# File classical.ML

    (*  Title:      FOLP/classical
        ID:         $Id: classical.ML,v 1.7 2005/09/20 06:21:50 haftmann Exp $
        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";  prths hazIs;
      writeln"Safe introduction rules";  prths safeIs;
      writeln"Elimination rules";  prths hazEs;
      writeln"Safe elimination rules";  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;