(*  Title:      TFL/post.ML

    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 define_i: bool -> theory -> claset -> simpset -> thm list -> thm list -> xstring ->

    term -> term list -> theory * {rules: (thm * int) list, induct: thm, tcs: term list}

  val define: bool -> 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 *)

val read_term = Thm.term_of oo (HOLogic.read_cterm o Theory.sign_of);

(*---------------------------------------------------------------------------

 * 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 strict cs ss wfs =

  Prim.postprocess strict

   {wf_tac     = REPEAT (ares_tac wfs 1),

    terminator = asm_simp_tac ss 1

                 THEN TRY (silent_arith_tac 1 ORELSE

                           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 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 U.ERR _ => false;

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 strict 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 strict 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 dummy = (Prim.trace_thms "solved =" solved;

                           Prim.trace_thms "simplified' =" simplified')

              val rewr = full_simplify (ss addsimps (solved @ simplified'));

              val dummy = Prim.trace_thms "Simplifying the induction rule..."

                                          [induction]

              val induction' = rewr induction

              val dummy = Prim.trace_thms "Simplifying the recursion rules..."

                                          [rules]

              val rules'     = rewr rules

              val _ = message "... Postprocessing finished";

          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 =

  SplitRule.split_rule_var (Term.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 tracing true _ = ()

  | tracing false msg = writeln msg;

fun simplify_defn strict 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 dummy = Prim.trace_thms "congs =" congs

       (*the next step has caused simplifier looping in some cases*)

       val {induction, rules, tcs} =

             proof_stage strict cs ss wfs theory

               {f = f, R = R, rules = rules,

                full_pats_TCs = full_pats_TCs,

                TCs = TCs}

       val rules' = map (standard o ObjectLogic.rulify_no_asm)

                        (R.CONJUNCTS rules)

         in  {induct = meta_outer (ObjectLogic.rulify_no_asm (induction RS spec')),

        rules = ListPair.zip(rules', rows),

        tcs = (termination_goals rules') @ tcs}

   end

  handle U.ERR {mesg,func,module} =>

               error (mesg ^

                      "\n    (In TFL function " ^ module ^ "." ^ func ^ ")");

(* Derive the initial equations from the case-split rules to meet the

users specification of the recursive function. 

 Note: We don't do this if the wf conditions fail to be solved, as each

case may have a different wf condition. We could group the conditions

together and say that they must be true to solve the general case,

but that would hide from the user which sub-case they were related

to. Probably this is not important, and it would work fine, but, for now, I

prefer leaving more fine-grain control to the user. 

-- Lucas Dixon, Aug 2004 *)

local

  fun get_related_thms i = 

      mapfilter ((fn (r,x) => if x = i then Some r else None));

  fun solve_eq (th, [], i) = 

      raise ERROR_MESSAGE "derive_init_eqs: missing rules"

    | solve_eq (th, [a], i) = [(a, i)]

    | solve_eq (th, splitths as (_ :: _), i) = 

      (writeln "Proving unsplit equation...";

      [((standard o ObjectLogic.rulify_no_asm)

          (CaseSplit.splitto splitths th), i)])

      (* if there's an error, pretend nothing happened with this definition 

         We should probably print something out so that the user knows...? *)

      handle ERROR_MESSAGE s => 

             (writeln ("WARNING:post.ML:solve_eq: " ^ s); map (fn x => (x,i)) splitths); 

in

fun derive_init_eqs sgn rules eqs = 

    let 

      val eqths = map (Thm.trivial o (Thm.cterm_of sgn) o HOLogic.mk_Trueprop) 

                      eqs

      fun countlist l = 

          (rev o snd o (foldl (fn ((i,L), e) => (i + 1,(e,i) :: L)))) ((0,[]), l)

    in

      flat (map (fn (e,i) => solve_eq (e, (get_related_thms i rules), i))

                (countlist eqths))

    end;

end;

(*---------------------------------------------------------------------------

 * Defining a function with an associated termination relation.

 *---------------------------------------------------------------------------*)

fun define_i strict 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

      val {induct, rules, tcs} = 

          simplify_defn strict thy cs ss congs wfs fid pats def

      val rules' = 

          if strict then derive_init_eqs (Theory.sign_of thy) rules eqs

          else rules

  in (thy, {rules = rules', induct = induct, tcs = tcs}) end;

fun define strict thy cs ss congs wfs fid R seqs =

  define_i strict thy cs ss congs wfs fid (read_term thy R) (map (read_term thy) seqs)

    handle U.ERR {mesg,...} => error mesg;

(*---------------------------------------------------------------------------

 *

 *     Definitions with synthesized termination relation

 *

 *---------------------------------------------------------------------------*)

fun func_of_cond_eqn tm =

  #1 (S.strip_comb (#lhs (S.dest_eq (#2 (S.strip_forall (#2 (S.strip_imp tm)))))));

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 U.ERR _ => rules));

     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 U.ERR {mesg,...} => error mesg;

end;

end;