in experiments, due to
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  greater search depth required.*)
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fun dup_step_tac (cs as (CS{dup_netpair,...})) = 
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  biresolve_from_nets_tac dup_netpair;
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(*Searching to depth m.*)
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fun depth_tac cs m i = STATE(fn state => 
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  SELECT_GOAL 
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    (REPEAT_DETERM1 (safe_step_tac cs 1) THEN_ELSE
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     (DEPTH_SOLVE (depth_tac cs m 1),
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      inst0_step_tac cs 1  APPEND
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      COND (K(m=0)) no_tac
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        ((instp_step_tac cs 1 APPEND dup_step_tac cs 1)
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	 THEN DEPTH_SOLVE (depth_tac cs (m-1) 1))))
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  i);
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   394
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(*Iterative deepening tactical.  Allows us to "deepen" any search tactic*)
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fun DEEPEN tacf m i = STATE(fn state => 
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   if has_fewer_prems i state then no_tac
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   else (writeln ("Depth = " ^ string_of_int m);
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	 tacf m i  ORELSE  DEEPEN tacf (m+2) i));
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   400
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fun safe_depth_tac cs m = 
681
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  SUBGOAL 
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   403
    (fn (prem,i) =>
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   404
      let val deti =
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   405
	  (*No Vars in the goal?  No need to backtrack between goals.*)
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   406
	  case term_vars prem of
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   407
	      []	=> DETERM 
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   408
	    | _::_	=> I
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   409
      in  SELECT_GOAL (TRY (safe_tac cs) THEN 
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bdc066781063 deepen_tac: modified due to outcome of experiments. Its
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   410
		       DEPTH_SOLVE (deti (depth_tac cs m 1))) i
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   411
      end);
681
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parents: 469
diff changeset
   412
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   413
fun deepen_tac cs = DEEPEN (safe_depth_tac cs);
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0
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end; 
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end;