TFL/post.sml

add \<le> to list of "good" symbols;

(*  Title:      TFL/post.sml
    ID:         $Id$
    Author:     Konrad Slind, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge

Second part of main module (postprocessing of TFL definitions).
*)

signature TFL =
sig
  val trace: bool ref
  val quiet_mode: bool ref
  val message: string -> unit
  val tgoalw: theory -> thm list -> thm list -> thm list
  val tgoal: theory -> thm list -> thm list
  val std_postprocessor: claset -> simpset -> thm list -> theory ->
    {induction: thm, rules: thm, TCs: term list list} ->
    {induction: thm, rules: thm, nested_tcs: thm list}
  val define_i: theory -> claset -> simpset -> thm list -> thm list -> xstring ->
    term -> term list -> theory * {rules: (thm * int) list, induct: thm, tcs: term list}
  val define: theory -> claset -> simpset -> thm list -> thm list -> xstring ->
    string -> string list -> theory * {rules: (thm * int) list, induct: thm, tcs: term list}
  val defer_i: theory -> thm list -> xstring -> term list -> theory * thm
  val defer: theory -> thm list -> xstring -> string list -> theory * thm
end;

structure Tfl: TFL =
struct

structure S = USyntax


(* messages *)

val trace = Prim.trace

val quiet_mode = ref false;
fun message s = if ! quiet_mode then () else writeln s;


(* misc *)

fun read_term thy = Sign.simple_read_term (Theory.sign_of thy) HOLogic.termT;


(*---------------------------------------------------------------------------
 * Extract termination goals so that they can be put it into a goalstack, or
 * have a tactic directly applied to them.
 *--------------------------------------------------------------------------*)
fun termination_goals rules =
    map (#1 o Type.freeze_thaw o HOLogic.dest_Trueprop)
      (foldr (fn (th,A) => union_term (prems_of th, A)) (rules, []));

(*---------------------------------------------------------------------------
 * Finds the termination conditions in (highly massaged) definition and
 * puts them into a goalstack.
 *--------------------------------------------------------------------------*)
fun tgoalw thy defs rules =
  case termination_goals rules of
      [] => error "tgoalw: no termination conditions to prove"
    | L  => goalw_cterm defs
              (Thm.cterm_of (Theory.sign_of thy)
                        (HOLogic.mk_Trueprop(USyntax.list_mk_conj L)));

fun tgoal thy = tgoalw thy [];

(*---------------------------------------------------------------------------
 * Three postprocessors are applied to the definition.  It
 * attempts to prove wellfoundedness of the given relation, simplifies the
 * non-proved termination conditions, and finally attempts to prove the
 * simplified termination conditions.
 *--------------------------------------------------------------------------*)
fun std_postprocessor cs ss wfs =
  Prim.postprocess
   {wf_tac     = REPEAT (ares_tac wfs 1),
    terminator = asm_simp_tac ss 1
                 THEN TRY (fast_tac (cs addSDs [not0_implies_Suc] addss ss) 1),
    simplifier = Rules.simpl_conv ss []};



val concl = #2 o Rules.dest_thm;

(*---------------------------------------------------------------------------
 * Postprocess a definition made by "define". This is a separate stage of
 * processing from the definition stage.
 *---------------------------------------------------------------------------*)
local
structure R = Rules
structure U = Utils

(* The rest of these local definitions are for the tricky nested case *)
val solved = not o U.can S.dest_eq o #2 o S.strip_forall o concl

fun id_thm th =
   let val {lhs,rhs} = S.dest_eq(#2(S.strip_forall(#2 (R.dest_thm th))))
   in lhs aconv rhs
   end handle _ => false (* FIXME do not handle _ !!! *)

fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
val P_imp_P_iff_True = prover "P --> (P= True)" RS mp;
val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
fun mk_meta_eq r = case concl_of r of
     Const("==",_)$_$_ => r
  |   _ $(Const("op =",_)$_$_) => r RS eq_reflection
  |   _ => r RS P_imp_P_eq_True

(*Is this the best way to invoke the simplifier??*)
fun rewrite L = rewrite_rule (map mk_meta_eq (filter(not o id_thm) L))

fun join_assums th =
  let val {sign,...} = rep_thm th
      val tych = cterm_of sign
      val {lhs,rhs} = S.dest_eq(#2 (S.strip_forall (concl th)))
      val cntxtl = (#1 o S.strip_imp) lhs  (* cntxtl should = cntxtr *)
      val cntxtr = (#1 o S.strip_imp) rhs  (* but union is solider *)
      val cntxt = gen_union (op aconv) (cntxtl, cntxtr)
  in
    R.GEN_ALL
      (R.DISCH_ALL
         (rewrite (map (R.ASSUME o tych) cntxt) (R.SPEC_ALL th)))
  end
  val gen_all = S.gen_all
in
fun proof_stage cs ss wfs theory {f, R, rules, full_pats_TCs, TCs} =
  let
    val _ = message "Proving induction theorem ..."
    val ind = Prim.mk_induction theory {fconst=f, R=R, SV=[], pat_TCs_list=full_pats_TCs}
    val _ = message "Postprocessing ...";
    val {rules, induction, nested_tcs} =
      std_postprocessor cs ss wfs theory {rules=rules, induction=ind, TCs=TCs}
  in
  case nested_tcs
  of [] => {induction=induction, rules=rules,tcs=[]}
  | L  => let val dummy = message "Simplifying nested TCs ..."
              val (solved,simplified,stubborn) =
               U.itlist (fn th => fn (So,Si,St) =>
                     if (id_thm th) then (So, Si, th::St) else
                     if (solved th) then (th::So, Si, St)
                     else (So, th::Si, St)) nested_tcs ([],[],[])
              val simplified' = map join_assums simplified
              val rewr = full_simplify (ss addsimps (solved @ simplified'));
              val induction' = rewr induction
              and rules'     = rewr rules
          in
          {induction = induction',
               rules = rules',
                 tcs = map (gen_all o S.rhs o #2 o S.strip_forall o concl)
                           (simplified@stubborn)}
          end
  end;


(*lcp: curry the predicate of the induction rule*)
fun curry_rule rl = split_rule_var
                        (head_of (HOLogic.dest_Trueprop (concl_of rl)),
                         rl);

(*lcp: put a theorem into Isabelle form, using meta-level connectives*)
val meta_outer =
    curry_rule o standard o
    rule_by_tactic (REPEAT
                    (FIRSTGOAL (resolve_tac [allI, impI, conjI]
                                ORELSE' etac conjE)));

(*Strip off the outer !P*)
val spec'= read_instantiate [("x","P::?'b=>bool")] spec;

fun simplify_defn thy cs ss congs wfs id pats def0 =
   let val def = freezeT def0 RS meta_eq_to_obj_eq
       val {theory,rules,rows,TCs,full_pats_TCs} = Prim.post_definition congs (thy, (def,pats))
       val {lhs=f,rhs} = S.dest_eq (concl def)
       val (_,[R,_]) = S.strip_comb rhs
       val {induction, rules, tcs} =
             proof_stage cs ss wfs theory
               {f = f, R = R, rules = rules,
                full_pats_TCs = full_pats_TCs,
                TCs = TCs}
       val rules' = map (standard o Rulify.rulify_no_asm) (R.CONJUNCTS rules)
   in  {induct = meta_outer (Rulify.rulify_no_asm (induction RS spec')),
        rules = ListPair.zip(rules', rows),
        tcs = (termination_goals rules') @ tcs}
   end
  handle Utils.ERR {mesg,func,module} =>
               error (mesg ^
                      "\n    (In TFL function " ^ module ^ "." ^ func ^ ")");

(*---------------------------------------------------------------------------
 * Defining a function with an associated termination relation.
 *---------------------------------------------------------------------------*)
fun define_i thy cs ss congs wfs fid R eqs =
  let val {functional,pats} = Prim.mk_functional thy eqs
      val (thy, def) = Prim.wfrec_definition0 thy (Sign.base_name fid) R functional
  in (thy, simplify_defn thy cs ss congs wfs fid pats def) end;

fun define thy cs ss congs wfs fid R seqs =
  define_i thy cs ss congs wfs fid (read_term thy R) (map (read_term thy) seqs)
    handle Utils.ERR {mesg,...} => error mesg;


(*---------------------------------------------------------------------------
 *
 *     Definitions with synthesized termination relation
 *
 *---------------------------------------------------------------------------*)

local open USyntax
in
fun func_of_cond_eqn tm =
  #1(strip_comb(#lhs(dest_eq(#2 (strip_forall(#2(strip_imp tm)))))))
end;

fun defer_i thy congs fid eqs =
 let val {rules,R,theory,full_pats_TCs,SV,...} =
             Prim.lazyR_def thy (Sign.base_name fid) congs eqs
     val f = func_of_cond_eqn (concl(R.CONJUNCT1 rules handle _ => rules)) (* FIXME do not handle _ !!! *)
     val dummy = message "Proving induction theorem ...";
     val induction = Prim.mk_induction theory
                        {fconst=f, R=R, SV=SV, pat_TCs_list=full_pats_TCs}
 in (theory,
     (*return the conjoined induction rule and recursion equations,
       with assumptions remaining to discharge*)
     standard (induction RS (rules RS conjI)))
 end

fun defer thy congs fid seqs =
  defer_i thy congs fid (map (read_term thy) seqs)
    handle Utils.ERR {mesg,...} => error mesg;
end;

end;