(** * Axiomatization of the helper functions *)

Definition external_implements (name: string) (sg: signature) (vargs: list val) (vres: val) : Prop :=
  forall F V (ge: Genv.t F V) m,
  external_call (EF_runtime name sg) ge vargs m E0 vres m.Definition builtin_implements (name: string) (sg: signature) (vargs: list val) (vres: val) : Prop :=
  forall F V (ge: Genv.t F V) m,
  external_call (EF_builtin name sg) ge vargs m E0 vres m.Axiom i64_helpers_correct :
    (forall x z, Val.longoffloat x = Some z -> external_implements "__compcert_i64_dtos" sig_f_l (x::nil) z)
 /\ (forall x z, Val.longuoffloat x = Some z -> external_implements "__compcert_i64_dtou" sig_f_l (x::nil) z)
 /\ (forall x z, Val.floatoflong x = Some z -> external_implements "__compcert_i64_stod" sig_l_f (x::nil) z)
 /\ (forall x z, Val.floatoflongu x = Some z -> external_implements "__compcert_i64_utod" sig_l_f (x::nil) z)
 /\ (forall x z, Val.singleoflong x = Some z -> external_implements "__compcert_i64_stof" sig_l_s (x::nil) z)
 /\ (forall x z, Val.singleoflongu x = Some z -> external_implements "__compcert_i64_utof" sig_l_s (x::nil) z)
 /\ (forall x, builtin_implements "__builtin_negl" sig_l_l (x::nil) (Val.negl x))
 /\ (forall x y, builtin_implements "__builtin_addl" sig_ll_l (x::y::nil) (Val.addl x y))
 /\ (forall x y, builtin_implements "__builtin_subl" sig_ll_l (x::y::nil) (Val.subl x y))
 /\ (forall x y, builtin_implements "__builtin_mull" sig_ii_l (x::y::nil) (Val.mull' x y))
 /\ (forall x y z, Val.divls x y = Some z -> external_implements "__compcert_i64_sdiv" sig_ll_l (x::y::nil) z)
 /\ (forall x y z, Val.divlu x y = Some z -> external_implements "__compcert_i64_udiv" sig_ll_l (x::y::nil) z)
 /\ (forall x y z, Val.modls x y = Some z -> external_implements "__compcert_i64_smod" sig_ll_l (x::y::nil) z)
 /\ (forall x y z, Val.modlu x y = Some z -> external_implements "__compcert_i64_umod" sig_ll_l (x::y::nil) z)
 /\ (forall x y, external_implements "__compcert_i64_shl" sig_li_l (x::y::nil) (Val.shll x y))
 /\ (forall x y, external_implements "__compcert_i64_shr" sig_li_l (x::y::nil) (Val.shrlu x y))
 /\ (forall x y, external_implements "__compcert_i64_sar" sig_li_l (x::y::nil) (Val.shrl x y))
 /\ (forall x y, external_implements "__compcert_i64_umulh" sig_ll_l (x::y::nil) (Val.mullhu x y))
 /\ (forall x y, external_implements "__compcert_i64_smulh" sig_ll_l (x::y::nil) (Val.mullhs x y)).Definition helper_functions_declared {F V: Type} (p: AST.program (AST.fundef F) V) (hf: helper_functions) : Prop :=
     helper_declared p i64_dtos "__compcert_i64_dtos" sig_f_l
  /\ helper_declared p i64_dtou "__compcert_i64_dtou" sig_f_l
  /\ helper_declared p i64_stod "__compcert_i64_stod" sig_l_f
  /\ helper_declared p i64_utod "__compcert_i64_utod" sig_l_f
  /\ helper_declared p i64_stof "__compcert_i64_stof" sig_l_s
  /\ helper_declared p i64_utof "__compcert_i64_utof" sig_l_s
  /\ helper_declared p i64_sdiv "__compcert_i64_sdiv" sig_ll_l
  /\ helper_declared p i64_udiv "__compcert_i64_udiv" sig_ll_l
  /\ helper_declared p i64_smod "__compcert_i64_smod" sig_ll_l
  /\ helper_declared p i64_umod "__compcert_i64_umod" sig_ll_l
  /\ helper_declared p i64_shl "__compcert_i64_shl" sig_li_l
  /\ helper_declared p i64_shr "__compcert_i64_shr" sig_li_l
  /\ helper_declared p i64_sar "__compcert_i64_sar" sig_li_l
  /\ helper_declared p i64_umulh "__compcert_i64_umulh" sig_ll_l
  /\ helper_declared p i64_smulh "__compcert_i64_smulh" sig_ll_l.Ltac UseHelper := decompose [Logic.and] i64_helpers_correct; eauto.Ltac DeclHelper := red in HELPERS; decompose [Logic.and] HELPERS; eauto.

Lemma eval_helper: forall le id name sg args vargs vres, eval_exprlist ge sp e m le args vargs -> helper_declared prog id name sg -> external_implements name sg vargs vres -> eval_expr ge sp e m le (Eexternal id sg args) vres.