src/Provers/classical.ML

tuned;
prepare ext;

(*  Title: 	Provers/classical
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1992  University of Cambridge

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.
*)

(*Should be a type abbreviation in signature CLASSICAL*)
type netpair = (int*(bool*thm)) Net.net * (int*(bool*thm)) Net.net;

signature CLASSICAL_DATA =
  sig
  val mp	: thm    	(* [| P-->Q;  P |] ==> Q *)
  val not_elim	: thm		(* [| ~P;  P |] ==> R *)
  val classical	: thm		(* (~P ==> P) ==> P *)
  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 delrules
        setSWrapper compSWrapper setWrapper compWrapper 
	addSbefore addSaltern addbefore addaltern;


signature CLASSICAL =
  sig
  type claset
  val empty_cs		: claset
  val merge_cs		: claset * claset -> 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 delrules		: claset * thm list -> claset
  val setSWrapper 	: claset * ((int -> tactic) -> (int -> tactic)) ->claset
  val compSWrapper 	: claset * ((int -> tactic) -> (int -> tactic)) ->claset
  val setWrapper 	: claset * ((int -> tactic) -> (int -> tactic)) ->claset
  val compWrapper 	: claset * ((int -> tactic) -> (int -> tactic)) ->claset
  val addSbefore 	: claset * (int -> tactic) -> claset
  val addSaltern 	: claset * (int -> tactic) -> claset
  val addbefore 	: claset * (int -> tactic) -> claset
  val addaltern	 	: claset * (int -> tactic) -> claset

  val print_cs		: claset -> unit
  val rep_claset	: 
      claset -> {safeIs: thm list, safeEs: thm list, 
		 hazIs: thm list, hazEs: thm list,
		 uwrapper: (int -> tactic) -> (int -> tactic),
		 swrapper: (int -> tactic) -> (int -> tactic),
		 safe0_netpair: netpair, safep_netpair: netpair,
		 haz_netpair: netpair, dup_netpair: netpair}
  val getWrapper	: claset -> (int -> tactic) -> (int -> tactic)
  val getSWrapper	: claset -> (int -> tactic) -> (int -> tactic)

  val fast_tac 		: claset -> int -> tactic
  val slow_tac 		: claset -> int -> tactic
  val weight_ASTAR	: int ref
  val astar_tac		: claset -> int -> tactic
  val slow_astar_tac 	: claset -> int -> tactic
  val best_tac 		: claset -> int -> tactic
  val slow_best_tac 	: claset -> int -> tactic
  val depth_tac		: claset -> int -> int -> tactic
  val deepen_tac	: claset -> int -> int -> tactic

  val contr_tac 	: int -> tactic
  val dup_elim		: thm -> thm
  val dup_intr		: thm -> thm
  val dup_step_tac	: claset -> int -> tactic
  val eq_mp_tac		: int -> tactic
  val haz_step_tac 	: claset -> 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 clarify_tac 	: claset -> int -> tactic
  val clarify_step_tac 	: claset -> int -> tactic
  val step_tac 		: claset -> int -> tactic
  val slow_step_tac	: claset -> int -> tactic
  val swap		: thm                 (* ~P ==> (~Q ==> P) ==> Q *)
  val swapify 		: thm list -> thm list
  val swap_res_tac 	: thm list -> int -> tactic
  val inst_step_tac 	: claset -> int -> tactic
  val inst0_step_tac 	: claset -> int -> tactic
  val instp_step_tac 	: claset -> int -> tactic

  val claset : claset ref
  val AddDs 		: thm list -> unit
  val AddEs 		: thm list -> unit
  val AddIs 		: thm list -> unit
  val AddSDs		: thm list -> unit
  val AddSEs		: thm list -> unit
  val AddSIs		: thm list -> unit
  val Delrules		: thm list -> unit
  val Safe_tac         	: tactic
  val Safe_step_tac	: int -> tactic
  val Clarify_tac 	: int -> tactic
  val Clarify_step_tac 	: int -> tactic
  val Step_tac 		: int -> tactic
  val Fast_tac 		: int -> tactic
  val Best_tac 		: int -> tactic
  val Slow_tac 		: int -> tactic
  val Slow_best_tac     : int -> tactic
  val Deepen_tac	: int -> int -> tactic

  end;


functor ClassicalFun(Data: CLASSICAL_DATA): CLASSICAL = 
struct

local open Data in

(*** Useful tactics for classical reasoning ***)

val imp_elim = (*cannot use bind_thm within a structure!*)
  store_thm ("imp_elim", make_elim mp);

(*Solve goal that assumes both P and ~P. *)
val contr_tac = eresolve_tac [not_elim]  THEN'  assume_tac;

(*Finds P-->Q and P in the assumptions, replaces implication by Q.
  Could do the same thing for P<->Q and P... *)
fun mp_tac i = eresolve_tac [not_elim, imp_elim] i  THEN  assume_tac i;

(*Like mp_tac but instantiates no variables*)
fun eq_mp_tac i = ematch_tac [not_elim, imp_elim] i  THEN  eq_assume_tac i;

val swap =
  store_thm ("swap", rule_by_tactic (etac thin_rl 1) (not_elim RS classical));

(*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 addrl (rl,brls) = (false, rl) :: (true, rl RSN (2,swap)) :: brls
    in  assume_tac 	ORELSE' 
	contr_tac 	ORELSE' 
        biresolve_tac (foldr addrl (rls,[]))
    end;

(*Duplication of hazardous rules, for complete provers*)
fun dup_intr th = zero_var_indexes (th RS classical);

fun dup_elim th = th RSN (2, revcut_rl) |> assumption 2 |> Sequence.hd |> 
                  rule_by_tactic (TRYALL (etac revcut_rl));


(**** Classical rule sets ****)

datatype claset =
  CS of {safeIs		: thm list,		(*safe introduction rules*)
	 safeEs		: thm list,		(*safe elimination rules*)
	 hazIs		: thm list,		(*unsafe introduction rules*)
	 hazEs		: thm list,		(*unsafe elimination rules*)
	 uwrapper	: (int -> tactic) ->
			  (int -> tactic),	(*for transforming step_tac*)
	 swrapper	: (int -> tactic) ->
			  (int -> tactic),	(*for transform. safe_step_tac*)
	 safe0_netpair	: netpair,		(*nets for trivial cases*)
	 safep_netpair	: netpair,		(*nets for >0 subgoals*)
	 haz_netpair  	: netpair,		(*nets for unsafe rules*)
	 dup_netpair	: netpair};		(*nets for duplication*)

(*Desired invariants are
	safe0_netpair = build safe0_brls,
	safep_netpair = build safep_brls,
	haz_netpair = build (joinrules(hazIs, hazEs)),
	dup_netpair = build (joinrules(map dup_intr hazIs, 
				       map dup_elim hazEs))}

where build = build_netpair(Net.empty,Net.empty), 
      safe0_brls contains all brules that solve the subgoal, and
      safep_brls contains all brules that generate 1 or more new subgoals.
The theorem lists are largely comments, though they are used in merge_cs.
Nets must be built incrementally, to save space and time.
*)

val empty_cs = 
  CS{safeIs	= [],
     safeEs	= [],
     hazIs	= [],
     hazEs	= [],
     uwrapper   = I,
     swrapper   = I,
     safe0_netpair = (Net.empty,Net.empty),
     safep_netpair = (Net.empty,Net.empty),
     haz_netpair   = (Net.empty,Net.empty),
     dup_netpair   = (Net.empty,Net.empty)};

fun print_cs (CS {safeIs, safeEs, hazIs, hazEs, ...}) =
  let val pretty_thms = map Display.pretty_thm in
    Pretty.writeln (Pretty.big_list "introduction rules:" (pretty_thms hazIs));
    Pretty.writeln (Pretty.big_list "safe introduction rules:" (pretty_thms safeIs));
    Pretty.writeln (Pretty.big_list "elimination rules:" (pretty_thms hazEs));
    Pretty.writeln (Pretty.big_list "safe elimination rules:" (pretty_thms safeEs))
  end;

fun rep_claset (CS args) = args;

fun getWrapper  (CS{uwrapper,...}) = uwrapper;

fun getSWrapper (CS{swrapper,...}) = swrapper;


(*** Adding (un)safe introduction or elimination rules.

    In case of overlap, new rules are tried BEFORE old ones!!
***)

(*For use with biresolve_tac.  Combines intr rules with swap to handle negated
  assumptions.  Pairs elim rules with true. *)
fun joinrules (intrs,elims) =  
    (map (pair true) (elims @ swapify intrs)  @
     map (pair false) intrs);

(*Priority: prefer rules with fewest subgoals, 
  then rules added most recently (preferring the head of the list).*)
fun tag_brls k [] = []
  | tag_brls k (brl::brls) =
      (1000000*subgoals_of_brl brl + k, brl) :: 
      tag_brls (k+1) brls;

fun insert_tagged_list kbrls netpr = foldr insert_tagged_brl (kbrls, netpr);

(*Insert into netpair that already has nI intr rules and nE elim rules.
  Count the intr rules double (to account for swapify).  Negate to give the
  new insertions the lowest priority.*)
fun insert (nI,nE) = insert_tagged_list o (tag_brls (~(2*nI+nE))) o joinrules;

fun delete_tagged_list brls netpr = foldr delete_tagged_brl (brls, netpr);

val delete = delete_tagged_list o joinrules;

val mem_thm = gen_mem eq_thm
and rem_thm = gen_rem eq_thm;

(*Warn if the rule is already present ELSEWHERE in the claset.  The addition
  is still allowed.*)
fun warn_dup th (CS{safeIs, safeEs, hazIs, hazEs, ...}) = 
       if mem_thm (th, safeIs) then 
	 warning ("rule already in claset as Safe Intr\n" ^ string_of_thm th)
  else if mem_thm (th, safeEs) then
         warning ("rule already in claset as Safe Elim\n" ^ string_of_thm th)
  else if mem_thm (th, hazIs) then 
         warning ("rule already in claset as unsafe Intr\n" ^ string_of_thm th)
  else if mem_thm (th, hazEs) then 
         warning ("rule already in claset as unsafe Elim\n" ^ string_of_thm th)
  else ();

(*** Safe rules ***)

fun addSI (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
	      safe0_netpair, safep_netpair, haz_netpair, dup_netpair},
	   th)  =
  if mem_thm (th, safeIs) then 
	 (warning ("ignoring duplicate Safe Intr\n" ^ string_of_thm th);
	  cs)
  else
  let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
          partition (fn rl => nprems_of rl=0) [th]
      val nI = length safeIs + 1
      and nE = length safeEs
  in warn_dup th cs;
     CS{safeIs	= th::safeIs,
        safe0_netpair = insert (nI,nE) (safe0_rls, []) safe0_netpair,
	safep_netpair = insert (nI,nE) (safep_rls, []) safep_netpair,
	safeEs	= safeEs,
	hazIs	= hazIs,
	hazEs	= hazEs,
	uwrapper     = uwrapper,
	swrapper     = swrapper,
	haz_netpair  = haz_netpair,
	dup_netpair  = dup_netpair}
  end;

fun addSE (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
		    safe0_netpair, safep_netpair, haz_netpair, dup_netpair},
	   th)  =
  if mem_thm (th, safeEs) then 
	 (warning ("ignoring duplicate Safe Elim\n" ^ string_of_thm th);
	  cs)
  else
  let val (safe0_rls, safep_rls) = (*0 subgoals vs 1 or more*)
          partition (fn rl => nprems_of rl=1) [th]
      val nI = length safeIs
      and nE = length safeEs + 1
  in warn_dup th cs;
     CS{safeEs	= th::safeEs,
        safe0_netpair = insert (nI,nE) ([], safe0_rls) safe0_netpair,
	safep_netpair = insert (nI,nE) ([], safep_rls) safep_netpair,
	safeIs	= safeIs,
	hazIs	= hazIs,
	hazEs	= hazEs,
	uwrapper     = uwrapper,
	swrapper     = swrapper,
	haz_netpair  = haz_netpair,
	dup_netpair  = dup_netpair}
  end;

fun rev_foldl f (e, l) = foldl f (e, rev l);

val op addSIs = rev_foldl addSI;
val op addSEs = rev_foldl addSE;

fun cs addSDs ths = cs addSEs (map make_elim ths);


(*** Hazardous (unsafe) rules ***)

fun addI (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
		   safe0_netpair, safep_netpair, haz_netpair, dup_netpair},
	  th)=
  if mem_thm (th, hazIs) then 
	 (warning ("ignoring duplicate unsafe Intr\n" ^ string_of_thm th);
	  cs)
  else
  let val nI = length hazIs + 1
      and nE = length hazEs
  in warn_dup th cs;
     CS{hazIs	= th::hazIs,
	haz_netpair = insert (nI,nE) ([th], []) haz_netpair,
	dup_netpair = insert (nI,nE) (map dup_intr [th], []) dup_netpair,
	safeIs 	= safeIs, 
	safeEs	= safeEs,
	hazEs	= hazEs,
	uwrapper      = uwrapper,
	swrapper      = swrapper,
	safe0_netpair = safe0_netpair,
	safep_netpair = safep_netpair}
  end;

fun addE (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
		   safe0_netpair, safep_netpair, haz_netpair, dup_netpair},
	  th) =
  if mem_thm (th, hazEs) then 
	 (warning ("ignoring duplicate unsafe Elim\n" ^ string_of_thm th);
	  cs)
  else
  let val nI = length hazIs 
      and nE = length hazEs + 1
  in warn_dup th cs;
     CS{hazEs	= th::hazEs,
	haz_netpair = insert (nI,nE) ([], [th]) haz_netpair,
	dup_netpair = insert (nI,nE) ([], map dup_elim [th]) dup_netpair,
	safeIs	= safeIs, 
	safeEs	= safeEs,
	hazIs	= hazIs,
	uwrapper      = uwrapper,
	swrapper      = swrapper,
	safe0_netpair = safe0_netpair,
	safep_netpair = safep_netpair}
  end;

val op addIs = rev_foldl addI;
val op addEs = rev_foldl addE;

fun cs addDs ths = cs addEs (map make_elim ths);


(*** Deletion of rules 
     Working out what to delete, requires repeating much of the code used
	to insert.
     Separate functions delSI, etc., are not exported; instead delrules
        searches in all the lists and chooses the relevant delXX functions.
***)

fun delSI th 
          (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
		    safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) =
 if mem_thm (th, safeIs) then
   let val (safe0_rls, safep_rls) = partition (fn rl => nprems_of rl=0) [th]
   in CS{safe0_netpair = delete (safe0_rls, []) safe0_netpair,
	 safep_netpair = delete (safep_rls, []) safep_netpair,
	 safeIs	= rem_thm (safeIs,th),
	 safeEs	= safeEs,
	 hazIs	= hazIs,
	 hazEs	= hazEs,
	 uwrapper     = uwrapper,
	 swrapper     = swrapper,
	 haz_netpair  = haz_netpair,
	 dup_netpair  = dup_netpair}
   end
 else cs;

fun delSE th
          (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
	            safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) =
 if mem_thm (th, safeEs) then
   let val (safe0_rls, safep_rls) = partition (fn rl => nprems_of rl=1) [th]
   in CS{safe0_netpair = delete ([], safe0_rls) safe0_netpair,
	 safep_netpair = delete ([], safep_rls) safep_netpair,
	 safeIs	= safeIs,
	 safeEs	= rem_thm (safeEs,th),
	 hazIs	= hazIs,
	 hazEs	= hazEs,
	 uwrapper     = uwrapper,
	 swrapper     = swrapper,
	 haz_netpair  = haz_netpair,
	 dup_netpair  = dup_netpair}
   end
 else cs;


fun delI th
         (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
	           safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) =
 if mem_thm (th, hazIs) then
     CS{haz_netpair = delete ([th], []) haz_netpair,
	dup_netpair = delete ([dup_intr th], []) dup_netpair,
	safeIs 	= safeIs, 
	safeEs	= safeEs,
	hazIs	= rem_thm (hazIs,th),
	hazEs	= hazEs,
	uwrapper      = uwrapper,
	swrapper      = swrapper,
	safe0_netpair = safe0_netpair,
	safep_netpair = safep_netpair}
 else cs;

fun delE th
	 (cs as CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper, 
	           safe0_netpair, safep_netpair, haz_netpair, dup_netpair}) =
 if mem_thm (th, hazEs) then
     CS{haz_netpair = delete ([], [th]) haz_netpair,
	dup_netpair = delete ([], [dup_elim th]) dup_netpair,
	safeIs	= safeIs, 
	safeEs	= safeEs,
	hazIs	= hazIs,
	hazEs	= rem_thm (hazEs,th),
	uwrapper      = uwrapper,
	swrapper      = swrapper,
	safe0_netpair = safe0_netpair,
	safep_netpair = safep_netpair}
 else cs;

(*Delete ALL occurrences of "th" in the claset (perhaps from several lists)*)
fun delrule (cs as CS{safeIs, safeEs, hazIs, hazEs, ...}, th) =
       if mem_thm (th, safeIs) orelse mem_thm (th, safeEs) orelse
	  mem_thm (th, hazIs)  orelse mem_thm (th, hazEs) 
       then delSI th (delSE th (delI th (delE th cs)))
       else (warning ("rule not in claset\n" ^ (string_of_thm th)); 
	     cs);

val op delrules = foldl delrule;


(*** Setting or modifying the wrapper tacticals ***)

(*Set a new uwrapper*)
fun (CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper,
	safe0_netpair, safep_netpair, haz_netpair, dup_netpair, ...}) 
    setWrapper new_uwrapper  =
  CS{safeIs	= safeIs,
     safeEs	= safeEs,
     hazIs	= hazIs,
     hazEs	= hazEs,
     uwrapper 	= new_uwrapper,
     swrapper   = swrapper,
     safe0_netpair = safe0_netpair,
     safep_netpair = safep_netpair,
     haz_netpair = haz_netpair,
     dup_netpair = dup_netpair};

(*Set a new swrapper*)
fun (CS{safeIs, safeEs, hazIs, hazEs, uwrapper, swrapper,
	safe0_netpair, safep_netpair, haz_netpair, dup_netpair, ...}) 
    setSWrapper new_swrapper  =
  CS{safeIs	= safeIs,
     safeEs	= safeEs,
     hazIs	= hazIs,
     hazEs	= hazEs,
     uwrapper   = uwrapper,
     swrapper   = new_swrapper,
     safe0_netpair = safe0_netpair,
     safep_netpair = safep_netpair,
     haz_netpair = haz_netpair,
     dup_netpair = dup_netpair};

(*Compose a tactical with the existing uwrapper*)
fun cs compWrapper  uwrapper' = cs setWrapper  (uwrapper' o getWrapper cs);

(*Compose a tactical with the existing swrapper*)
fun cs compSWrapper swrapper' = cs setSWrapper (swrapper' o getSWrapper cs);

(*compose a safe tactic sequentially before/alternatively after safe_step_tac*)
fun cs addSbefore tac1 = cs compSWrapper (fn tac2 => tac1 THEN_MAYBE' tac2);
fun cs addSaltern tac2 = cs compSWrapper (fn tac1 => tac1 ORELSE'     tac2);

(*compose a tactic sequentially before/alternatively after the step tactic*)
fun cs addbefore  tac1 = cs compWrapper  (fn tac2 => tac1 THEN_MAYBE' tac2);
fun cs addaltern  tac2 = cs compWrapper  (fn tac1 => tac1 APPEND'     tac2);

(*Merge works by adding all new rules of the 2nd claset into the 1st claset.
  Merging the term nets may look more efficient, but the rather delicate
  treatment of priority might get muddled up.*)
fun merge_cs
    (cs as CS{safeIs, safeEs, hazIs, hazEs, ...},
     CS{safeIs=safeIs2, safeEs=safeEs2, hazIs=hazIs2, hazEs=hazEs2,...}) =
  let val safeIs' = gen_rems eq_thm (safeIs2,safeIs)
      val safeEs' = gen_rems eq_thm (safeEs2,safeEs)
      val  hazIs' = gen_rems eq_thm ( hazIs2, hazIs)
      val  hazEs' = gen_rems eq_thm ( hazEs2, hazEs)
  in cs addSIs safeIs'
        addSEs safeEs'
        addIs  hazIs'
        addEs  hazEs'
  end;


(**** Simple tactics for theorem proving ****)

(*Attack subgoals using safe inferences -- matching, not resolution*)
fun safe_step_tac (cs as CS{safe0_netpair,safep_netpair,...}) = 
  getSWrapper cs (FIRST' [
	eq_assume_tac,
	eq_mp_tac,
	bimatch_from_nets_tac safe0_netpair,
	FIRST' hyp_subst_tacs,
	bimatch_from_nets_tac safep_netpair]);

(*Repeatedly attack subgoals using safe inferences -- it's deterministic!*)
fun safe_tac cs = REPEAT_DETERM_FIRST 
	(fn i => COND (has_fewer_prems i) no_tac (safe_step_tac cs i));


(*** Clarify_tac: do safe steps without causing branching ***)

fun nsubgoalsP n (k,brl) = (subgoals_of_brl brl = n);

(*version of bimatch_from_nets_tac that only applies rules that
  create precisely n subgoals.*)
fun n_bimatch_from_nets_tac n = 
    biresolution_from_nets_tac (orderlist o filter (nsubgoalsP n)) true;

fun eq_contr_tac i = ematch_tac [not_elim] i  THEN  eq_assume_tac i;
val eq_assume_contr_tac = eq_assume_tac ORELSE' eq_contr_tac;

(*Two-way branching is allowed only if one of the branches immediately closes*)
fun bimatch2_tac netpair i =
    n_bimatch_from_nets_tac 2 netpair i THEN
    (eq_assume_contr_tac i ORELSE eq_assume_contr_tac (i+1));

(*Attack subgoals using safe inferences -- matching, not resolution*)
fun clarify_step_tac (cs as CS{safe0_netpair,safep_netpair,...}) = 
  getSWrapper cs (FIRST' [
	eq_assume_contr_tac,
	bimatch_from_nets_tac safe0_netpair,
	FIRST' hyp_subst_tacs,
	n_bimatch_from_nets_tac 1 safep_netpair,
        bimatch2_tac safep_netpair]);

fun clarify_tac cs = SELECT_GOAL (REPEAT_DETERM (clarify_step_tac cs 1));


(*** Unsafe steps instantiate variables or lose information ***)

(*But these unsafe steps at least solve a subgoal!*)
fun inst0_step_tac (CS{safe0_netpair,safep_netpair,...}) =
  assume_tac 			  APPEND' 
  contr_tac 			  APPEND' 
  biresolve_from_nets_tac safe0_netpair;

(*These are much worse since they could generate more and more subgoals*)
fun instp_step_tac (CS{safep_netpair,...}) =
  biresolve_from_nets_tac safep_netpair;

(*These steps could instantiate variables and are therefore unsafe.*)
fun inst_step_tac cs = inst0_step_tac cs APPEND' instp_step_tac cs;

fun haz_step_tac (CS{haz_netpair,...}) = 
  biresolve_from_nets_tac haz_netpair;

(*Single step for the prover.  FAILS unless it makes progress. *)
fun step_tac cs i = getWrapper cs 
	(K (safe_tac cs) ORELSE' (inst_step_tac cs ORELSE' haz_step_tac cs)) i;

(*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 i = getWrapper cs 
	(K (safe_tac cs) ORELSE' (inst_step_tac cs APPEND' haz_step_tac cs)) 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));

fun slow_tac cs = SELECT_GOAL (DEPTH_SOLVE (slow_step_tac cs 1));

fun slow_best_tac cs = 
  SELECT_GOAL (BEST_FIRST (has_fewer_prems 1, sizef) (slow_step_tac cs 1));


(***ASTAR with weight weight_ASTAR, by Norbert Voelker*) 
val weight_ASTAR = ref 5; 

fun astar_tac cs = 
  SELECT_GOAL ( ASTAR (has_fewer_prems 1
	      , fn level =>(fn thm =>size_of_thm thm + !weight_ASTAR *level)) 
	      (step_tac cs 1));

fun slow_astar_tac cs = 
  SELECT_GOAL ( ASTAR (has_fewer_prems 1
	      , fn level =>(fn thm =>size_of_thm thm + !weight_ASTAR *level)) 
	      (slow_step_tac cs 1));

(**** Complete tactic, loosely based upon LeanTaP.  This tactic is the outcome
  of much experimentation!  Changing APPEND to ORELSE below would prove
  easy theorems faster, but loses completeness -- and many of the harder
  theorems such as 43. ****)

(*Non-deterministic!  Could always expand the first unsafe connective.
  That's hard to implement and did not perform better in experiments, due to
  greater search depth required.*)
fun dup_step_tac (cs as (CS{dup_netpair,...})) = 
  biresolve_from_nets_tac dup_netpair;

(*Searching to depth m.*)
fun depth_tac cs m i state = 
  SELECT_GOAL 
   (getWrapper cs
    (fn i => REPEAT_DETERM1 (COND (has_fewer_prems i) no_tac
			     (safe_step_tac cs i)) THEN_ELSE
     (DEPTH_SOLVE (depth_tac cs m i),
      inst0_step_tac cs i  APPEND
      COND (K(m=0)) no_tac
        ((instp_step_tac cs i APPEND dup_step_tac cs i)
	 THEN DEPTH_SOLVE (depth_tac cs (m-1) i)))) 1)
  i state;

(*Search, with depth bound m.  
  This is the "entry point", which does safe inferences first.*)
fun safe_depth_tac cs m = 
  SUBGOAL 
    (fn (prem,i) =>
      let val deti =
	  (*No Vars in the goal?  No need to backtrack between goals.*)
	  case term_vars prem of
	      []	=> DETERM 
	    | _::_	=> I
      in  SELECT_GOAL (TRY (safe_tac cs) THEN 
		       DEPTH_SOLVE (deti (depth_tac cs m 1))) i
      end);

fun deepen_tac cs = DEEPEN (2,10) (safe_depth_tac cs);

val claset = ref empty_cs;

fun AddDs ts = (claset := !claset addDs ts);

fun AddEs ts = (claset := !claset addEs ts);

fun AddIs ts = (claset := !claset addIs ts);

fun AddSDs ts = (claset := !claset addSDs ts);

fun AddSEs ts = (claset := !claset addSEs ts);

fun AddSIs ts = (claset := !claset addSIs ts);

fun Delrules ts = (claset := !claset delrules ts);

(** The abstraction over the proof state delays the dereferencing **)

fun Safe_tac st		= safe_tac (!claset) st;

fun Safe_step_tac i st	= safe_step_tac (!claset) i st; 

fun Clarify_step_tac i st	= clarify_step_tac (!claset) i st;

fun Clarify_tac i st	= clarify_tac (!claset) i st;

fun Step_tac i st	= step_tac (!claset) i st; 

fun Fast_tac i st	= fast_tac (!claset) i st; 

fun Best_tac i st	= best_tac (!claset) i st; 

fun Slow_tac i st	= slow_tac (!claset) i st; 

fun Slow_best_tac i st	= slow_best_tac (!claset) i st; 

fun Deepen_tac m	= deepen_tac (!claset) m; 

end; 
end;