src/Pure/tctical.ML

add lemmas abs_hnorm_cancel, hnorm_of_hypreal

(*  Title:      Pure/tctical.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

Tacticals.
*)

infix 1 THEN THEN' THEN_ALL_NEW;
infix 0 ORELSE APPEND INTLEAVE ORELSE' APPEND' INTLEAVE';
infix 0 THEN_ELSE;


signature TACTICAL =
sig
  type tactic  (* = thm -> thm Seq.seq*)
  val all_tac           : tactic
  val ALLGOALS          : (int -> tactic) -> tactic
  val APPEND            : tactic * tactic -> tactic
  val APPEND'           : ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
  val CHANGED           : tactic -> tactic
  val CHANGED_PROP      : tactic -> tactic
  val CHANGED_GOAL      : (int -> tactic) -> int -> tactic
  val COND              : (thm -> bool) -> tactic -> tactic -> tactic
  val DETERM            : tactic -> tactic
  val EVERY             : tactic list -> tactic
  val EVERY'            : ('a -> tactic) list -> 'a -> tactic
  val EVERY1            : (int -> tactic) list -> tactic
  val FILTER            : (thm -> bool) -> tactic -> tactic
  val FIRST             : tactic list -> tactic
  val FIRST'            : ('a -> tactic) list -> 'a -> tactic
  val FIRST1            : (int -> tactic) list -> tactic
  val FIRSTGOAL         : (int -> tactic) -> tactic
  val INTLEAVE          : tactic * tactic -> tactic
  val INTLEAVE'         : ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
  val METAHYPS          : (thm list -> tactic) -> int -> tactic
  val metahyps_thms     : int -> thm -> thm list option
  val no_tac            : tactic
  val ORELSE            : tactic * tactic -> tactic
  val ORELSE'           : ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
  val pause_tac         : tactic
  val print_tac         : string -> tactic
  val PRIMITIVE         : (thm -> thm) -> tactic
  val PRIMSEQ           : (thm -> thm Seq.seq) -> tactic
  val RANGE             : (int -> tactic) list -> int -> tactic
  val REPEAT            : tactic -> tactic
  val REPEAT1           : tactic -> tactic
  val REPEAT_FIRST      : (int -> tactic) -> tactic
  val REPEAT_SOME       : (int -> tactic) -> tactic
  val REPEAT_DETERM_N   : int -> tactic -> tactic
  val REPEAT_DETERM     : tactic -> tactic
  val REPEAT_DETERM1    : tactic -> tactic
  val REPEAT_DETERM_FIRST: (int -> tactic) -> tactic
  val REPEAT_DETERM_SOME: (int -> tactic) -> tactic
  val DETERM_UNTIL      : (thm -> bool) -> tactic -> tactic
  val SINGLE            : tactic -> thm -> thm option
  val SOMEGOAL          : (int -> tactic) -> tactic
  val strip_context     : term -> (string * typ) list * term list * term
  val SUBGOAL           : ((term*int) -> tactic) -> int -> tactic
  val suppress_tracing  : bool ref
  val THEN              : tactic * tactic -> tactic
  val THEN'             : ('a -> tactic) * ('a -> tactic) -> 'a -> tactic
  val THEN_ALL_NEW      : (int -> tactic) * (int -> tactic) -> int -> tactic
  val REPEAT_ALL_NEW    : (int -> tactic) -> int -> tactic
  val THEN_ELSE         : tactic * (tactic*tactic) -> tactic
  val traced_tac        : (thm -> (thm * thm Seq.seq) option) -> tactic
  val tracify           : bool ref -> tactic -> tactic
  val trace_REPEAT      : bool ref
  val TRY               : tactic -> tactic
  val TRYALL            : (int -> tactic) -> tactic
end;


structure Tactical : TACTICAL =
struct

(**** Tactics ****)

(*A tactic maps a proof tree to a sequence of proof trees:
    if length of sequence = 0 then the tactic does not apply;
    if length > 1 then backtracking on the alternatives can occur.*)

type tactic = thm -> thm Seq.seq;


(*** LCF-style tacticals ***)

(*the tactical THEN performs one tactic followed by another*)
fun (tac1 THEN tac2) st = Seq.maps tac2 (tac1 st);


(*The tactical ORELSE uses the first tactic that returns a nonempty sequence.
  Like in LCF, ORELSE commits to either tac1 or tac2 immediately.
  Does not backtrack to tac2 if tac1 was initially chosen. *)
fun (tac1 ORELSE tac2) st =
    case Seq.pull(tac1 st) of
        NONE       => tac2 st
      | sequencecell => Seq.make(fn()=> sequencecell);


(*The tactical APPEND combines the results of two tactics.
  Like ORELSE, but allows backtracking on both tac1 and tac2.
  The tactic tac2 is not applied until needed.*)
fun (tac1 APPEND tac2) st =
  Seq.append (tac1 st) (Seq.make(fn()=> Seq.pull (tac2 st)));

(*Like APPEND, but interleaves results of tac1 and tac2.*)
fun (tac1 INTLEAVE tac2) st =
    Seq.interleave(tac1 st,
                        Seq.make(fn()=> Seq.pull (tac2 st)));

(*Conditional tactic.
        tac1 ORELSE tac2 = tac1 THEN_ELSE (all_tac, tac2)
        tac1 THEN tac2   = tac1 THEN_ELSE (tac2, no_tac)
*)
fun (tac THEN_ELSE (tac1, tac2)) st =
    case Seq.pull(tac st) of
        NONE    => tac2 st                                   (*failed; try tactic 2*)
      | seqcell => Seq.maps tac1 (Seq.make(fn()=> seqcell)); (*succeeded; use tactic 1*)


(*Versions for combining tactic-valued functions, as in
     SOMEGOAL (resolve_tac rls THEN' assume_tac) *)
fun (tac1 THEN' tac2) x = tac1 x THEN tac2 x;
fun (tac1 ORELSE' tac2) x = tac1 x ORELSE tac2 x;
fun (tac1 APPEND' tac2) x = tac1 x APPEND tac2 x;
fun (tac1 INTLEAVE' tac2) x = tac1 x INTLEAVE tac2 x;

(*passes all proofs through unchanged;  identity of THEN*)
fun all_tac st = Seq.single st;

(*passes no proofs through;  identity of ORELSE and APPEND*)
fun no_tac st  = Seq.empty;


(*Make a tactic deterministic by chopping the tail of the proof sequence*)
fun DETERM tac = Seq.DETERM tac;

(*Conditional tactical: testfun controls which tactic to use next.
  Beware: due to eager evaluation, both thentac and elsetac are evaluated.*)
fun COND testfun thenf elsef = (fn prf =>
    if testfun prf then  thenf prf   else  elsef prf);

(*Do the tactic or else do nothing*)
fun TRY tac = tac ORELSE all_tac;

(*** List-oriented tactics ***)

local
  (*This version of EVERY avoids backtracking over repeated states*)

  fun EVY (trail, []) st =
        Seq.make (fn()=> SOME(st,
                        Seq.make (fn()=> Seq.pull (evyBack trail))))
    | EVY (trail, tac::tacs) st =
          case Seq.pull(tac st) of
              NONE    => evyBack trail              (*failed: backtrack*)
            | SOME(st',q) => EVY ((st',q,tacs)::trail, tacs) st'
  and evyBack [] = Seq.empty (*no alternatives*)
    | evyBack ((st',q,tacs)::trail) =
          case Seq.pull q of
              NONE        => evyBack trail
            | SOME(st,q') => if Thm.eq_thm (st',st)
                             then evyBack ((st',q',tacs)::trail)
                             else EVY ((st,q',tacs)::trail, tacs) st
in

(* EVERY [tac1,...,tacn]   equals    tac1 THEN ... THEN tacn   *)
fun EVERY tacs = EVY ([], tacs);
end;


(* EVERY' [tac1,...,tacn] i  equals    tac1 i THEN ... THEN tacn i   *)
fun EVERY' tacs i = EVERY (map (fn f => f i) tacs);

(*Apply every tactic to 1*)
fun EVERY1 tacs = EVERY' tacs 1;

(* FIRST [tac1,...,tacn]   equals    tac1 ORELSE ... ORELSE tacn   *)
fun FIRST tacs = foldr (op ORELSE) no_tac tacs;

(* FIRST' [tac1,...,tacn] i  equals    tac1 i ORELSE ... ORELSE tacn i   *)
fun FIRST' tacs = foldr (op ORELSE') (K no_tac) tacs;

(*Apply first tactic to 1*)
fun FIRST1 tacs = FIRST' tacs 1;

(*Apply tactics on consecutive subgoals*)
fun RANGE [] _ = all_tac
  | RANGE (tac :: tacs) i = RANGE tacs (i + 1) THEN tac i;


(*** Tracing tactics ***)

(*Print the current proof state and pass it on.*)
fun print_tac msg =
    (fn st =>
     (tracing msg;
      tracing ((Pretty.string_of o Pretty.chunks o 
                 Display.pretty_goals (! Display.goals_limit)) st); 
      Seq.single st));

(*Pause until a line is typed -- if non-empty then fail. *)
fun pause_tac st =
  (tracing "** Press RETURN to continue:";
   if TextIO.inputLine TextIO.stdIn = "\n" then Seq.single st
   else (tracing "Goodbye";  Seq.empty));

exception TRACE_EXIT of thm
and TRACE_QUIT;

(*Tracing flags*)
val trace_REPEAT= ref false
and suppress_tracing = ref false;

(*Handle all tracing commands for current state and tactic *)
fun exec_trace_command flag (tac, st) =
   case TextIO.inputLine(TextIO.stdIn) of
       "\n" => tac st
     | "f\n" => Seq.empty
     | "o\n" => (flag:=false;  tac st)
     | "s\n" => (suppress_tracing:=true;  tac st)
     | "x\n" => (tracing "Exiting now";  raise (TRACE_EXIT st))
     | "quit\n" => raise TRACE_QUIT
     | _     => (tracing
"Type RETURN to continue or...\n\
\     f    - to fail here\n\
\     o    - to switch tracing off\n\
\     s    - to suppress tracing until next entry to a tactical\n\
\     x    - to exit at this point\n\
\     quit - to abort this tracing run\n\
\** Well? "     ;  exec_trace_command flag (tac, st));


(*Extract from a tactic, a thm->thm seq function that handles tracing*)
fun tracify flag tac st =
  if !flag andalso not (!suppress_tracing)
           then (Display.print_goals (! Display.goals_limit) st;
                 tracing "** Press RETURN to continue:";
                 exec_trace_command flag (tac,st))
  else tac st;

(*Create a tactic whose outcome is given by seqf, handling TRACE_EXIT*)
fun traced_tac seqf st =
    (suppress_tracing := false;
     Seq.make (fn()=> seqf st
                         handle TRACE_EXIT st' => SOME(st', Seq.empty)));


(*Deterministic DO..UNTIL: only retains the first outcome; tail recursive.
  Forces repitition until predicate on state is fulfilled.*)
fun DETERM_UNTIL p tac =
let val tac = tracify trace_REPEAT tac
    fun drep st = if p st then SOME (st, Seq.empty)
                          else (case Seq.pull(tac st) of
                                  NONE        => NONE
                                | SOME(st',_) => drep st')
in  traced_tac drep  end;

(*Deterministic REPEAT: only retains the first outcome;
  uses less space than REPEAT; tail recursive.
  If non-negative, n bounds the number of repetitions.*)
fun REPEAT_DETERM_N n tac =
  let val tac = tracify trace_REPEAT tac
      fun drep 0 st = SOME(st, Seq.empty)
        | drep n st =
           (case Seq.pull(tac st) of
                NONE       => SOME(st, Seq.empty)
              | SOME(st',_) => drep (n-1) st')
  in  traced_tac (drep n)  end;

(*Allows any number of repetitions*)
val REPEAT_DETERM = REPEAT_DETERM_N ~1;

(*General REPEAT: maintains a stack of alternatives; tail recursive*)
fun REPEAT tac =
  let val tac = tracify trace_REPEAT tac
      fun rep qs st =
        case Seq.pull(tac st) of
            NONE       => SOME(st, Seq.make(fn()=> repq qs))
          | SOME(st',q) => rep (q::qs) st'
      and repq [] = NONE
        | repq(q::qs) = case Seq.pull q of
            NONE       => repq qs
          | SOME(st,q) => rep (q::qs) st
  in  traced_tac (rep [])  end;

(*Repeat 1 or more times*)
fun REPEAT_DETERM1 tac = DETERM tac THEN REPEAT_DETERM tac;
fun REPEAT1 tac = tac THEN REPEAT tac;


(** Filtering tacticals **)

fun FILTER pred tac st = Seq.filter pred (tac st);

(*Accept only next states that change the theorem somehow*)
fun CHANGED tac st =
  let fun diff st' = not (Thm.eq_thm (st, st'));
  in Seq.filter diff (tac st) end;

(*Accept only next states that change the theorem's prop field
  (changes to signature, hyps, etc. don't count)*)
fun CHANGED_PROP tac st =
  let fun diff st' = not (Thm.eq_thm_prop (st, st'));
  in Seq.filter diff (tac st) end;


(*** Tacticals based on subgoal numbering ***)

(*For n subgoals, performs tac(n) THEN ... THEN tac(1)
  Essential to work backwards since tac(i) may add/delete subgoals at i. *)
fun ALLGOALS tac st =
  let fun doall 0 = all_tac
        | doall n = tac(n) THEN doall(n-1)
  in  doall(nprems_of st)st  end;

(*For n subgoals, performs tac(n) ORELSE ... ORELSE tac(1)  *)
fun SOMEGOAL tac st =
  let fun find 0 = no_tac
        | find n = tac(n) ORELSE find(n-1)
  in  find(nprems_of st)st  end;

(*For n subgoals, performs tac(1) ORELSE ... ORELSE tac(n).
  More appropriate than SOMEGOAL in some cases.*)
fun FIRSTGOAL tac st =
  let fun find (i,n) = if i>n then no_tac else  tac(i) ORELSE find (i+1,n)
  in  find(1, nprems_of st)st  end;

(*Repeatedly solve some using tac. *)
fun REPEAT_SOME tac = REPEAT1 (SOMEGOAL (REPEAT1 o tac));
fun REPEAT_DETERM_SOME tac = REPEAT_DETERM1 (SOMEGOAL (REPEAT_DETERM1 o tac));

(*Repeatedly solve the first possible subgoal using tac. *)
fun REPEAT_FIRST tac = REPEAT1 (FIRSTGOAL (REPEAT1 o tac));
fun REPEAT_DETERM_FIRST tac = REPEAT_DETERM1 (FIRSTGOAL (REPEAT_DETERM1 o tac));

(*For n subgoals, tries to apply tac to n,...1  *)
fun TRYALL tac = ALLGOALS (TRY o tac);


(*Make a tactic for subgoal i, if there is one.  *)
fun SUBGOAL goalfun i st =
  (case try Logic.nth_prem (i, Thm.prop_of st) of
    SOME goal => goalfun (goal, i) st
  | NONE => Seq.empty);

(*Returns all states that have changed in subgoal i, counted from the LAST
  subgoal.  For stac, for example.*)
fun CHANGED_GOAL tac i st =
    let val np = nprems_of st
        val d = np-i                 (*distance from END*)
        val t = List.nth(prems_of st, i-1)
        fun diff st' =
            nprems_of st' - d <= 0   (*the subgoal no longer exists*)
            orelse
             not (Pattern.aeconv (t,
                                  List.nth(prems_of st',
                                           nprems_of st' - d - 1)))
    in  Seq.filter diff (tac i st)  end
    handle Subscript => Seq.empty  (*no subgoal i*);

fun (tac1 THEN_ALL_NEW tac2) i st =
  st |> (tac1 i THEN (fn st' => Seq.INTERVAL tac2 i (i + nprems_of st' - nprems_of st) st'));

(*repeatedly dig into any emerging subgoals*)
fun REPEAT_ALL_NEW tac =
  tac THEN_ALL_NEW (TRY o (fn i => REPEAT_ALL_NEW tac i));


(*Strips assumptions in goal yielding  ( [x1,...,xm], [H1,...,Hn], B )
    H1,...,Hn are the hypotheses;  x1...xm are variants of the parameters.
  Main difference from strip_assums concerns parameters:
    it replaces the bound variables by free variables.  *)
fun strip_context_aux (params, Hs, Const("==>", _) $ H $ B) =
        strip_context_aux (params, H::Hs, B)
  | strip_context_aux (params, Hs, Const("all",_)$Abs(a,T,t)) =
        let val (b,u) = Syntax.variant_abs(a,T,t)
        in  strip_context_aux ((b,T)::params, Hs, u)  end
  | strip_context_aux (params, Hs, B) = (rev params, rev Hs, B);

fun strip_context A = strip_context_aux ([],[],A);


(**** METAHYPS -- tactical for using hypotheses as meta-level assumptions
       METAHYPS (fn prems => tac prems) i

converts subgoal i, of the form !!x1...xm. [| A1;...;An] ==> A into a new
proof state A==>A, supplying A1,...,An as meta-level assumptions (in
"prems").  The parameters x1,...,xm become free variables.  If the
resulting proof state is [| B1;...;Bk] ==> C (possibly assuming A1,...,An)
then it is lifted back into the original context, yielding k subgoals.

Replaces unknowns in the context by Frees having the prefix METAHYP_
New unknowns in [| B1;...;Bk] ==> C are lifted over x1,...,xm.
DOES NOT HANDLE TYPE UNKNOWNS.
****)

local

  (*Left-to-right replacements: ctpairs = [...,(vi,ti),...].
    Instantiates distinct free variables by terms of same type.*)
  fun free_instantiate ctpairs =
      forall_elim_list (map snd ctpairs) o forall_intr_list (map fst ctpairs);

  fun free_of s ((a,i), T) =
        Free(s ^ (case i of 0 => a | _ => a ^ "_" ^ string_of_int i),
             T)

  fun mk_inst (var as Var(v,T))  = (var,  free_of "METAHYP1_" (v,T))
in

(*Common code for METAHYPS and metahyps_thms*)
fun metahyps_split_prem prem =
  let (*find all vars in the hyps -- should find tvars also!*)
      val hyps_vars = foldr add_term_vars [] (Logic.strip_assums_hyp prem)
      val insts = map mk_inst hyps_vars
      (*replace the hyps_vars by Frees*)
      val prem' = subst_atomic insts prem
      val (params,hyps,concl) = strip_context prem'
  in (insts,params,hyps,concl)  end;

fun metahyps_aux_tac tacf (prem,gno) state =
  let val (insts,params,hyps,concl) = metahyps_split_prem prem 
      val {thy = sign,maxidx,...} = rep_thm state
      val cterm = cterm_of sign
      val chyps = map cterm hyps
      val hypths = map assume chyps
      val subprems = map (forall_elim_vars 0) hypths
      val fparams = map Free params
      val cparams = map cterm fparams
      fun swap_ctpair (t,u) = (cterm u, cterm t)
      (*Subgoal variables: make Free; lift type over params*)
      fun mk_subgoal_inst concl_vars (var as Var(v,T)) =
          if member (op =) concl_vars var
          then (var, true, free_of "METAHYP2_" (v,T))
          else (var, false,
                free_of "METAHYP2_" (v, map #2 params --->T))
      (*Instantiate subgoal vars by Free applied to params*)
      fun mk_ctpair (t,in_concl,u) =
          if in_concl then (cterm t,  cterm u)
          else (cterm t,  cterm (list_comb (u,fparams)))
      (*Restore Vars with higher type and index*)
      fun mk_subgoal_swap_ctpair
                (t as Var((a,i),_), in_concl, u as Free(_,U)) =
          if in_concl then (cterm u, cterm t)
          else (cterm u, cterm(Var((a, i+maxidx), U)))
      (*Embed B in the original context of params and hyps*)
      fun embed B = list_all_free (params, Logic.list_implies (hyps, B))
      (*Strip the context using elimination rules*)
      fun elim Bhyp = implies_elim_list (forall_elim_list cparams Bhyp) hypths
      (*A form of lifting that discharges assumptions.*)
      fun relift st =
        let val prop = Thm.prop_of st
            val subgoal_vars = (*Vars introduced in the subgoals*)
                  foldr add_term_vars [] (Logic.strip_imp_prems prop)
            and concl_vars = add_term_vars (Logic.strip_imp_concl prop, [])
            val subgoal_insts = map (mk_subgoal_inst concl_vars) subgoal_vars
            val st' = Thm.instantiate ([], map mk_ctpair subgoal_insts) st
            val emBs = map (cterm o embed) (prems_of st')
            val Cth  = implies_elim_list st' (map (elim o assume) emBs)
        in  (*restore the unknowns to the hypotheses*)
            free_instantiate (map swap_ctpair insts @
                              map mk_subgoal_swap_ctpair subgoal_insts)
                (*discharge assumptions from state in same order*)
                (implies_intr_list emBs
                  (forall_intr_list cparams (implies_intr_list chyps Cth)))
        end
      (*function to replace the current subgoal*)
      fun next st = bicompose false (false, relift st, nprems_of st)
                    gno state
  in Seq.maps next (tacf subprems (trivial (cterm concl))) end;

end;

(*Returns the theorem list that METAHYPS would supply to its tactic*)
fun metahyps_thms i state =
  let val prem = Logic.nth_prem (i, Thm.prop_of state) 
      val (insts,params,hyps,concl) = metahyps_split_prem prem 
      val cterm = cterm_of (Thm.theory_of_thm state)
  in SOME (map (forall_elim_vars 0 o assume o cterm) hyps) end
  handle TERM ("nth_prem", [A]) => NONE;

local

fun print_vars_terms thy (n,thm) =
  let
    fun typed ty = " has type: " ^ Sign.string_of_typ thy ty;
    fun find_vars thy (Const (c, ty)) =
        (case Term.typ_tvars ty
         of [] => I
          | _ => insert (op =) (c ^ typed ty))
      | find_vars thy (Var (xi, ty)) = insert (op =) (Term.string_of_vname xi ^ typed ty)
      | find_vars _ (Free _) = I
      | find_vars _ (Bound _) = I
      | find_vars thy (Abs (_, _, t)) = find_vars thy t
      | find_vars thy (t1 $ t2) = 
          find_vars thy t1 #> find_vars thy t1;
    val prem = Logic.nth_prem (n, Thm.prop_of thm)
    val tms = find_vars thy prem []
  in
    (warning "Found schematic vars in assumptions:"; warning (cat_lines tms))
  end;

in

fun METAHYPS tacf n thm = SUBGOAL (metahyps_aux_tac tacf) n thm
    handle THM("assume: variables",_,_) => (print_vars_terms (theory_of_thm thm) (n,thm); Seq.empty)

end; (*local*)

(*Makes a tactic whose effect on a state is given by thmfun: thm->thm seq.*)
fun PRIMSEQ thmfun st =  thmfun st handle THM _ => Seq.empty;

(*Makes a tactic whose effect on a state is given by thmfun: thm->thm.*)
fun PRIMITIVE thmfun = PRIMSEQ (Seq.single o thmfun);

(* Inverse (more or less) of PRIMITIVE *)
fun SINGLE tacf = Option.map fst o Seq.pull o tacf

end;

open Tactical;