AffineValidation.v 58.3 KB
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Require Import Coq.QArith.QArith Coq.QArith.Qreals QArith.Qminmax Coq.Lists.List Coq.micromega.Psatz.
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Require Import Recdef.
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Require Import Flover.AffineForm Flover.AffineArithQ Flover.AffineArith.
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Require Import Flover.Infra.Abbrevs Flover.Infra.RationalSimps Flover.Infra.RealRationalProps.
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Require Import Flover.Infra.Ltacs Flover.Infra.RealSimps Flover.Typing Flover.ssaPrgs.
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Require Import Flover.IntervalValidation.
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Definition updateExpMapIncr e new_af noise (emap: expressionsAffine) intv incr :=
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  let new_iv := toIntv new_af in
  if isSupersetIntv new_iv intv then
    Some (FloverMap.add e new_af emap, (noise + incr)%nat)
  else None.

Definition updateExpMap e af noise emap intv :=
  updateExpMapIncr e af noise emap intv 0.

Definition updateExpMapSucc e af noise emap intv :=
  updateExpMapIncr e af noise emap intv 1.

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Definition nozeroiv iv :=
  ((Qleb (ivhi iv) 0) && (negb (Qeq_bool (ivhi iv) 0))) ||
            ((Qleb 0 (ivlo iv)) && (negb (Qeq_bool (ivlo iv) 0))).

Fixpoint validAffineBounds (e: exp Q) (A: analysisResult) (P: precond) (validVars: NatSet.t)
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           (exprsAf: expressionsAffine) (currentMaxNoise: nat): option (expressionsAffine * nat) :=
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  (* match FloverMap.find e exprsAf with *)
  (* | Some _ => *)
  (*   (* expression has already been checked; we do not want to introduce *) *)
  (*   (*      a new affine polynomial for the same expression *) *)
  (*   Some (exprsAf, currentMaxNoise) *)
  (* | None => *)
  (*   (* We see it for the first time; update the expressions map *) *)
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    olet ares := FloverMap.find e A in
    let (intv, _) := ares in
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    match e with
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    | Var _ v =>
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      if NatSet.mem v validVars then
        Some (exprsAf, currentMaxNoise)
      else
        let af := fromIntv (P v) currentMaxNoise in
        if isSupersetIntv (toIntv af) intv then
          Some (FloverMap.add e af exprsAf, (currentMaxNoise + 1)%nat)
        else None
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    | Const _ c => if isSupersetIntv (c, c) intv then
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                    let af := fromIntv (c,c) currentMaxNoise in
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                    Some (FloverMap.add e af exprsAf, currentMaxNoise)
                  else None
    | Unop o e' =>
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      olet valid := validAffineBounds e' A P validVars exprsAf currentMaxNoise in
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      let (exprsAf', n') := valid in
      olet af := FloverMap.find e' exprsAf' in
      match o with 
      | Neg =>
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        updateExpMap e (AffineArithQ.negate_aff af) n' exprsAf' intv
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      | Inv =>
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        let iv := toIntv af in
        if nozeroiv iv
        then
          updateExpMapSucc e (AffineArithQ.inverse_aff af n') n' exprsAf' intv
        else None
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      end
    | Binop o e1 e2 =>
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      olet valid1 := validAffineBounds e1 A P validVars exprsAf currentMaxNoise in
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      let (exprsAf1, n1) := valid1 in
      olet af1 := FloverMap.find e1 exprsAf1 in
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      olet valid2 := validAffineBounds e2 A P validVars exprsAf1 n1 in
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      let (exprsAf2, n2) := valid2 in
      olet af2 := FloverMap.find e2 exprsAf2 in
      match o with
      | Plus =>
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        updateExpMap e (AffineArithQ.plus_aff af1 af2) n2 exprsAf2 intv
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      | Sub =>
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        updateExpMap e (AffineArithQ.subtract_aff af1 af2) n2 exprsAf2 intv
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      | Mult =>
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        updateExpMapSucc e (AffineArithQ.mult_aff af1 af2 n2) n2 exprsAf2 intv
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      | Div =>
          let iv2 := toIntv af2 in
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          if nozeroiv iv2
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          then
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            updateExpMapIncr e (AffineArithQ.divide_aff af1 af2 n2) n2 exprsAf2 intv 2
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          else None
      end
    | Fma e1 e2 e3 =>
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      olet valid1 := validAffineBounds e1 A P validVars exprsAf currentMaxNoise in
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      let (exprsAf1, n1) := valid1 in
      olet af1 := FloverMap.find e1 exprsAf1 in
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      olet valid2 := validAffineBounds e2 A P validVars exprsAf1 n1 in
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      let (exprsAf2, n2) := valid2 in
      olet af2 := FloverMap.find e2 exprsAf2 in
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      olet valid3 := validAffineBounds e3 A P validVars exprsAf2 n2 in
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      let (exprsAf3, n3) := valid3 in
      olet af3 := FloverMap.find e3 exprsAf3 in
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        updateExpMapSucc e (AffineArithQ.plus_aff af1 (AffineArithQ.mult_aff af2 af3 n3)) n3 exprsAf3 intv
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    | Downcast _ e' => 
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      olet valid' := validAffineBounds e' A P validVars exprsAf currentMaxNoise in
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      let (exprsAf', n') := valid' in
      olet asubres := FloverMap.find e' A in
      let (iv, _) := asubres in
      olet af' := FloverMap.find e' exprsAf' in
      if (isSupersetIntv intv iv) && (isSupersetIntv iv intv) then
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        Some (FloverMap.add e af' exprsAf', n')
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      else None
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    (* end *)
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  end.
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Fixpoint afQ2R (af: affine_form Q): affine_form R := match af with
| AffineForm.Const c => AffineForm.Const (Q2R c)
| Noise n v af' => Noise n (Q2R v) (afQ2R af')
end.

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Lemma afQ2R_const v:
  afQ2R (AffineForm.Const v) = AffineForm.Const (Q2R v).
Proof.
  trivial.
Qed.

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Lemma afQ2R_get_const a:
  get_const (afQ2R a) = Q2R (get_const a).
Proof.
  induction a; auto.
Qed.

Lemma afQ2R_radius a:
  radius (afQ2R a) = Q2R (AffineArithQ.radius a).
Proof.
  induction a; try (simpl; lra).
  simpl.
  rewrite Q2R_plus.
  rewrite Rabs_eq_Qabs.
  now f_equal.
Qed.

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Lemma afQ2R_get_max_index a:
  get_max_index (afQ2R a) = get_max_index a.
Proof.
  unfold get_max_index.
  functional induction (get_max_index_aux 0 a); try auto;simpl; rewrite e0; auto.
Qed.

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Lemma to_interval_to_intv a:
  (Q2R (fst (toIntv a)), Q2R (snd (toIntv a))) = toInterval (afQ2R a).
Proof.
  unfold toIntv, toInterval.
  unfold mkInterval, mkIntv.
  simpl fst; simpl snd.
  rewrite afQ2R_get_const.
  rewrite afQ2R_radius.
  f_equal; try apply Q2R_minus; try apply Q2R_plus.
Qed.
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Lemma afQ2R_plus_aff af1 af2:
  afQ2R (AffineArithQ.plus_aff af1 af2) = plus_aff (afQ2R af1) (afQ2R af2).
Proof.
  unfold AffineArithQ.plus_aff, plus_aff.
  remember (af1, af2) as a12.
  assert (fst a12 = af1 /\ snd a12 = af2) as Havals by now rewrite Heqa12.
  destruct Havals as [Heqa1 Heqa2].
  rewrite <- Heqa1, <- Heqa2.
  clear Heqa1 Heqa2 Heqa12 af1 af2.
  functional induction (AffineArithQ.plus_aff_tuple a12); simpl; rewrite plus_aff_tuple_equation.
  - f_equal; apply Q2R_plus.
  - f_equal.
    assumption.
  - f_equal.
    assumption.
  - rewrite e2.
    f_equal; try apply Q2R_plus.
    assumption.
  - rewrite e2.
    rewrite e3.
    f_equal; try apply Q2R_plus.
    assumption.
  - rewrite e2.
    rewrite e3.
    f_equal; try apply Q2R_plus.
    assumption.
Qed.

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Lemma afQ2R_mult_aff_aux af1 af2:
  afQ2R (AffineArithQ.mult_aff_aux (af1, af2)) = mult_aff_aux (afQ2R af1, afQ2R af2).
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Proof.
  unfold AffineArithQ.mult_aff, mult_aff.
  remember (af1, af2) as a12.
  assert (fst a12 = af1 /\ snd a12 = af2) as Havals by now rewrite Heqa12.
  destruct Havals as [Heqa1 Heqa2].
  rewrite <- Heqa1, <- Heqa2.
  clear Heqa1 Heqa2 Heqa12 af1 af2.
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  functional induction (AffineArithQ.mult_aff_aux a12); simpl in *;
    rewrite mult_aff_aux_equation;
    try (f_equal; try apply Q2R_mult; assumption).
  {
    rewrite e2.
    f_equal; try assumption.
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    simpl.
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    rewrite Q2R_plus; do 2 rewrite Q2R_mult.
    do 2 rewrite afQ2R_get_const.
    reflexivity.
  }
  all: rewrite e2; rewrite e3.
  all: f_equal; try assumption.
  all: simpl.
  all: rewrite Q2R_mult.
  all: rewrite afQ2R_get_const.
  all: reflexivity.
Qed.

Lemma afQ2R_mult_aff af1 af2 n:
  afQ2R (AffineArithQ.mult_aff af1 af2 n) = mult_aff (afQ2R af1) (afQ2R af2) n.
Proof.
  unfold AffineArithQ.mult_aff, mult_aff.
  destruct (Qeq_bool (AffineArithQ.radius af1) 0) eqn: Heq.
  - rewrite orb_true_l.
    rewrite Qeq_bool_iff in Heq.
    apply Qeq_eqR in Heq.
    rewrite <- afQ2R_radius in Heq.
    rewrite Q2R0_is_0 in Heq.
    destruct Req_dec_sum as [Heq' | Heq']; rewrite Heq in Heq'; try lra.
    apply afQ2R_mult_aff_aux.
  - rename Heq into Heq1.
    destruct (Qeq_bool (AffineArithQ.radius af2) 0) eqn: Heq2.
    + rewrite orb_true_r.
      rewrite Qeq_bool_iff in Heq2.
      apply Qeq_eqR in Heq2.
      rewrite <- afQ2R_radius in Heq2.
      rewrite Q2R0_is_0 in Heq2.
      destruct Req_dec_sum as [Heq' | Heq']; rewrite Heq2 in Heq'; try lra.
      apply afQ2R_mult_aff_aux.
    + apply RMicromega.Qeq_false in Heq1.
      apply RMicromega.Qeq_false in Heq2.
      rewrite Q2R0_is_0 in Heq1.
      rewrite Q2R0_is_0 in Heq2.
      rewrite <- afQ2R_radius in Heq1.
      rewrite <- afQ2R_radius in Heq2.
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      simpl.
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      destruct Req_dec_sum as [Heq' | Heq'];
        try (apply Rmult_integral in Heq'; destruct Heq'; try lra).
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      f_equal.
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      * rewrite Q2R_mult.
        do 2 rewrite afQ2R_radius.
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        reflexivity.
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      * apply afQ2R_mult_aff_aux.
Qed.
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Lemma afQ2R_negate_aff af:
  afQ2R (AffineArithQ.negate_aff af) = negate_aff (afQ2R af).
Proof.
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  unfold AffineArithQ.negate_aff.
  unfold AffineArithQ.mult_aff_const.
  rewrite afQ2R_mult_aff.
  simpl.
  unfold negate_aff.
  unfold mult_aff_const.
  f_equal.
  - f_equal.
    rewrite Q2R_opp.
    lra.
  - f_equal.
    unfold get_max_index.
    functional induction (get_max_index_aux 0 af); try auto; simpl; now rewrite e0.
Qed.

Lemma afQ2R_subtract_aff af1 af2:
  afQ2R (AffineArithQ.subtract_aff af1 af2) = subtract_aff (afQ2R af1) (afQ2R af2).
Proof.
  unfold AffineArithQ.subtract_aff.
  rewrite afQ2R_plus_aff.
  rewrite afQ2R_negate_aff.
  reflexivity.
Qed.

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Lemma afQ2R_inverse_aff af n:
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  above_zero (afQ2R af) \/ below_zero (afQ2R af) ->
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  afQ2R (AffineArithQ.inverse_aff af n) = inverse_aff (afQ2R af) n.
Proof.
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  intros above_below.
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  unfold AffineArithQ.inverse_aff.
  unfold inverse_aff.
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  unfold above_zero, below_zero in above_below.
  unfold ivhi, IVhi, ivlo, IVlo in *.
  rewrite <- to_interval_to_intv in above_below.
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  rewrite <- to_interval_to_intv.
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  simpl fst in above_below |-*.
  simpl snd in above_below |-*.
  assert (AffineArithQ.radius af >= 0) by eauto using AffineArithQ.radius_nonneg.
  assert (get_const af - AffineArithQ.radius af <= get_const af + AffineArithQ.radius af) as Hrel by lra.
  replace 0%R with (Q2R 0) in above_below by lra.
  destruct above_below as [Heq | Heq]; apply Rlt_Qlt in Heq.
  - assert (get_const af + AffineArithQ.radius af > 0) as Hposhi by lra.
    destruct (Qlt_bool (get_const af + AffineArithQ.radius af) 0) eqn: H'';
      try rewrite Qlt_bool_iff in H''; try lra.
    apply Qlt_Rlt in Hposhi.
    destruct (Rlt_dec (Q2R (get_const af + AffineArithQ.radius af)) 0); try lra.
    apply Rlt_Qlt in Hposhi.
    assert (~ get_const af - AffineArithQ.radius af == 0) as minus_nonzero by lra.
    assert (~ get_const af + AffineArithQ.radius af == 0) as plus_nonzero by lra.
    assert (~ -(get_const af - AffineArithQ.radius af) == 0) as minus_nonzero' by lra.
    assert (~ -(get_const af + AffineArithQ.radius af) == 0) as plus_nonzero' by lra.
    assert (~ maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) *
            maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Q.max_r; try lra.
      try rewrite Q.min_l; try lra.
      unfold not.
      intros H'.
      apply Qmult_integral in H'; lra.
    }
    assert (~ maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Q.max_r; try lra.
    }
    assert (~ minAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Qabs.Qabs_pos; try lra.
      try rewrite Q.max_r; try lra.
      try rewrite Q.min_l; try lra.
    }
    simpl.
    f_equal; unfold toIntv; simpl;
      repeat rewrite minAbs_impl_RminAbs; repeat rewrite maxAbs_impl_RmaxAbs;
      replace 1%R with (Q2R (1%Q)) by lra;
      replace 2%R with (Q2R (2#1)) by lra;
      repeat (repeat rewrite <- Q2R_plus; repeat rewrite <- Q2R_mult; repeat rewrite <- Q2R_div;
              repeat rewrite <- Q2R_minus; repeat rewrite <- Q2R_opp); try lra.
      unfold mult_aff_const, plus_aff_const.
      unfold AffineArithQ.mult_aff_const, AffineArithQ.plus_aff_const.
      repeat rewrite <- afQ2R_const.
      rewrite <- afQ2R_mult_aff.
      rewrite <- afQ2R_plus_aff.
      repeat f_equal.
      now rewrite afQ2R_get_max_index.
  - rewrite <- Qlt_bool_iff in Heq.
    rewrite Heq.
    rewrite Qlt_bool_iff in Heq.
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    apply Qlt_Rlt in Heq.
    destruct (Rlt_dec (Q2R (get_const af + AffineArithQ.radius af)) 0); try lra.
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    apply Rlt_Qlt in Heq.
    assert (get_const af - AffineArithQ.radius af < 0) as minus_neg by lra.
    assert (~ get_const af - AffineArithQ.radius af == 0) as minus_nonzero by lra.
    assert (~ get_const af + AffineArithQ.radius af == 0) as plus_nonzero by lra.
    assert (~ -(get_const af - AffineArithQ.radius af) == 0) as minus_nonzero' by lra.
    assert (~ -(get_const af + AffineArithQ.radius af) == 0) as plus_nonzero' by lra.
    assert (~ maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) *
            maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_neg; try lra.
      try rewrite Qabs.Qabs_neg; try lra.
      try rewrite Q.max_l; try lra.
      try rewrite Q.min_r; try lra.
      unfold not.
      intros H'.
      apply Qmult_integral in H'; lra.
    }
    assert (~ maxAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_neg; try lra.
      try rewrite Qabs.Qabs_neg; try lra.
      rewrite Q.max_l; try lra.
    }
    assert (~ minAbs (get_const af - AffineArithQ.radius af, get_const af + AffineArithQ.radius af) == 0) as ?.
    {
      unfold maxAbs.
      unfold minAbs.
      simpl fst.
      simpl snd.
      try rewrite Qabs.Qabs_neg; try lra.
      try rewrite Qabs.Qabs_neg; try lra.
      try rewrite Q.max_l; try lra.
      try rewrite Q.min_r; try lra.
    }
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    simpl.
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    f_equal; unfold toIntv; simpl;
      repeat rewrite minAbs_impl_RminAbs; repeat rewrite maxAbs_impl_RmaxAbs;
      replace 1%R with (Q2R (1%Q)) by lra;
      replace 2%R with (Q2R (2#1)) by lra;
      repeat (repeat rewrite <- Q2R_plus; repeat rewrite <- Q2R_mult; repeat rewrite <- Q2R_div;
              repeat rewrite <- Q2R_minus; repeat rewrite <- Q2R_opp); try lra.
      unfold mult_aff_const, plus_aff_const.
      unfold AffineArithQ.mult_aff_const, AffineArithQ.plus_aff_const.
      repeat rewrite <- afQ2R_const.
      rewrite <- afQ2R_mult_aff.
      rewrite <- afQ2R_plus_aff.
      repeat f_equal.
      now rewrite afQ2R_get_max_index.
Qed.
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Lemma afQ2R_divide_aff af1 af2 n:
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  above_zero (afQ2R af2) \/ below_zero (afQ2R af2) ->
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  afQ2R (AffineArithQ.divide_aff af1 af2 n) = divide_aff (afQ2R af1) (afQ2R af2) n.
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Proof.
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  intros.
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  unfold AffineArithQ.divide_aff.
  rewrite afQ2R_mult_aff.
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  rewrite afQ2R_inverse_aff; auto.
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Qed.
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Lemma afQ2R_fresh n a:
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  fresh n a <-> fresh n (afQ2R a).
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Proof.
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  split; induction a; intros *. 
  all: try (unfold fresh, get_max_index; rewrite get_max_index_aux_equation; now simpl).
  all: intros A.
  all: remember A as A' eqn:tmp; clear tmp.
  all: apply fresh_noise_gt in A.
  all: apply fresh_noise_compat in A'.
  all: specialize (IHa A').
  all: apply fresh_noise; assumption.
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Qed.

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Lemma nozero_above_below af:
  nozeroiv (toIntv af) = true ->
  above_zero (afQ2R af) \/ below_zero (afQ2R af).
Proof.
  intros noz % orb_prop.
  destruct noz as [below | above].
  - right.
    unfold below_zero.
    apply andb_prop in below as [below notz].
    rewrite negb_true_iff in notz.
    apply Qeq_bool_neq in notz.
    rewrite Qle_bool_iff in below.
    assert (IVhi (toInterval (afQ2R af)) = Q2R (ivhi (toIntv af))) as Heq
        by (rewrite <- to_interval_to_intv; trivial).
    rewrite Heq.
    assert (ivhi (toIntv af) < 0) as Hlt by lra.
    apply Qlt_Rlt in Hlt.
    lra.
  - left.
    unfold above_zero.
    apply andb_prop in above as [above notz].
    rewrite negb_true_iff in notz.
    apply Qeq_bool_neq in notz.
    rewrite Qle_bool_iff in above.
    assert (IVlo (toInterval (afQ2R af)) = Q2R (ivlo (toIntv af))) as Heq
        by (rewrite <- to_interval_to_intv; trivial).
    rewrite Heq.
    assert (ivlo (toIntv af) > 0) as Hlt by lra.
    apply Qlt_Rlt in Hlt.
    lra.
Qed.

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Definition affine_dVars_range_valid (dVars: NatSet.t) (E: env) (A: analysisResult) noise exprAfs map1: Prop :=
  forall v, NatSet.In v dVars ->
       exists af vR iv err,
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         isSupersetIntv (toIntv af) iv = true /\
         FloverMap.find (elt:=affine_form Q) (Var Q v) exprAfs = Some af /\
         fresh noise af /\
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         (forall n, (n >= noise)%nat -> map1 n = None) /\
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         FloverMap.find (Var Q v) A = Some (iv, err) /\
         E v = Some vR /\
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         af_evals (afQ2R af) vR map1.
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Definition affine_fVars_P_sound (fVars:NatSet.t) (E:env) (P:precond) :Prop :=
  forall v,
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    NatSet.In v fVars ->
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    exists vR, E v = Some vR /\
          (Q2R (fst (P v)) <= vR <= Q2R (snd (P v)))%R.
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Definition affine_vars_typed (S: NatSet.t) (Gamma: nat -> option mType) :=
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  forall v, NatSet.In v S ->
       exists m: mType, Gamma v = Some m.

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Definition checked_expressions (A: analysisResult) E Gamma iexpmap map1 inoise :=
      forall (e: exp Q),
          (exists (af: affine_form Q), FloverMap.find e iexpmap = Some af) ->
          exists af vR aiv aerr,
            FloverMap.find e A = Some (aiv, aerr) /\
            isSupersetIntv (toIntv af) aiv = true /\
            FloverMap.find e iexpmap = Some af /\
            fresh inoise af /\
            (forall n, (n >= inoise)%nat -> map1 n = None) /\
            eval_exp E (toRMap Gamma) (toREval (toRExp e)) vR M0 /\
            af_evals (afQ2R af) vR map1.
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Lemma validAffineBounds_sound (e: exp Q) (A: analysisResult) (P: precond)
      fVars dVars (E: env) Gamma exprAfs noise iexpmap inoise map1: 
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      (* checked_expressions A E Gamma iexpmap -> *)
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      (inoise > 0)%nat ->
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      (forall n, (n >= inoise)%nat -> map1 n = None) ->
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      validAffineBounds e A P dVars iexpmap inoise = Some (exprAfs, noise) ->
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      affine_dVars_range_valid dVars E A inoise iexpmap map1 ->
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      NatSet.Subset (NatSet.diff (Expressions.usedVars e) dVars) fVars ->
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      affine_fVars_P_sound fVars E P ->
      affine_vars_typed (NatSet.union fVars dVars) Gamma ->
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      exists map2 af vR aiv aerr,
        contained_map map1 map2 /\
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        contained_flover_map iexpmap exprAfs /\
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        FloverMap.find e A = Some (aiv, aerr) /\
        isSupersetIntv (toIntv af) aiv = true /\
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        FloverMap.find e exprAfs = Some af /\
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        fresh noise af /\
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        (forall n, (n >= noise)%nat -> map2 n = None) /\
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        (noise >= inoise)%nat /\
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        eval_exp E (toRMap Gamma) (toREval (toRExp e)) vR M0 /\
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        af_evals (afQ2R af) vR map2.
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Proof.
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  revert noise exprAfs inoise iexpmap map1.
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  induction e;
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    intros * inoisegtz validmap1 validBounds dVarsValid varsDisjoint fVarsSound varsTyped;
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    simpl in validBounds.
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  -
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    specialize (dVarsValid n).
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    specialize (fVarsSound n).
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    specialize (varsTyped n).
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    (* specialize (visitedExpr (Var Q n) map1 exprAfs noise inoise). *)
    (* destruct (FloverMap.find (elt:=affine_form Q) (Var Q n) iexpmap) eqn: Hvisited; eauto using visitedExpr. *)
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    destruct (FloverMap.find (elt:=intv * error) (Var Q n) A) as [p | p] eqn: Hares; simpl in validBounds; try congruence.
    destruct p as [aiv aerr].
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    destruct (n mem dVars) eqn: Hmem.
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    + rewrite NatSet.mem_spec in Hmem.
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      specialize (dVarsValid Hmem).
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      assert (n  fVars  dVars) as H by intuition.
      specialize (varsTyped H) as [m varsTyped].
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      destruct dVarsValid as [af [vR [iv [err dVarsValid]]]]; try reflexivity.
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      inversion validBounds; subst; clear validBounds.
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      exists map1, af, vR, iv, err.
      intuition; try reflexivity.
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      constructor; auto.
      simpl.
      rewrite varsTyped.
      reflexivity.
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    + destruct (isSupersetIntv (toIntv (fromIntv (P n) inoise)) aiv) eqn: Hsup; try congruence.
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      inversion validBounds; subst; clear validBounds.
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      apply not_in_not_mem in Hmem.
      assert (n  fVars  dVars) as H by intuition.
      specialize (varsTyped H) as [m varsTyped].
      assert (n  fVars) as H' by intuition.
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      specialize (fVarsSound H') as [vR [eMap interval_containment]].
      destruct (Qeq_bool (ivlo (P n)) (ivhi (P n))) eqn: Heq.
      * exists map1, (fromIntv (P n) inoise), vR, aiv, aerr.
        repeat split; auto.
        -- reflexivity.
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        -- apply contained_flover_map_extension.
           admit.
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        -- rewrite FloverMapFacts.P.F.add_eq_o; try auto.
           apply Q_orderedExps.expCompare_refl.
        -- unfold fresh, fromIntv, get_max_index.
           rewrite Heq.
           simpl; lia.
        -- intros n' Hn'.
           apply validmap1.
           lia.
        -- lia.
        -- constructor; auto.
           simpl.
           rewrite varsTyped.
           reflexivity.
        -- assert (fromIntv (P n) inoise = (AffineForm.Const (ivhi (P n) / (2 # 1) + ivlo (P n) / (2 # 1))%Q)) as HfromIntv
              by (unfold fromIntv; now rewrite Heq).
           pose proof Heq as Heq'.
           apply Qeq_bool_iff in Heq'.
           simpl in Heq'.
           apply Qeq_eqR in Heq'.
           rewrite Heq' in interval_containment.
           assert (vR = Q2R (snd (P n))) as HvR by lra.
           rewrite HfromIntv.
           unfold af_evals.
           simpl.
           rewrite Q2R_plus.
           repeat rewrite Q2R_div by lra.
           rewrite Heq'.
           rewrite HvR.
           lra.
      * assert (exists q, af_evals (afQ2R (fromIntv (P n) inoise)) vR (updMap map1 inoise q))
          as [q Hevals].
        {
          unfold af_evals, fromIntv.
          rewrite Heq.
          apply Qeq_bool_neq in Heq.
          simpl in Heq.
          simpl.
          setoid_rewrite upd_sound.
          simpl.
          apply Q.max_case_strong.
          - intros x y Hxy.
            apply Qeq_eqR in Hxy.
            rewrite Hxy.
            auto.
          - intros Hmax.
            apply Qle_Rle in Hmax.
            repeat rewrite Q2R_minus.
            repeat rewrite Q2R_plus.
            repeat rewrite Q2R_div; try lra.
            replace (Q2R (2#1)) with 2%R by lra.
            repeat rewrite Q2R_minus in Hmax.
            repeat rewrite Q2R_plus in Hmax.
            rewrite Q2R_div in Hmax; try lra.
            rewrite Q2R_div in Hmax; try lra.
            replace (Q2R (2#1)) with 2%R in Hmax by lra.
            pose (l := (Q2R (fst (P n)))).
            pose (h := (Q2R (snd (P n)))).
            fold l h in Hmax, interval_containment |-*.
            pose (noise_expression := ((vR - h / 2 - l / 2) / (h / 2 + l / 2 - l))%R).
            assert (-(1) <= noise_expression <= 1)%R as Hnoise.
            {
              unfold noise_expression.
              apply Rabs_Rle_condition.
              destruct (Rle_lt_dec (h / 2 + l / 2 - l) 0)%R as [Hle0 | Hle0].
              - apply Rle_lt_or_eq_dec in Hle0; destruct Hle0 as [Hlt | Hlt];
                  try (field_simplify in Hlt; assert (h = l) as Hz by lra; apply eqR_Qeq in Hz; lra).
              - rewrite Rdiv_abs_le_bounds; try lra.
                assert (0 < h - l)%R as H1 by lra.
                Rrewrite (vR - h / 2 - l /2 = vR - (h + l) / 2)%R.
                Rrewrite (1 * (h / 2 + l / 2 - l) = (h - l) / 2)%R.
                apply Rabs_Rle_condition; lra.
            }
            pose (noise := exist (fun x => -(1) <= x <= 1)%R noise_expression Hnoise).
            exists noise.
            unfold noise, noise_expression.
            simpl.
            field.
            intros Hnotz.
            field_simplify in Hnotz.
            assert (h = l) as Hz by lra.
            apply eqR_Qeq in Hz.
            lra.
          - intros Hmax.
            apply Qle_Rle in Hmax.
            repeat rewrite Q2R_minus.
            repeat rewrite Q2R_plus.
            repeat rewrite Q2R_div; try lra.
            replace (Q2R (2#1)) with 2%R by lra.
            repeat rewrite Q2R_minus in Hmax.
            repeat rewrite Q2R_plus in Hmax.
            rewrite Q2R_div in Hmax; try lra.
            rewrite Q2R_div in Hmax; try lra.
            replace (Q2R (2#1)) with 2%R in Hmax by lra.
            pose (l := (Q2R (fst (P n)))).
            pose (h := (Q2R (snd (P n)))).
            fold l h in Hmax, interval_containment |-*.
            pose (noise_expression := ((vR - h / 2 - l / 2) / (h / 2 + l / 2 - l))%R).
            assert (-(1) <= noise_expression <= 1)%R as Hnoise.
            {
              unfold noise_expression.
              apply Rabs_Rle_condition.
              destruct (Rle_lt_dec (h / 2 + l / 2 - l) 0)%R as [Hle0 | Hle0].
              - apply Rle_lt_or_eq_dec in Hle0; destruct Hle0 as [Hlt | Hlt];
                  try (field_simplify in Hlt; assert (h = l) as Hz by lra; apply eqR_Qeq in Hz; lra).
              - rewrite Rdiv_abs_le_bounds; try lra.
                assert (0 < h - l)%R as H1 by lra.
                Rrewrite (vR - h / 2 - l /2 = vR - (h + l) / 2)%R.
                Rrewrite (1 * (h / 2 + l / 2 - l) = (h - l) / 2)%R.
                apply Rabs_Rle_condition; lra.
            }
            pose (noise := exist (fun x => -(1) <= x <= 1)%R noise_expression Hnoise).
            exists noise.
            unfold noise, noise_expression.
            simpl.
            field.
            intros Hnotz.
            field_simplify in Hnotz.
            assert (h = l) as Hz by lra.
            apply eqR_Qeq in Hz.
            lra.
        }
        exists (updMap map1 inoise q), (fromIntv (P n) inoise), vR, aiv, aerr.
        repeat split; auto.
        -- apply contained_map_extension.
           apply validmap1; lia.
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        -- apply contained_flover_map_extension.
           admit.
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        -- rewrite FloverMapFacts.P.F.add_eq_o; try auto.
           apply Q_orderedExps.expCompare_refl.
        -- unfold fresh, fromIntv, get_max_index.
           rewrite Heq.
           simpl; lia.
        -- intros n' Hn'.
           unfold updMap.
           destruct (n' =? inoise) eqn: Hneq.
           ++ apply beq_nat_true in Hneq.
              lia.
           ++ apply validmap1.
              lia.
        -- lia.
        -- constructor; auto.
           simpl.
           rewrite varsTyped.
           reflexivity.
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  - clear varsTyped varsDisjoint dVarsValid fVarsSound fVars.
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    (* specialize (visitedExpr (Const m v)). *)
    (* destruct (FloverMap.find (elt:=affine_form Q) (Const m v) iexpmap) eqn: Hvisited; *)
    (*   eauto using visitedExpr. *)
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    destruct (FloverMap.find (elt:=intv * error) (Const m v) A) eqn: Hares;
      simpl in validBounds; try congruence.
    destruct p as [i e].
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    destruct (isSupersetIntv (v, v) i) eqn: Hsup; try congruence.
    assert (isSupersetIntv (v, v) i = true) as Hsup' by assumption.
    apply andb_prop in Hsup' as [L R].
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    rewrite Qle_bool_iff in L, R.
    simpl ivlo in L, R.
    simpl ivhi in L, R.
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    assert (fst i <= v) as L' by assumption.
    assert (v <= snd i) as R' by assumption.
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    apply Qle_Rle in L.
    apply Qle_Rle in R.
    inversion validBounds; subst; clear validBounds.
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    exists map1, (fromIntv (v, v) noise), (perturb (Q2R v) 0), i, e.
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    repeat split.
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    + reflexivity.
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    + apply contained_flover_map_extension.
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      admit.
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    + unfold fromIntv, toIntv.
      simpl.
      rewrite Qeq_bool_refl.
      apply andb_true_intro.
      split; rewrite Qle_bool_iff; simpl; field_simplify; Qrewrite ((2#1) * v * / (2#1) == v).
      * now Qrewrite (fst i / 1 == fst i).
      * now Qrewrite (snd i / 1 == snd i).
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    + rewrite FloverMapFacts.P.F.add_eq_o; try auto.
      apply Q_orderedExps.expCompare_refl.
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    + unfold fromIntv.
      simpl ivlo.
      simpl ivhi.
      rewrite Qeq_bool_refl.
      unfold fresh, get_max_index; rewrite get_max_index_aux_equation. lia.
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    + assumption.
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    + lia.
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    + constructor; apply Rabs_0_equiv.
    + unfold perturb.
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      unfold fromIntv.
      simpl.
      rewrite Qeq_bool_refl.
      simpl.
      rewrite Q2R_plus.
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      rewrite Q2R_div; try lra.
      unfold af_evals, Ropt_eq; simpl.
      lra.
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  -
    (* pose proof visitedExpr as visitedExpr'. *)
    (* specialize (visitedExpr (Unop u e)). *)
    (* destruct (FloverMap.find (elt:=affine_form Q) (Unop u e) iexpmap) eqn: Hvisited; *)
    (*   eauto using visitedExpr. *)
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    unfold updateExpMap, updateExpMapSucc, updateExpMapIncr in validBounds.
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    destruct (FloverMap.find (elt:=intv * error) (Unop u e) A) as [p | p] eqn: Hares; simpl in validBounds; try congruence.
    destruct p as [aiv aerr].
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    destruct (validAffineBounds e A P dVars iexpmap inoise) eqn: Hsubvalid; simpl in validBounds; try congruence.
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    destruct p as [subexprAff subnoise].
    destruct (FloverMap.find (elt:=affine_form Q) e subexprAff) as [af | af] eqn: He; simpl in validBounds; try congruence.
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    destruct (IHe subnoise subexprAff inoise iexpmap map1)
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      as [ihmap [af' [vR [subaiv [subaerr [Hcont [Hcontf [Hsubares [Hsubsup [Haf [subfresh [Hsubvalidmap [Hsubnoise [subeval subaff]]]]]]]]]]]]]];
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      auto; clear IHe varsTyped varsDisjoint dVarsValid fVarsSound fVars.
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    assert (af' = af) by congruence; subst.
    destruct u.
    + destruct (isSupersetIntv (toIntv (AffineArithQ.negate_aff af)) aiv) eqn: Hsup; try congruence.
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      exists ihmap, (AffineArithQ.negate_aff af), (-vR)%R, aiv, aerr.
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      inversion validBounds; subst; clear validBounds.
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      repeat split; auto.
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      * etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * rewrite plus_0_r.
        unfold AffineArithQ.negate_aff.
        now apply AffineArithQ.fresh_mult_aff_const.
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      * rewrite plus_0_r.
        assumption.
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      * lia.
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      * now constructor.
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      * rewrite afQ2R_negate_aff.
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        now apply negate_aff_sound.
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    + destruct (nozeroiv (toIntv af)) eqn: noZeroAf; try congruence.
      destruct (isSupersetIntv (toIntv (AffineArithQ.inverse_aff af subnoise)) aiv) eqn: Hsup; try congruence.
      inversion validBounds; subst; clear validBounds.
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      pose proof (inverse_aff_sound) as invsound.
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      assert (fresh subnoise (afQ2R af)) as H' by now apply afQ2R_fresh.
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      assert (above_zero (afQ2R af) \/ below_zero (afQ2R af)) as H'' by eauto using nozero_above_below.
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      specialize (invsound (afQ2R af) vR ihmap subnoise H' H'' subaff) as [qInv inveval].
      exists (updMap ihmap subnoise qInv), (AffineArithQ.inverse_aff af subnoise),
      (perturb (evalUnop Inv vR) 0)%R, aiv, aerr.
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      repeat split; try auto.
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      * eapply contained_map_trans; try exact Hcont.
        apply contained_map_extension.
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        apply Hsubvalidmap; lia.
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      * etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * now apply AffineArithQ.fresh_inverse_aff.
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      * intros n H.
        unfold updMap.
        assert (n <> subnoise) as Hneq by lia.
        rewrite <- Nat.eqb_neq in Hneq.
        rewrite Hneq.
        apply Hsubvalidmap.
        lia.
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      * lia.
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      * constructor; try apply Rabs_0_equiv; auto.
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        eauto using above_below_nonzero.
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      * unfold perturb.
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        rewrite Rplus_0_r.
        rewrite Rmult_1_r.
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        simpl evalUnop.
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        replace (/ vR)%R with (1 / vR)%R by lra.
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        now rewrite afQ2R_inverse_aff.
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  -
    (* pose proof visitedExpr as visitedExpr'. *)
    (* specialize (visitedExpr (Binop b e1 e2)). *)
    (* destruct (FloverMap.find (elt:=affine_form Q) (Binop b e1 e2) iexpmap) eqn: Hvisited; *)
    (*   eauto using visitedExpr. *)
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    unfold updateExpMap, updateExpMapSucc, updateExpMapIncr in validBounds.
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    destruct (FloverMap.find (elt:=intv * error) (Binop b e1 e2) A) as [p | _] eqn: Hares; simpl in validBounds; try congruence.
    destruct p as [aiv aerr].
    destruct (validAffineBounds e1 A P dVars iexpmap inoise) eqn: Hsubvalid1; simpl in validBounds; try congruence.
    destruct p as [subexprAff1 subnoise1].
    destruct (FloverMap.find (elt:=affine_form Q) e1 subexprAff1) as [af1 | _] eqn: He1; simpl in validBounds; try congruence.
    destruct (validAffineBounds e2 A P dVars subexprAff1 subnoise1) eqn: Hsubvalid2; simpl in validBounds; try congruence.
    destruct p as [subexprAff2 subnoise2].
    destruct (FloverMap.find (elt:=affine_form Q) e2 subexprAff2) as [af2 | _] eqn: He2; simpl in validBounds; try congruence.
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    destruct (IHe1 subnoise1 subexprAff1 inoise iexpmap map1) as [ihmap1 [af1' [vR1 [ihaiv1 [ihaerr1 [ihcont1 [ihcontf1 [ihares1 [ihsup1 [Haf1 [subfresh1 [Hsubvalidmap1 [Hsubnoise1 [subeval1 subaff1]]]]]]]]]]]]]];
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      try auto; clear IHe1.
    { repeat set_tac. }
    assert (af1' = af1) by congruence; subst.
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    destruct (IHe2 subnoise2 subexprAff2 subnoise1 subexprAff1 ihmap1) as [ihmap2 [af2' [vR2 [_ [_ [ihcont2 [ihcontf2 [_ [_ [Haf2 [subfresh2 [Hsubvalidmap2 [Hsubnoise2 [subeval2 subaff2]]]]]]]]]]]]]];
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      try auto; try lra; clear IHe2; eauto using fresh_n_gt_O.
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    {
      unfold affine_dVars_range_valid in dVarsValid |-*.
      intros v vin.
      specialize (dVarsValid v vin).
      destruct dVarsValid as [af [vR [iv [err dVarsValid]]]].
      exists af, vR, iv, err; intuition.
      - eapply fresh_monotonic; try eassumption.
      - eapply af_evals_map_extension; try eassumption.
    }
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    { repeat set_tac. }
    assert (af2' = af2) by congruence; subst.
    clear varsTyped varsDisjoint dVarsValid fVarsSound fVars.
    destruct b.
    + destruct (isSupersetIntv (toIntv (AffineArithQ.plus_aff af1 af2)) aiv) eqn: Hsup; try congruence.
      inversion validBounds; subst; clear validBounds.
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      exists ihmap2, (AffineArithQ.plus_aff af1 af2), (perturb (evalBinop Plus vR1 vR2) 0)%R, aiv, aerr.
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      repeat split; auto.
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      * etransitivity; try exact ihcont1.
        etransitivity; try exact ihcont2.
        reflexivity.
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      * etransitivity; try eassumption.
        etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * apply AffineArithQ.plus_aff_preserves_fresh; rewrite plus_0_r; auto.
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        eapply fresh_monotonic.
        exact Hsubnoise2.
        assumption.
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      * rewrite plus_0_r; assumption.
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      * lia.
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      * replace M0 with (join M0 M0) by trivial.
        apply Binop_dist; try apply Rabs_0_equiv; auto; congruence.
      * unfold perturb.
        rewrite Rplus_0_r.
        rewrite Rmult_1_r.
        simpl evalBinop.
        rewrite afQ2R_plus_aff.
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        apply plus_aff_sound; auto.
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        eauto using af_evals_map_extension.
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    + destruct (isSupersetIntv (toIntv (AffineArithQ.subtract_aff af1 af2)) aiv) eqn: Hsup; try congruence.
      inversion validBounds; subst; clear validBounds.
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      exists ihmap2, (AffineArithQ.subtract_aff af1 af2), (perturb (evalBinop Sub vR1 vR2) 0)%R, aiv, aerr.
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      repeat split; auto.
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      * etransitivity; try exact ihcont1.
        etransitivity; try exact ihcont2.
        reflexivity.
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      * etransitivity; try eassumption.
        etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * unfold AffineArithQ.subtract_aff.
        rewrite plus_0_r.
        apply AffineArithQ.plus_aff_preserves_fresh.
        {
          eapply fresh_monotonic.
          exact Hsubnoise2.
          assumption.
        }
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        unfold AffineArithQ.negate_aff.
        now apply AffineArithQ.fresh_mult_aff_const.
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      * rewrite plus_0_r; assumption.
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      * lia.
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      * replace M0 with (join M0 M0) by trivial.
        apply Binop_dist; try apply Rabs_0_equiv; auto; congruence.
      * unfold perturb.
        rewrite Rplus_0_r.
        rewrite Rmult_1_r.
        simpl evalBinop.
        rewrite afQ2R_subtract_aff.
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        apply subtract_aff_sound; auto.
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        eauto using af_evals_map_extension.
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    + destruct (isSupersetIntv (toIntv (AffineArithQ.mult_aff af1 af2 subnoise2)) aiv) eqn: Hsup; try congruence.
      inversion validBounds; subst; clear validBounds.
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      pose proof mult_aff_sound as multsound.
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      assert (fresh subnoise2 (afQ2R af1)) as fresh1
          by (apply afQ2R_fresh; eauto using fresh_monotonic).
      assert (fresh subnoise2 (afQ2R af2)) as fresh2 by now apply afQ2R_fresh.
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      assert (af_evals (afQ2R af1) vR1 ihmap2) as subaff1' by eauto using af_evals_map_extension.
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      specialize (multsound (afQ2R af1) (afQ2R af2) vR1 vR2 ihmap2 subnoise2 fresh1 fresh2 subaff1' subaff2) as [qMult multsound].
      exists (updMap ihmap2 subnoise2 qMult), (AffineArithQ.mult_aff af1 af2 subnoise2),
      (perturb (evalBinop Mult vR1 vR2) 0)%R, aiv, aerr.
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      repeat split; auto.
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      * etransitivity; try exact ihcont1.
        etransitivity; try exact ihcont2.
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        apply contained_map_extension; auto.
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      * etransitivity; try eassumption.
        etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * unfold AffineArithQ.mult_aff.
        destruct Qeq_bool.
        -- rewrite orb_true_l.
           apply AffineArithQ.mult_aff_aux_preserves_fresh;
             apply fresh_inc; now rewrite afQ2R_fresh.
        -- destruct Qeq_bool.
           ++ rewrite orb_true_r.
              apply AffineArithQ.mult_aff_aux_preserves_fresh;
                apply fresh_inc; now rewrite afQ2R_fresh.
           ++ simpl.
              apply fresh_noise; try lia.
              apply AffineArithQ.mult_aff_aux_preserves_fresh;
                apply fresh_inc; now rewrite afQ2R_fresh.
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      * intros n H.
        assert (n <> subnoise2) as Hneq by lia.
        rewrite <- Nat.eqb_neq in Hneq.
        unfold updMap.
        rewrite Hneq.
        apply Hsubvalidmap2.
        lia.
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      * lia.
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      * replace M0 with (join M0 M0) by trivial.
        apply Binop_dist; try apply Rabs_0_equiv; auto; congruence.
      * unfold perturb.
        rewrite Rplus_0_r.
        rewrite Rmult_1_r.
        simpl evalBinop.
        rewrite afQ2R_mult_aff.
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        assumption.
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    + destruct (nozeroiv (toIntv af2)) eqn: Hnozero; try congruence.
      destruct (isSupersetIntv (toIntv (AffineArithQ.divide_aff af1 af2 subnoise2)) aiv) eqn: Hsup; try congruence.
      inversion validBounds; subst; clear validBounds.
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      pose proof divide_aff_sound as divsound.
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      assert (fresh subnoise2 (afQ2R af1)) as fresh1
          by now apply afQ2R_fresh; eauto using fresh_monotonic.
      assert (fresh subnoise2 (afQ2R af2)) as fresh2 by now apply afQ2R_fresh.
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      assert (af_evals (afQ2R af1) vR1 ihmap2) as subaff1' by eauto using af_evals_map_extension.
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      assert (above_zero (afQ2R af2) \/ below_zero (afQ2R af2)) as nozero' by eauto using nozero_above_below.
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      specialize (divsound (afQ2R af1) (afQ2R af2) vR1 vR2 subnoise2 ihmap2 fresh1 fresh2 nozero' subaff1' subaff2) as [qInv [qMult multsound]].
      exists (updMap (updMap ihmap2 subnoise2 qInv) (subnoise2 + 1) qMult),
      (AffineArithQ.divide_aff af1 af2 subnoise2), (perturb (evalBinop Div vR1 vR2) 0)%R, aiv, aerr.
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      repeat split; auto.
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      * etransitivity; try exact ihcont1.
        etransitivity; try exact ihcont2.
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        apply contained_map_extension2; apply Hsubvalidmap2; lia.
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      * etransitivity; try eassumption.
        etransitivity; try eassumption.
        apply contained_flover_map_extension.
        admit.
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      * rewrite FloverMapFacts.P.F.add_eq_o; try auto.
        apply Q_orderedExps.expCompare_refl.
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      * unfold AffineArithQ.divide_aff, AffineArithQ.mult_aff.
        destruct Qeq_bool.
        -- rewrite orb_true_l.
           apply AffineArithQ.mult_aff_aux_preserves_fresh.
           { 
             apply fresh_monotonic with (n := subnoise2);
             try lia; try apply afQ2R_fresh; auto.
           }
           apply fresh_monotonic with (n := (subnoise2 + 1)%nat); try lia.
           now apply AffineArithQ.fresh_inverse_aff.
        -- destruct Qeq_bool.
           ++ rewrite orb_true_r.
              apply AffineArithQ.mult_aff_aux_preserves_fresh.
              { 
                apply fresh_monotonic with (n := subnoise2);
                  try lia; try apply afQ2R_fresh; auto.
              }
              apply fresh_monotonic with (n := (subnoise2 + 1)%nat); try lia.
              now apply AffineArithQ.fresh_inverse_aff.
           ++ simpl.
              apply fresh_noise; try lia.
              apply AffineArithQ.mult_aff_aux_preserves_fresh.
              { 
                apply fresh_monotonic with (n := subnoise2);
                  try lia; try apply afQ2R_fresh; auto.
              }
              apply fresh_monotonic with (n := (subnoise2 + 1)%nat); try lia.
              now apply AffineArithQ.fresh_inverse_aff.
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      * intros n H.
        assert (n <> (subnoise2 + 1)%nat) as Hneq by lia.
        assert (n <> subnoise2) as Hneq' by lia.
        rewrite <- Nat.eqb_neq in Hneq.
        rewrite <- Nat.eqb_neq in Hneq'.
        unfold updMap.
        rewrite Hneq', Hneq.
        apply Hsubvalidmap2.
        lia.
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      * lia.
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      * replace M0 with (join M0 M0) by trivial.
        apply Binop_dist; try apply Rabs_0_equiv; auto.
        intros _.
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        eauto using above_below_nonzero.
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      * unfold perturb.
        rewrite Rplus_0_r.
        rewrite Rmult_1_r.
        simpl evalBinop.
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        rewrite afQ2R_divide_aff; auto.
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  -
    (* pose proof visitedExpr as visitedExpr'. *)
    (* specialize (visitedExpr (Fma e1 e2 e3)). *)
    (* destruct (FloverMap.find (elt:=affine_form Q) (Fma e1 e2 e3) iexpmap) eqn: Hvisited; *)
    (*   eauto using visitedExpr. *)
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    unfold updateExpMap, updateExpMapSucc, updateExpMapIncr in validBounds.
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    destruct (FloverMap.find (elt:=intv * error) (Fma e1 e2 e3) A) as [p | _] eqn: Hares; simpl in validBounds; try congruence.
    destruct p as [aiv aerr].
    destruct (validAffineBounds e1 A P dVars iexpmap inoise) eqn: Hsubvalid1; simpl in validBounds; try congruence.
    destruct p as [subexprAff1 subnoise1].
    destruct (FloverMap.find (elt:=affine_form Q) e1 subexprAff1) as [af1 | _] eqn: He1; simpl in validBounds; try congruence.
    destruct (validAffineBounds e2 A P dVars subexprAff1 subnoise1) eqn: Hsubvalid2; simpl in validBounds; try congruence.
    destruct p as [subexprAff2 subnoise2].
    destruct (FloverMap.find (elt:=affine_form Q) e2 subexprAff2) as [af2 | _] eqn: He2; simpl in validBounds; try congruence.
    destruct (validAffineBounds e3 A P dVars subexprAff2 subnoise2) eqn: Hsubvalid3; simpl in validBounds; try congruence.
    destruct p as [subexprAff3 subnoise3].
    destruct (FloverMap.find (elt:=affine_form Q) e3 subexprAff3) as [af3 | _] eqn: He3; simpl in validBounds; try congruence.
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    destruct (IHe1 subnoise1 subexprAff1 inoise iexpmap map1) as [ihmap1 [af1' [vR1 [_ [_ [ihcont1 [ihcontf1 [_ [_ [Haf1 [subfresh1 [Hsubmapvalid1 [Hsubnoise1 [subeval1 subaff1]]]]]]]]]]]]]];
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      try auto; clear IHe1.
    { repeat set_tac. }
    assert (af1' = af1) by congruence; subst.
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    destruct (IHe2 subnoise2 subexprAff2 subnoise1 subexprAff1 ihmap1) as [ihmap2 [af2' [vR2 [_ [_ [ihcont2 [ihcontf2 [_ [_ [Haf2 [subfresh2 [Hsubmapvalid2 [Hsubnoise2 [subeval2 subaff2]]]]]]]]]]]]]];
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      try auto; clear IHe2; try lia.
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    {
      unfold affine_dVars_range_valid in dVarsValid |-*.
      intros v vin.
      specialize (dVarsValid v vin).
      destruct dVarsValid as [af [vR [iv [err dVarsValid]]]].
      exists af, vR, iv, err; intuition.
      - eapply fresh_monotonic; try eassumption.
      - eapply af_evals_map_extension; try eassumption.
    }
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    { repeat set_tac. }
    assert (af2' = af2) by congruence; subst.
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    destruct (IHe3 subnoise3 subexprAff3 subnoise2 subexprAff2 ihmap2) as [ihmap3 [af3' [vR3 [_ [_ [ihcont3 [ihcontf3 [_ [_ [Haf3 [subfresh3 [Hsubmapvalid3 [Hsubnoise3 [subeval3 subaff3]]]]]]]]]]]]]];
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      try auto; clear IHe3; try lia.
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    {
      unfold affine_dVars_range_valid in dVarsValid |-*.
      intros v vin.
      specialize (dVarsValid v vin).
      destruct dVarsValid as [af [vR [iv [err dVarsValid]]]].
      exists af, vR, iv, err; intuition.
      - eapply fresh_monotonic; try eassumption.
        eapply fresh_monotonic; try eassumption.
      - eapply af_evals_map_extension; try eassumption.
        eapply af_evals_map_extension; try eassumption.
    }
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    { repeat set_tac. }
    assert (af3' = af3) by congruence; subst.
    clear varsTyped varsDisjoint dVarsValid fVarsSound fVars.
    destruct (isSupersetIntv (toIntv (AffineArithQ.plus_aff af1 (AffineArithQ.mult_aff af2 af3 subnoise3))) aiv) eqn: Hsup; try congruence.
    inversion validBounds; subst; clear validBounds.
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    pose proof mult_aff_sound as multsound.
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    assert (fresh subnoise3 (afQ2R af1)) as fresh1 by now apply afQ2R_fresh;
      eauto using fresh_monotonic.
    assert (fresh subnoise3 (afQ2R af2)) as fresh2 by now apply afQ2R_fresh;
      eauto using fresh_monotonic.
    assert (fresh subnoise3 (afQ2R af3)) as fresh3 by now apply afQ2R_fresh.
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    assert (af_evals (afQ2R af1) vR1 ihmap3) as subaff1' by eauto using af_evals_map_extension. 
    assert (af_evals (afQ2R af2) vR2 ihmap3) as subaff2' by eauto using af_evals_map_extension. 
    assert (af_evals (afQ2R af3) vR3 ihmap3) as subaff3' by eauto using af_evals_map_extension. 
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    specialize (multsound (afQ2R af2) (afQ2R af3) vR2 vR3 ihmap3 subnoise3 fresh2 fresh3 subaff2' subaff3') as [qMult multsound].
    exists (updMap ihmap3 subnoise3 qMult),
    (AffineArithQ.plus_aff af1 (AffineArithQ.mult_aff af2 af3 subnoise3)),