(*  Title:      FOLP/classical.ML

    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: Proof.context -> 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 Classical(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 ctxt (CS{safeIs,safeEs,hazIs,hazEs,...}) =

  writeln (cat_lines

   (["Introduction rules"] @ map (Display.string_of_thm ctxt) hazIs @

    ["Safe introduction rules"] @ map (Display.string_of_thm ctxt) safeIs @

    ["Elimination rules"] @ map (Display.string_of_thm ctxt) hazEs @

    ["Safe elimination rules"] @ map (Display.string_of_thm ctxt) 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;