IEEE_connection.v 51.5 KB
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Require Import Coq.Reals.Reals Coq.QArith.QArith Coq.QArith.Qabs Coq.micromega.Psatz
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        Coq.QArith.Qreals.
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Require Import Flover.Expressions Flover.Infra.RationalSimps Flover.Typing
        Flover.IntervalValidation Flover.ErrorValidation Flover.CertificateChecker
        Flover.FPRangeValidator Flover.Environments Flover.Infra.RealRationalProps
        Flover.Commands Flover.Infra.Ltacs.
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Require Import Flocq.Appli.Fappli_IEEE_bits Flocq.Appli.Fappli_IEEE
        Flocq.Core.Fcore_Raux Flocq.Prop.Fprop_relative.
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Definition dmode := mode_NE.
Definition fl64:Type := binary_float 53 1024.

Definition normal_or_zero v :=
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   (v = 0 \/ (Q2R (minValue_pos M64)) <= (Rabs v))%R.
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Definition updFlEnv x v E :=
  fun y => if y =? x
        then Some (A:=(binary_float 53 1024)) v
        else E y.

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Fixpoint eval_expr_float (e:expr (binary_float 53 1024)) (E:nat -> option fl64):=
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  match e with
  | Var _ x => E x
  | Const m v => Some v
  | Unop Neg e =>
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    match eval_expr_float e E with
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    |Some v1 => Some (b64_opp v1)
    |_ => None
    end
  | Unop Inv e => None
  | Binop b e1 e2 =>
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    match eval_expr_float e1 E, eval_expr_float e2 E with
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    | Some f1, Some f2 =>
      match b with
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      | Plus => Some (b64_plus dmode f1 f2)
      | Sub => Some (b64_minus dmode f1 f2)
      | Mult => Some (b64_mult dmode f1 f2)
      | Div => Some (b64_div dmode f1 f2)
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      end
    |_ , _ => None
    end
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  | Fma e1 e2 e3 =>
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    match eval_expr_float e1 E, eval_expr_float e2 E, eval_expr_float e3 E with
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      (* | Some f1, Some f2, Some f3 => Some (b64_plus dmode f1 (b64_mult dmode f2 f3)) *)
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      | _, _, _ => None
    end
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  | _ => None
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  end.

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Fixpoint bstep_float f E :option fl64 :=
  match f with
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  | Let m x e g =>
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    olet res :=  eval_expr_float e E in
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        bstep_float g (updFlEnv x res E)
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  | Ret e => eval_expr_float e E
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  end.

Definition isValid e :=
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  plet v := e in
        normal_or_zero (B2R 53 1024 v).
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Fixpoint eval_expr_valid (e:expr fl64) E :=
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  match e with
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  | Var _ x => True (*isValid (eval_expr_float (Var n) E)*)
  | Const m v => True (*isValid (eval_expr_float (Const m v) E)*)
  | Unop u e => eval_expr_valid e E
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  | Binop b e1 e2 =>
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    (eval_expr_valid e1 E) /\ (eval_expr_valid e2 E) /\
    (let e1_res := eval_expr_float e1 E in
     let e2_res := eval_expr_float e2 E in
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     optionBind e1_res
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                (fun v1 =>
                   let v1_real := B2R 53 1024 v1 in
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                   optionBind e2_res
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                              (fun v2 =>
                                 let v2_real := B2R 53 1024 v2 in
                                 normal_or_zero (evalBinop b v1_real v2_real))
                              True)
                True)
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  | Fma e1 e2 e3 =>
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    (eval_expr_valid e1 E) /\ (eval_expr_valid e2 E) /\ (eval_expr_valid e3 E) /\
    (let e1_res := eval_expr_float e1 E in
     let e2_res := eval_expr_float e2 E in
     let e3_res := eval_expr_float e3 E in
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     optionBind e1_res
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                (fun v1 =>
                   let v1_real := B2R 53 1024 v1 in
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                   optionBind e2_res
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                              (fun v2 =>
                                 let v2_real := B2R 53 1024 v2 in
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                                 optionBind e3_res
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                                            (fun v3 =>
                                               let v3_real := B2R 53 1024 v3 in
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                                               (* No support for fma yet *)
                                               (* normal_or_zero (evalFma v1_real v2_real v3_real)) *)
                                               False)
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                                            True)
                              True)
                True)
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  | Downcast m e => eval_expr_valid e E
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  end.

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Fixpoint bstep_valid f E :=
  match f with
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  | Let m x e g =>
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    eval_expr_valid e E /\
    (optionBind (eval_expr_float e E)
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                (fun v_e => bstep_valid g (updFlEnv x v_e E))
                True)
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  | Ret e => eval_expr_valid e E
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  end.

Definition bpowQ (r:radix) (e: Z) :=
  match e with
  |0%Z => 1%Q
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  | Zpos p => (Z.pow_pos r p) #1
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  | Z.neg p => Qinv ((Z.pow_pos r p)#1)
  end.

Definition B2Q :=
  fun prec emax : Z =>
    let emin := (3 - emax - prec)%Z in
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    let fexpr := Fcore_FLT.FLT_exp emin prec in
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    fun f : binary_float prec emax =>
      match f with
      | B754_zero _ _ _ => 0%Q
      | B754_infinity _ _ _ => (bpowQ radix2 emax) +1%Q
      | B754_nan _ _ _ _ => (bpowQ radix2 emax) +1%Q
      | B754_finite _ _ s m e _ =>
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        let f_new: Fcore_defs.float radix2 := {| Fcore_defs.Fnum := cond_Zopp s (Zpos m); Fcore_defs.Fexp := e |} in
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        (Fcore_defs.Fnum f_new # 1) * bpowQ radix2 (Fcore_defs.Fexp f_new)
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      end.

Lemma B2Q_B2R_eq :
  forall v,
    is_finite 53 1024 v = true ->
    Q2R (B2Q v) = B2R 53 1024 v.
Proof.
  intros; unfold B2Q, B2R, is_finite in *.
  destruct v eqn:?; try congruence;
    try rewrite Q2R0_is_0; try lra.
  unfold Fcore_defs.F2R.
  rewrite Q2R_mult.
  f_equal.
  - unfold Z2R, Q2R.
    simpl. rewrite RMicromega.Rinv_1.
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    destruct (cond_Zopp b (Zpos m)); unfold IZR;
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      try rewrite P2R_INR, INR_IPR; lra.
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  - unfold Q2R; simpl.
    unfold bpow, bpowQ.
    destruct e; simpl; try lra.
    + rewrite RMicromega.Rinv_1.
      unfold Z2R, IZR.
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      destruct (Z.pow_pos 2 p); try rewrite P2R_INR, INR_IPR; auto.
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    + unfold Z2R, IZR. unfold Qinv; simpl.
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      destruct (Z.pow_pos 2 p) eqn:? ; try rewrite P2R_INR, INR_IPR; simpl; try lra.
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      * unfold bounded in e0.  simpl in e0. unfold canonic_mantissa in e0.
        simpl in e0.
        pose proof (Is_true_eq_left _ e0).
        apply Is_true_eq_true in H0; andb_to_prop H0.
        assert (0 < Z.pow_pos 2 p)%Z.
        { apply Zpower_pos_gt_0. cbv; auto. }
        rewrite Heqz in H0. inversion H0.
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      * unfold IPR at 1. rewrite Rmult_1_l; auto.
      * rewrite <- Ropp_mult_distr_l, Ropp_mult_distr_r, Ropp_inv_permute;
        [ unfold IPR at 1; lra | ].
        hnf; intros. pose proof (pos_INR_nat_of_P p0).
        rewrite INR_IPR in *.
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        rewrite H0 in H1; lra.
Qed.

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Fixpoint B2Qexpr (e: expr fl64) :=
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  match e with
  | Var _ x =>  Var Q x
  | Const m v => Const m (B2Q v)
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  | Unop u e => Unop u (B2Qexpr e)
  | Binop b e1 e2 => Binop b (B2Qexpr e1) (B2Qexpr e2)
  | Fma e1 e2 e3 => Fma (B2Qexpr e1) (B2Qexpr e2) (B2Qexpr e3)
  | Downcast m e => Downcast m (B2Qexpr e)
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  end.

Fixpoint B2Qcmd (f:cmd fl64) :=
  match f with
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  | Let m x e g => Let m x (B2Qexpr e) (B2Qcmd g)
  | Ret e => Ret (B2Qexpr e)
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  end.

Definition toREnv (E: nat -> option fl64) (x:nat):option R :=
  match E x with
  |Some v => Some (Q2R (B2Q v))
  |_ => None
  end.

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Fixpoint is64BitEval (V:Type) (e:expr V) :=
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  match e with
  | Var _ x => True
  | Const m e => m = M64
  | Unop u e => is64BitEval e
  | Binop b e1 e2 => is64BitEval e1 /\ is64BitEval e2
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  | Fma e1 e2 e3 => is64BitEval e1 /\ is64BitEval e2 /\ is64BitEval e3
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  | Downcast m e => m = M64 /\ is64BitEval e
  end.

Fixpoint is64BitBstep (V:Type) (f:cmd V) :=
  match f with
  | Let m x e g => is64BitEval e /\ m = M64 /\ is64BitBstep g
  | Ret e => is64BitEval e
  end.

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Fixpoint noDowncast (V:Type) (e:expr V) :=
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  match e with
  | Var _ x => True
  | Const m e => True
  | Unop u e => noDowncast e
  | Binop b e1 e2 => noDowncast e1 /\ noDowncast e2
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  | Fma e1 e2 e3 => noDowncast e1 /\ noDowncast e2 /\ noDowncast e3
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  | Downcast m e => False
  end.

Fixpoint noDowncastFun (V:Type) (f:cmd V) :=
  match f with
  | Let m x e g => noDowncast e /\ noDowncastFun g
  | Ret e => noDowncast e
  end.

Opaque mTypeToQ.

Lemma validValue_is_finite v:
  validFloatValue (Q2R (B2Q v)) M64 -> is_finite 53 1024 v = true.
Proof.
  intros validVal.
  unfold is_finite.
  unfold validFloatValue, B2Q in *.
  destruct v; try auto;
    destruct validVal; unfold Normal in *; unfold Denormal in *;
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      unfold maxValue, minValue_pos, maxExponent, minExponentPos in*;
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      rewrite Q2R_inv in *; unfold bpowQ in *.
  - assert (Z.pow_pos radix2 1024 = 179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216%Z)
      by (vm_compute;auto).
    rewrite H0 in H; destruct H; try lra.
    assert (Z.pow_pos 2 1023 = 89884656743115795386465259539451236680898848947115328636715040578866337902750481566354238661203768010560056939935696678829394884407208311246423715319737062188883946712432742638151109800623047059726541476042502884419075341171231440736956555270413618581675255342293149119973622969239858152417678164812112068608%Z)
      by (vm_compute; auto).
    rewrite H2 in *.
    clear H0 H2.
    rewrite Rabs_right in H1.
    apply Rle_Qle in H1.
    + rewrite <- Qle_bool_iff in H1.
      cbv in H1; try congruence.
    + rewrite <- Q2R0_is_0.
      apply Rle_ge. apply Qle_Rle; rewrite <- Qle_bool_iff; cbv; auto.
  - vm_compute; intros; congruence.
  - destruct H.
    + destruct H. rewrite Rabs_right in H.
      * rewrite <- Q2R_inv in H.
        apply Rlt_Qlt in H.
        vm_compute in H.
        congruence.
        vm_compute; congruence.
      * rewrite <- Q2R0_is_0.
        apply Rle_ge. apply Qle_Rle; rewrite <- Qle_bool_iff; cbv; auto.
    + rewrite <- Q2R0_is_0 in H.
      apply eqR_Qeq in H.
      vm_compute in H; congruence.
  - vm_compute; congruence.
  - destruct H.
    rewrite Rabs_right in H0.
    + apply Rle_Qle in H0.
      rewrite <- Qle_bool_iff in H0.
      vm_compute in H0; auto.
    + rewrite <- Q2R0_is_0.
      apply Rle_ge. apply Qle_Rle; rewrite <- Qle_bool_iff; cbv; auto.
  - vm_compute; congruence.
  - destruct H.
    + rewrite Rabs_right in H.
      * destruct H. rewrite <- Q2R_inv in H.
        { apply Rlt_Qlt in H. rewrite Qlt_alt in H.
          vm_compute in H. congruence. }
        { vm_compute; congruence. }
      * rewrite <- Q2R0_is_0.
        apply Rle_ge. apply Qle_Rle; rewrite <- Qle_bool_iff; cbv; auto.
    + rewrite <- Q2R0_is_0 in H.
      apply eqR_Qeq in H. vm_compute in H; congruence.
  - vm_compute; congruence.
Qed.

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Lemma typing_expr_64_bit e:
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  forall Gamma tMap,
    noDowncast e ->
    is64BitEval e ->
    typeCheck e Gamma tMap = true ->
    (forall v,
        NatSet.In v (usedVars e) -> Gamma v = Some M64) ->
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    FloverMap.find e tMap = Some M64.
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Proof.
  induction e; intros * noDowncast_e is64BitEval_e typecheck_e types_valid;
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    cbn in *; try inversion noDowncast_e;
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      subst; Flover_compute; try congruence;
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        type_conv; subst.
  - rewrite types_valid in *; try set_tac.
  - destruct m; try congruence.
  - erewrite IHe in *; eauto.
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  - repeat (match goal with
            |H: _ /\ _ |- _=> destruct H
            end).
      erewrite IHe1 in *; eauto.
    + erewrite IHe2 in *; eauto.
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      * unfold join in *.
        destr_factorize.
        rewrite <- isMorePrecise_morePrecise.
        rewrite isMorePrecise_refl. inversion Heqo0; auto.
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      * intros.
        apply types_valid. set_tac.
    + intros; apply types_valid; set_tac.
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  - repeat (match goal with
            |H: _ /\ _ |- _=> destruct H
            end).
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    erewrite IHe1 in *; eauto; try (intros; apply types_valid; set_tac; fail).
    erewrite IHe2 in *; eauto; try (intros; apply types_valid; set_tac; fail).
    unfold join3, join in *.
    erewrite IHe3 in *; eauto; try (intros; apply types_valid; set_tac; fail).
    repeat destr_factorize.
    repeat rewrite <- isMorePrecise_morePrecise.
    repeat rewrite isMorePrecise_refl;
      type_conv; subst; auto.
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Qed.

Lemma typing_cmd_64_bit f:
  forall Gamma tMap,
    noDowncastFun f ->
    is64BitBstep f ->
    typeCheckCmd f Gamma tMap = true ->
    (forall v,
        NatSet.In v (freeVars f) -> Gamma v = Some M64) ->
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    FloverMap.find (getRetExp f) tMap = Some M64.
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Proof.
  induction f; intros * noDowncast_f is64BitEval_f typecheck_f types_valid;
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    cbn in *;
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    subst; try eauto using typing_expr_64_bit;
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      Flover_compute; try congruence.
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  destruct noDowncast_f; destruct is64BitEval_f as [Ha [Hb Hc]].
  eapply IHf; eauto.
  intros. unfold updDefVars.
  destruct (v =? n) eqn:?.
  - type_conv; auto.
  - apply types_valid.
    rewrite NatSet.remove_spec, NatSet.union_spec.
    split; try auto.
    hnf; intros; subst. rewrite Nat.eqb_neq in Heqb.
    congruence.
Qed.

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Lemma typing_agrees_expr e:
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  forall E Gamma tMap v m1 m2,
    typeCheck e Gamma tMap = true ->
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    eval_expr E Gamma (toRExp e) v m1 ->
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    FloverMap.find e tMap = Some m2 ->
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    m1 = m2.
Proof.
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  induction e; intros * typeCheck_e eval_e tMap_e; cbn in *;
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    rewrite tMap_e in *;
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    inversion eval_e; subst; cbn in *;
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      Flover_compute; try congruence; type_conv; subst; try auto.
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  - eapply IHe; eauto.
  - eapply IHe; eauto.
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  - assert (m0 = m) by eauto using IHe1.
    assert (m3 = m1) by eauto using IHe2.
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    subst; auto.
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  - assert (m0 = m) by eauto using IHe1.
    assert (m3 = m1) by eauto using IHe2.
    assert (m4 = m5) by eauto using IHe3.
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    subst; auto.
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Qed.

Lemma typing_agrees_cmd f:
  forall E Gamma tMap v m1 m2,
    typeCheckCmd f Gamma tMap = true ->
    bstep (toRCmd f) E Gamma v m1 ->
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    FloverMap.find (getRetExp f) tMap = Some m2 ->
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    m1 = m2.
Proof.
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  induction f; intros * typeCheck_f eval_f tMap_f; cbn in *;
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    Flover_compute; try congruence; type_conv; subst.
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  - inversion eval_f; subst; simpl in *.
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    specialize (IHf (updEnv n v0 E) (updDefVars n m3 Gamma) tMap v m1 m2).
    apply IHf; auto.
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  - inversion eval_f; subst; eapply typing_agrees_expr; eauto.
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Qed.

Lemma round_0_zero:
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  (Fcore_generic_fmt.round radix2 (Fcore_FLT.FLT_exp (3 - 1024 - 53) 53)
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                           (round_mode mode_NE) 0) = 0%R.
Proof.
  unfold Fcore_generic_fmt.round. simpl.
  unfold Fcore_generic_fmt.scaled_mantissa.
  rewrite Rmult_0_l.
  unfold Fcore_generic_fmt.Znearest.
  unfold Zfloor.
  assert (up 0 = 1%Z).
  { symmetry. apply tech_up; lra. }
  rewrite H.
  simpl. rewrite Rsub_eq_Ropp_Rplus. rewrite Rplus_opp_r.
  assert (Rcompare (0) (/ 2) = Lt).
  { apply Rcompare_Lt. lra. }
  rewrite H0.
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  unfold Fcore_generic_fmt.canonic_exp.
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  unfold Fcore_defs.F2R; simpl.
  rewrite Rmult_0_l. auto.
Qed.

Lemma validValue_bounded b v_e1 v_e2:
  (Normal (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2)) M64\/
   ((evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2)) = 0)%R) ->
  (forall eps, (Rabs eps <= / 2 * bpow radix2 (- 53 + 1))%R ->
  validFloatValue ((evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2)) * (1 + eps)) M64) ->
  Rlt_bool
    (Rabs
       (Fcore_generic_fmt.round
          radix2
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          (Fcore_FLT.FLT_exp (3 - 1024 - 53) 53)
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          (round_mode mode_NE)
          (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2))))
    (bpow radix2 1024) = true.
Proof.
  simpl.
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  pose proof (fexp_correct 53 1024 eq_refl) as fexpr_corr.
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  assert (forall k : Z, (-1022 < k)%Z ->
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                     (53 <= k - Fcore_FLT.FLT_exp (3 - 1024 - 53) 53 k)%Z)
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    as expr_valid.
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  { intros k k_pos.
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    unfold Fcore_FLT.FLT_exp; simpl.
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    destruct (Z.max_spec_le (k - 53) (-1074)); omega. }
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  pose proof (relative_error_N_ex radix2 (Fcore_FLT.FLT_exp (3 -1024 - 53) 53)
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                                  (-1022)%Z 53%Z expr_valid)
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    as rel_error_exists.
  intros [normal_v | zero_v] validVal;
  apply Rlt_bool_true.
  - unfold Normal in *; destruct normal_v.
    specialize (rel_error_exists (fun x => negb (Zeven x))
                                 (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2))%R).
    destruct (rel_error_exists) as [eps [bounded_eps round_eq]].
    + eapply Rle_trans; eauto.
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      unfold minValue_pos, Z.pow_pos; simpl.
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      rewrite Q2R_inv.
      * apply Rinv_le.
        { rewrite <- Q2R0_is_0. apply Qlt_Rlt.
          apply Qlt_alt. vm_compute; auto. }
        { unfold Q2R.
          unfold Qnum, Qden. lra. }
      *  vm_compute; congruence.
    + simpl in *.
      rewrite round_eq.
      destruct (validVal eps) as [normal_v | [denormal_v | zero_v]]; try auto.
      * unfold Normal in *. destruct normal_v.
        eapply Rle_lt_trans; eauto.
        unfold maxValue, bpow. unfold maxExponent. unfold Q2R.
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        unfold Qnum, Qden. rewrite <- Z2R_IZR. unfold IZR.
        repeat rewrite <- INR_IPR. simpl. lra.
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      * unfold Denormal in *. destruct denormal_v.
        eapply Rlt_trans; eauto.
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        unfold minValue_pos, minExponentPos, bpow.
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        rewrite Q2R_inv.
        { unfold Q2R, Qnum, Qden.
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          rewrite <- Z2R_IZR; unfold IZR.
          repeat rewrite <- INR_IPR; simpl; lra. }
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        { vm_compute; congruence. }
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      * rewrite zero_v. simpl. rewrite Rabs_R0. lra.
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  - rewrite zero_v.
    pose proof round_0_zero. simpl in H. rewrite H.
    rewrite Rabs_R0.
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    lra.
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Qed.

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(* (fexpr_correct 53 1024 eq_refl) as fexpr_corr. *)
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(* (relative_error_N_ex radix2 (Fcore_FLT.FLT_exp (3 -1024 - 53) 53) *)
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(*                                     (-1022)%Z 53%Z) *)
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Lemma eval_expr_gives_IEEE (e:expr fl64) :
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  forall E1 E2 E2_real Gamma tMap vR A P fVars dVars,
    (forall x, (toREnv E2) x = E2_real x) ->
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    typeCheck (B2Qexpr e) Gamma tMap = true ->
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    approxEnv E1 Gamma A fVars dVars E2_real ->
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    validIntervalbounds (B2Qexpr e) A P dVars = true ->
    validErrorbound (B2Qexpr e) tMap A dVars = true ->
    FPRangeValidator (B2Qexpr e) A tMap dVars = true ->
    eval_expr (toREnv E2) Gamma (toRExp (B2Qexpr e)) vR M64 ->
    NatSet.Subset ((usedVars (B2Qexpr e)) -- dVars) fVars ->
    is64BitEval (B2Qexpr e) ->
    noDowncast (B2Qexpr e) ->
    eval_expr_valid e E2 ->
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    dVars_range_valid dVars E1 A ->
    fVars_P_sound fVars E1 P ->
    vars_typed (NatSet.union fVars dVars) Gamma ->
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      (forall v,
        NatSet.In v dVars ->
        exists vF m,
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        (E2_real v = Some vF /\ FloverMap.find (Var Q v) tMap = Some m /\
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        validFloatValue vF m)) ->
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      (forall v, NatSet.In v (usedVars (B2Qexpr e)) -> Gamma v = Some M64) ->
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      exists v,
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        eval_expr_float e E2 = Some v /\
        eval_expr (toREnv E2) Gamma (toRExp (B2Qexpr e)) (Q2R (B2Q v)) M64.
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Proof.
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   induction e; simpl in *;
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    intros * envs_eq typecheck_e approxEnv_E1_E2_real valid_rangebounds
                     valid_roundoffs valid_float_ranges eval_e_float
                     usedVars_sound is64BitEval_e noDowncast_e eval_IEEE_valid_e
                     fVars_defined vars_typed dVars_sound dVars_valid
                     usedVars_64bit;
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    (match_pat (eval_expr _ _ _ _ _) (fun H => inversion H; subst; simpl in *));
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     revert eval_IEEE_valid_e;
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     Flover_compute_asm; try congruence; type_conv; subst;
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     unfold optionBind;
     intros eval_IEEE_valid_e.
     - unfold toREnv in *.
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    destruct (E2 n) eqn:HE2; try congruence.
    exists f; split; try eauto.
    eapply Var_load; try auto. rewrite HE2; auto.
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     - eexists; split; try eauto.
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    eapply (Const_dist') with (delta:=0%R); eauto.
    + rewrite Rabs_R0; apply mTypeToR_pos_R.
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    + unfold perturb. lra.
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     - edestruct IHe as [v_e [eval_float_e eval_rel_e]]; eauto.
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    assert (is_finite 53 1024 v_e = true).
    { apply validValue_is_finite.
      eapply FPRangeValidator_sound; eauto.
      eapply eval_eq_env; eauto. }
    rewrite eval_float_e.
    exists (b64_opp v_e); split; try auto.
    unfold b64_opp. rewrite <- (is_finite_Bopp _ _ pair) in H.
    rewrite B2Q_B2R_eq; auto. rewrite B2R_Bopp.
    eapply Unop_neg'; eauto.
    unfold evalUnop. rewrite is_finite_Bopp in H. rewrite B2Q_B2R_eq; auto.
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     - repeat (match goal with
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            |H: _ /\ _ |- _ => destruct H
            end).
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       assert (FloverMap.find (B2Qexpr e1) tMap = Some M64 /\
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            FloverMap.find (B2Qexpr e2) tMap = Some M64 /\
            FloverMap.find (Binop b (B2Qexpr e1) (B2Qexpr e2)) tMap = Some M64)
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           as [tMap_e1 [tMap_e2 tMap_b]].
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    { repeat split; apply (typing_expr_64_bit _ Gamma); simpl; auto.
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      - intros; apply usedVars_64bit; set_tac.
      - intros; apply usedVars_64bit; set_tac.
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      - rewrite Heqo, Heqo4, Heqo6.
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        apply Is_true_eq_true; apply andb_prop_intro; split.
        + apply andb_prop_intro; split; apply Is_true_eq_left; auto.
          apply mTypeEq_refl.
        + apply Is_true_eq_left; auto. }
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    repeat destr_factorize.
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    assert (m1 = M64).
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    { eapply (typing_agrees_expr (B2Qexpr e1)); eauto. }
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    assert (m2 = M64).
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    { eapply typing_agrees_expr; eauto. }
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    subst.
    destruct (IHe1 E1 E2 E2_real Gamma tMap v1 A P fVars dVars)
      as [v_e1 [eval_float_e1 eval_rel_e1]];
      try auto; try set_tac;
        [ intros; apply usedVars_64bit ; set_tac | ].
    destruct (IHe2 E1 E2 E2_real Gamma tMap v2 A P fVars dVars)
      as [v_e2 [eval_float_e2 eval_rel_e2]];
      try auto; try set_tac;
        [ intros; apply usedVars_64bit ; set_tac | ].
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    edestruct (validIntervalbounds_sound (B2Qexpr e2))
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      as [iv_2 [err_2 [nR2 [map_e2 [eval_real_e2 e2_bounded_real]]]]];
      eauto; set_tac.
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    rewrite eval_float_e1, eval_float_e2.
    inversion Heqo1; subst.
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    destr_factorize.
    destruct iv_2 as [ivlo_2 ivhi_2].
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    assert (forall vF2 m2,
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               eval_expr E2_real Gamma (toRExp (B2Qexpr e2)) vF2 m2 ->
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               (Rabs (nR2 - vF2) <= Q2R err_2))%R.
    { eapply validErrorbound_sound; try eauto; try set_tac. }
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    assert (contained (Q2R (B2Q v_e2))
                      (widenInterval
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                         (Q2R ivlo_2, Q2R ivhi_2) (Q2R err_2))).
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    { eapply distance_gives_iv.
      - simpl. eapply e2_bounded_real.
      - eapply H11. instantiate(1:=M64).
        eapply eval_eq_env; eauto. }
    assert (b = Div -> (Q2R (B2Q v_e2)) <> 0%R).
    { intros; subst; simpl in *.
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      andb_to_prop R3.
      apply le_neq_bool_to_lt_prop in L3.
      destruct L3; hnf; intros.
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      - rewrite H15 in *.
        apply Qlt_Rlt in H14.
        rewrite Q2R0_is_0, Q2R_plus in H14. lra.
      - rewrite H15 in *.
        apply Qlt_Rlt in H14.
        rewrite Q2R0_is_0, Q2R_minus in H14; lra. }
    assert (validFloatValue
              (evalBinop b (Q2R (B2Q v_e1)) (Q2R (B2Q v_e2))) M64).
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    { eapply (FPRangeValidator_sound (Binop b (B2Qexpr e1) (B2Qexpr e2)));
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        try eauto; set_tac.
      - eapply eval_eq_env; eauto.
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        eapply Binop_dist' with (delta:=0%R); eauto.
        + rewrite Rabs_R0. apply mTypeToR_pos_R.
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        + unfold perturb; lra.
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      - Flover_compute.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro); split; try auto using Is_true_eq_left.
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      - Flover_compute.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro); split; try auto using Is_true_eq_left.
        apply andb_prop_intro; split; auto using Is_true_eq_left.
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      - Flover_compute.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro; split); try auto using Is_true_eq_left.
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        apply Is_true_eq_left. inversion Heqo2; subst. auto.
      - rewrite Heqo, Heqo0.
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        apply Is_true_eq_true.
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        inversion Heqo1; inversion Heqo2; subst.
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        repeat (apply andb_prop_intro; split); try auto using Is_true_eq_left. }
    assert (validFloatValue (Q2R (B2Q v_e1)) M64).
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    { eapply (FPRangeValidator_sound (B2Qexpr e1)); try eauto; try set_tac.
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      eapply eval_eq_env; eauto. }
    assert (validFloatValue (Q2R (B2Q v_e2)) M64).
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    { eapply (FPRangeValidator_sound (B2Qexpr e2)); try eauto; try set_tac.
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      - eapply eval_eq_env; eauto. }
    assert (is_finite 53 1024 v_e1 = true) as finite_e1.
    { apply validValue_is_finite; simpl; auto. }
    assert (is_finite 53 1024 v_e2 = true) as finite_e2.
    { apply validValue_is_finite; simpl; auto. }
    assert (forall eps,
               (Rabs eps <= / 2 * bpow radix2 (- 53 + 1))%R ->
               validFloatValue
                 (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2) * (1 + eps)) M64).
    { intros.
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      eapply FPRangeValidator_sound with (e:=Binop b (B2Qexpr e1) (B2Qexpr e2)); eauto.
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      - eapply eval_eq_env; eauto.
        eapply Binop_dist' with (delta:=eps); eauto.
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        + simpl in H2. Transparent mTypeToQ. unfold mTypeToQ.
          eapply Rle_trans; eauto.
          simpl. lra.
        + unfold perturb. repeat rewrite B2Q_B2R_eq; try auto.
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      - cbn. Flover_compute.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro; split; try auto using Is_true_eq_left).
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      - cbn. Flover_compute.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro; split; try auto using Is_true_eq_left).
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      - cbn. Flover_compute.
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        inversion Heqo1; inversion Heqo2; subst.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro; split; try auto using Is_true_eq_left).
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      - cbn. Flover_compute.
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        inversion Heqo1; inversion Heqo2; subst.
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        apply Is_true_eq_true.
        repeat (apply andb_prop_intro; split; try auto using Is_true_eq_left). }
    assert (b = Div -> (Q2R (B2Q v_e2)) <> 0%R) as no_div_zero_float.
    { intros; subst; simpl in *.
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      andb_to_prop R3.
      apply le_neq_bool_to_lt_prop in L3.
      destruct L3 as [case_low | case_high]; hnf; intros.
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      - rewrite H19 in *.
        apply Qlt_Rlt in case_low.
        rewrite Q2R0_is_0, Q2R_plus in case_low. lra.
      - rewrite H19 in *.
        apply Qlt_Rlt in case_high.
        rewrite Q2R0_is_0, Q2R_minus in case_high; lra. }
    clear H2 H12 dVars_sound dVars_valid usedVars_64bit vars_typed fVars_defined
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    usedVars_sound R2 R1 L1 L R6 L0 R3 R4 L2 R5 R7 L5 Heqo Heqo0 Heqo1 IHe1
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    IHe2.
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    pose proof (fexp_correct 53 1024 eq_refl) as fexpr_corr.
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    assert (forall k : Z, (-1022 < k)%Z ->
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                     (53 <= k - Fcore_FLT.FLT_exp (3 - 1024 - 53) 53 k)%Z)
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      as expr_valid.
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    { intros k k_pos.
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      unfold Fcore_FLT.FLT_exp; simpl.
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      destruct (Z.max_spec_le (k - 53) (-1074)); omega. }
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    pose proof (relative_error_N_ex radix2 (Fcore_FLT.FLT_exp (3 -1024 - 53) 53)
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                                    (-1022)%Z 53%Z expr_valid)
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      as rel_error_exists.
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    rewrite eval_float_e1, eval_float_e2 in H1.
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    unfold optionBind, normal_or_zero in *; simpl in *.
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    assert (Normal (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2)) M64 \/
            (evalBinop b (B2R 53 1024 v_e1) (B2R 53 1024 v_e2)) = 0)%R.
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    { revert H1; intros case_val.
      destruct case_val as [eval_is_zero | eval_normal]; try auto.
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      left; unfold Normal, Denormal in H15; unfold Normal;
        destruct H15 as [normal_b | [denormal_b |zero_b]].
      - repeat rewrite <- B2Q_B2R_eq; try auto.
      - destruct denormal_b.
        assert ((Rabs (evalBinop b (Q2R (B2Q v_e1)) (Q2R (B2Q v_e2)))) < (Rabs (evalBinop b (Q2R (B2Q v_e1)) (Q2R (B2Q v_e2)))))%R.
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        { eapply Rlt_le_trans; eauto.
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          repeat rewrite B2Q_B2R_eq; auto. }
        lra.
      - rewrite B2Q_B2R_eq in zero_b; auto.
        rewrite B2Q_B2R_eq in zero_b; auto.
        rewrite zero_b in *.
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        rewrite Rabs_R0 in eval_normal.
        unfold minValue_pos, minExponentPos in eval_normal.
        rewrite Q2R_inv in eval_normal; [|vm_compute; congruence].
        unfold Q2R, Qnum, Qden in eval_normal.
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        assert (Z.pow_pos 2 1022 = 44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304%Z)
          by (vm_compute; auto).
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        rewrite H1 in eval_normal. rewrite <- Z2R_IZR in eval_normal.
        unfold IZR in eval_normal.
        simpl in eval_normal. rewrite <- INR_IPR in eval_normal.
        simpl in eval_normal. lra. }
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    pose proof (validValue_bounded b v_e1 v_e2 H2 H18) as cond_valid.
    destruct b; revert H1; intros case_eval.
    (* Addition *)
    + unfold evalBinop in *. unfold b64_plus.
      pose proof (Bplus_correct 53 1024 eq_refl eq_refl binop_nan_pl64 mode_NE
                                v_e1 v_e2 finite_e1 finite_e2)
        as addition_correct.
      rewrite cond_valid in addition_correct.
      destruct addition_correct as [add_round [finite_res _]].
      destruct case_eval as [eval_zero | eval_normal].
      (* resutl is zero *)
      * rewrite eval_zero in *.
        rewrite round_0_zero in *.
        exists (Bplus 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2).
        split; try auto.
        rewrite B2Q_B2R_eq; try auto.
        unfold dmode; rewrite add_round.
        eapply Binop_dist' with (delta:=0%R); eauto.
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        { rewrite Rabs_R0; apply mTypeToR_pos_R. }
        { unfold perturb, evalBinop. cbn.
        repeat rewrite B2Q_B2R_eq; try auto; lra. }
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      * simpl in *.
        destruct (rel_error_exists
                    (fun x => negb (Zeven x))
                    (B2R 53 1024 v_e1 + B2R 53 1024 v_e2)%R)
          as [eps [eps_bounded round_eq]].
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        { eapply Rle_trans; eauto. unfold minValue_pos, minExponentPos.
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          rewrite Q2R_inv; [ | vm_compute; congruence].
          unfold Q2R, Qnum, Qden. rewrite <- Z2R_IZR.
          vm_compute. lra. }
        { exists (Bplus 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2);
            split; try auto.
          rewrite B2Q_B2R_eq; try auto.
          unfold dmode.
          eapply Binop_dist' with (delta:=eps); eauto.
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          - cbn; lra.
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          - unfold perturb, evalBinop.
            repeat rewrite B2Q_B2R_eq; try auto.
            rewrite <- round_eq. rewrite <- add_round; auto. }
    (* Subtraction *)
    + unfold evalBinop in *. unfold b64_minus.
      pose proof (Bminus_correct 53 1024 eq_refl eq_refl binop_nan_pl64 mode_NE
                                v_e1 v_e2 finite_e1 finite_e2)
        as subtraction_correct.
      rewrite cond_valid in subtraction_correct.
      destruct subtraction_correct as [add_round [finite_res _]].
      destruct case_eval as [eval_zero | eval_normal].
      (* resutl is zero *)
      * rewrite eval_zero in *.
        rewrite round_0_zero in *.
        exists (Bminus 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2).
        split; try auto.
        rewrite B2Q_B2R_eq; try auto.
        unfold dmode; rewrite add_round.
        eapply Binop_dist' with (delta:=0%R); eauto.
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        rewrite Rabs_R0; apply mTypeToR_pos_R.
        unfold perturb, evalBinop; cbn.
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        repeat rewrite B2Q_B2R_eq; try auto; lra.
      * simpl in *.
        destruct (rel_error_exists
                    (fun x => negb (Zeven x))
                    (B2R 53 1024 v_e1 - B2R 53 1024 v_e2)%R)
          as [eps [eps_bounded round_eq]].
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        { eapply Rle_trans; eauto. unfold minValue_pos, minExponentPos.
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          rewrite Q2R_inv; [ | vm_compute; congruence].
          unfold Q2R, Qnum, Qden. rewrite <- Z2R_IZR.
          vm_compute. lra. }
        { exists (Bminus 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2);
            split; try auto.
          rewrite B2Q_B2R_eq; try auto.
          unfold dmode.
          eapply Binop_dist' with (delta:=eps); eauto.
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          - cbn; lra.
          - unfold perturb, evalBinop; cbn.
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            repeat rewrite B2Q_B2R_eq; try auto.
            rewrite <- round_eq. rewrite <- add_round; auto. }
    (* Multiplication *)
    + unfold evalBinop in *. unfold b64_mult.
      pose proof (Bmult_correct 53 1024 eq_refl eq_refl binop_nan_pl64 mode_NE
                                v_e1 v_e2)
        as mult_correct.
      rewrite cond_valid in mult_correct.
      destruct mult_correct as [mult_round [finite_res _]].
      destruct case_eval as [eval_zero | eval_normal].
      (* resutl is zero *)
      * rewrite eval_zero in *.
        rewrite round_0_zero in *.
        exists (Bmult 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2).
        split; try auto.
        rewrite B2Q_B2R_eq; try auto.
        unfold dmode; rewrite mult_round.
        eapply Binop_dist' with (delta:=0%R); eauto.
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        rewrite Rabs_R0; apply mTypeToR_pos_R.
        unfold perturb, evalBinop; cbn.
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        repeat rewrite B2Q_B2R_eq; try auto; lra.
        rewrite finite_e1, finite_e2 in finite_res.
        auto.
      * simpl in *.
        destruct (rel_error_exists
                    (fun x => negb (Zeven x))
                    (B2R 53 1024 v_e1 * B2R 53 1024 v_e2)%R)
          as [eps [eps_bounded round_eq]].
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        { eapply Rle_trans; eauto. unfold minValue_pos, minExponentPos.
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          rewrite Q2R_inv; [ | vm_compute; congruence].
          unfold Q2R, Qnum, Qden. rewrite <- Z2R_IZR.
          vm_compute. lra. }
        { exists (Bmult 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2);
            split; try auto.
          rewrite B2Q_B2R_eq; try auto.
          unfold dmode.
          eapply Binop_dist' with (delta:=eps); eauto.
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          - cbn; lra.
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          - unfold perturb, evalBinop.
            repeat rewrite B2Q_B2R_eq; try auto.
            rewrite <- round_eq. rewrite <- mult_round; auto.
          - rewrite finite_e1, finite_e2 in finite_res; auto. }
    (* Division *)
    + unfold evalBinop in *. unfold b64_div.
      pose proof (Bdiv_correct 53 1024 eq_refl eq_refl binop_nan_pl64 mode_NE
                                v_e1 v_e2)
        as division_correct.
      rewrite cond_valid in division_correct.
      destruct division_correct as [div_round [finite_res _]].
      rewrite <- B2Q_B2R_eq; auto.
      destruct case_eval as [eval_zero | eval_normal].
      (* resutl is zero *)
      * rewrite eval_zero in *.
        rewrite round_0_zero in *.
        exists (Bdiv 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2).
        split; try auto.
        rewrite B2Q_B2R_eq; try auto.
        unfold dmode; rewrite div_round.
        eapply Binop_dist' with (delta:=0%R); eauto.
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        rewrite Rabs_R0; apply mTypeToR_pos_R.
        unfold perturb, evalBinop; cbn.
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        repeat rewrite B2Q_B2R_eq; try auto; lra.
        rewrite finite_e1 in finite_res; auto.
      * simpl in *.
        destruct (rel_error_exists
                    (fun x => negb (Zeven x))
                    (B2R 53 1024 v_e1 / B2R 53 1024 v_e2)%R)
          as [eps [eps_bounded round_eq]].
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        { eapply Rle_trans; eauto. unfold minValue_pos, minExponentPos.
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          rewrite Q2R_inv; [ | vm_compute; congruence].
          unfold Q2R, Qnum, Qden. rewrite <- Z2R_IZR.
          vm_compute. lra. }
        { exists (Bdiv 53 1024 eq_refl eq_refl binop_nan_pl64 dmode v_e1 v_e2);
            split; try auto.
          rewrite B2Q_B2R_eq; try auto.
          unfold dmode.
          eapply Binop_dist' with (delta:=eps); eauto.
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          - cbn; lra.
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          - unfold perturb, evalBinop.
            repeat rewrite B2Q_B2R_eq; try auto.
            rewrite <- round_eq. rewrite <- div_round; auto.
          - rewrite finite_e1 in finite_res; auto. }
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  - repeat (match goal with
            |H: _ /\ _ |- _ => destruct H
            end).
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    assert (FloverMap.find (B2Qexpr e1) tMap = Some M64 /\
            FloverMap.find (B2Qexpr e2) tMap = Some M64 /\
            FloverMap.find (B2Qexpr e3) tMap = Some M64 /\
            FloverMap.find (Fma (B2Qexpr e1) (B2Qexpr e2) (B2Qexpr e3)) tMap = Some M64)
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           as [tMap_e1 [tMap_e2 [tMap_e3 tMap_fma]]].
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    { repeat split; apply (typing_expr_64_bit _ Gamma); simpl; auto.
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      - intros; apply usedVars_64bit; set_tac.
      - intros; apply usedVars_64bit; set_tac.
      - intros; apply usedVars_64bit; set_tac.
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      - rewrite Heqo, Heqo4, Heqo6, Heqo8.
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        apply Is_true_eq_true; apply andb_prop_intro; split.
        + apply andb_prop_intro; split.
          * apply andb_prop_intro; split.
            ++ apply Is_true_eq_left; auto.
               apply mTypeEq_refl.
            ++ apply Is_true_eq_left; auto.
          * apply Is_true_eq_left; auto.
        + apply Is_true_eq_left; auto. }
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    repeat destr_factorize.
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    inversion Heqo; inversion Heqo0; inversion Heqo1; inversion Heqo2; subst.
    assert (m1 = M64).
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    { eapply (typing_agrees_expr (B2Qexpr e1)); eauto. }
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    assert (m2 = M64).
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    { eapply (typing_agrees_expr (B2Qexpr e2)); eauto. }
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    assert (m3 = M64).
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    { eapply (typing_agrees_expr (B2Qexpr e3)); eauto. }
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    subst.
    destruct (IHe1 E1 E2 E2_real Gamma tMap v1 A P fVars dVars)
      as [v_e1 [eval_float_e1 eval_rel_e1]];
      try auto; try set_tac;
        [ intros; apply usedVars_64bit ; set_tac | ].
    destruct (IHe2 E1 E2 E2_real Gamma tMap v2 A P fVars dVars)
      as [v_e2 [eval_float_e2 eval_rel_e2]];
      try auto; try set_tac;
        [ intros; apply usedVars_64bit ; set_tac | ].
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    destruct (IHe3 E1 E2 E2_real Gamma tMap v3 A P fVars dVars)
      as [v_e3 [eval_float_e3 eval_rel_e3]];
      try auto; try set_tac;
        [ intros; apply usedVars_64bit ; set_tac | ].
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    unfold optionBind in H4.
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    rewrite eval_float_e1, eval_float_e2, eval_float_e3 in H4.
    contradiction H4.
  - inversion noDowncast_e.
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Qed.

Lemma bstep_gives_IEEE (f:cmd fl64) :
  forall E1 E2 E2_real Gamma tMap vR vF A P fVars dVars outVars,
    (forall x, (toREnv E2) x = E2_real x) ->
    approxEnv E1 Gamma A fVars dVars E2_real ->
    ssa (B2Qcmd f) (NatSet.union fVars dVars) outVars ->
    typeCheckCmd (B2Qcmd f) Gamma tMap = true ->
    validIntervalboundsCmd (B2Qcmd f) A P dVars = true ->
    validErrorboundCmd (B2Qcmd f) tMap A dVars = true ->
    FPRangeValidatorCmd (B2Qcmd f) A tMap dVars = true ->
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    bstep (toREvalCmd (toRCmd (B2Qcmd f))) E1 (toRMap Gamma) vR REAL ->
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    bstep (toRCmd (B2Qcmd f)) (toREnv E2) Gamma vF M64 ->
    NatSet.Subset (NatSet.diff (freeVars (B2Qcmd f)) dVars) fVars ->
    is64BitBstep (B2Qcmd f) ->
    noDowncastFun (B2Qcmd f) ->
    bstep_valid f E2 ->
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    dVars_range_valid dVars E1 A ->
    fVars_P_sound fVars E1 P ->
    vars_typed (NatSet.union fVars dVars) Gamma ->
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      (forall v,
        NatSet.In v dVars ->
        exists vF m,
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        (E2_real v = Some vF /\ FloverMap.find (Var Q v) tMap= Some m /\
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        validFloatValue vF m)) ->
      (forall v, NatSet.In v (freeVars (B2Qcmd f)) -> Gamma v = Some M64) ->
      exists v,
        bstep_float f E2 = Some v /\
        bstep (toRCmd (B2Qcmd f)) (toREnv E2) Gamma (Q2R (B2Q v)) M64.
Proof.
  induction f;
    intros * envs_eq approxEnv_E1_E2_real ssa_f typeCheck_f valid_ranges_f
                     valid_roundoffs_f valid_float_ranges bstep_real bstep_float
                     freeVars_sound is64_eval nodowncast_f bstep_sound
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                     dVars_sound fVars_defined vars_typed dVars_valid
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                     freeVars_typed;
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    cbn in *;
    revert bstep_sound;
    Flover_compute_asm; try congruence; type_conv; subst;
      intros bstep_sound;
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      unfold Ltacs.optionBind;
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      inversion bstep_float; inversion bstep_real;
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      inversion ssa_f; subst; simpl in *;
        repeat (match goal with
                | H: _ = true |- _ => andb_to_prop H
                end).
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  - assert (FloverMap.find (B2Qexpr e) tMap= Some M64).
    { eapply typing_expr_64_bit; try eauto.
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      simpl in *; destruct nodowncast_f; auto.
      destruct is64_eval; auto.
      intros; apply freeVars_typed.
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      set_tac.
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      split; [ set_tac | ].
      hnf; intros; subst.
      apply H26.
      apply (H25 n H).  }
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    assert (m1 = M64).
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    { eapply typing_agrees_expr; eauto. }
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    subst.
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    assert (exists v_e, eval_expr_float e E2 = Some v_e /\
                   eval_expr (toREnv E2) Gamma (toRExp (B2Qexpr e)) (Q2R (B2Q v_e)) M64)
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           as eval_float_e.
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    { eapply eval_expr_gives_IEEE; try eauto.
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      - hnf; intros. rewrite NatSet.diff_spec in H0.
        destruct H0.
        specialize (H25 a H0). rewrite NatSet.union_spec in H25.
        destruct H25; try congruence; auto.
      - destruct is64_eval; auto.
      - destruct nodowncast_f; auto.
      - destruct bstep_sound; auto.
      - intros. apply freeVars_typed.
        rewrite NatSet.remove_spec, NatSet.union_spec.
        split; try auto.
        hnf; intros; subst.
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        specialize (H25 n H0).
        set_tac.  apply H26; set_tac. }
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    destruct eval_float_e as [v_e [eval_float_e eval_rel_e]].
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    assert (forall v m, eval_expr E2_real Gamma (toRExp (B2Qexpr e)) v m ->
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                   Rabs (v0 - v) <= Q2R e2)%R
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      as err_e_valid.
    { eapply validErrorbound_sound; try eauto.
      - hnf; intros. rewrite NatSet.diff_spec in H0.
        destruct H0. specialize (H25 a H0). rewrite NatSet.union_spec in H25.
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        destruct H25; try auto; congruence. }
    assert (Rabs (v0 - (Q2R (B2Q v_e))) <= Q2R e2)%R.
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    { eapply err_e_valid. eapply eval_eq_env; eauto. }
    (* Now construct a new evaluation according to our big-step semantics
       using lemma validErrorboundCmd_gives_eval *)
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    (* destruct (A (getRetExp (B2Qcmd f))) as [iv_f err_f] eqn:A_f. *)
    (* destruct iv_f as [ivlo_f ivhi_f]. *)