IntervalValidation.v 14.3 KB
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(**
Interval arithmetic checker and its soundness proof
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Explained in section 4 of the paper, used in CertificateChecker.v to build full checker,
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**)
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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 Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealRationalProps.
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Require Import Daisy.Infra.ExpressionAbbrevs Daisy.IntervalArithQ Daisy.IntervalArith Daisy.Infra.RealSimps.
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Import Lists.List.ListNotations.

Fixpoint freeVars (V:Type) (e:exp V) : list nat:=
  match e with
  |Const n => []
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  |Var _ v => []
  |Param _ v => [v]
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  |Binop b e1 e2 => (freeVars V e1) ++ (freeVars V e2)
  end.

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Fixpoint validIntervalbounds (e:exp Q) (absenv:analysisResult) (P:precond):=
  let (intv, _) := absenv e in
    match e with
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    | Var _ v => false
    | Param _ v =>
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      isSupersetIntv (P v) intv
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    | Const n =>
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      isSupersetIntv (n,n) intv
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    | Binop b e1 e2 =>
      let rec := andb (validIntervalbounds e1 absenv P) (validIntervalbounds e2 absenv P) in
      let (iv1,_) := absenv e1 in
      let (iv2,_) := absenv e2 in
      let opres :=
          match b with
          | Plus =>
            let new_iv := addIntv iv1 iv2 in
            isSupersetIntv new_iv intv
          | Sub =>
            let new_iv := subtractIntv iv1 iv2 in
            isSupersetIntv new_iv intv
          | Mult =>
            let new_iv := multIntv iv1 iv2 in
            isSupersetIntv new_iv intv
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          | Div =>
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            let nodiv0 := orb
                            (andb (Qleb (ivhi iv2) 0) (negb (Qeq_bool (ivhi iv2) 0)))
                            (andb (Qleb 0 (ivlo iv2)) (negb (Qeq_bool (ivlo iv2) 0))) in
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            let new_iv := divideIntv iv1 iv2 in
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            andb (isSupersetIntv new_iv intv) nodiv0
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          end
      in
      andb rec opres
    end.

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Theorem ivbounds_approximatesPrecond_sound e absenv P:
  validIntervalbounds e absenv P = true ->
  forall v, In v (freeVars Q e) ->
       Is_true(isSupersetIntv (P v) (fst (absenv (Param Q v)))).
Proof.
  induction e; unfold validIntervalbounds.
  - intros approx_true v v_in_fV; simpl in *; contradiction.
  - intros approx_true v v_in_fV; simpl in *.
    unfold isSupersetIntv.
    destruct v_in_fV; try contradiction.
    subst.
    destruct (P v); destruct (absenv (Param Q v)); simpl in *.
    destruct i; simpl in *.
    apply Is_true_eq_left in approx_true; auto.
  - intros approx_true v0 v_in_fV; simpl in *; contradiction.
  - intros approx_bin_true v v_in_fV.
    unfold freeVars in v_in_fV.
    apply in_app_or in v_in_fV.
    apply Is_true_eq_left in approx_bin_true.
    destruct (absenv (Binop b e1 e2)); destruct (absenv e1); destruct (absenv e2); simpl in *.
    apply andb_prop_elim in approx_bin_true.
    destruct approx_bin_true.
    apply andb_prop_elim in H.
    destruct H.
    destruct v_in_fV as [v_in_fV_e1 | v_in_fV_e2].
    + apply IHe1; auto.
      apply Is_true_eq_true; auto.
    + apply IHe2; auto.
      apply Is_true_eq_true; auto.
Qed.

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Corollary Q2R_max4 a b c d:
  Q2R (IntervalArithQ.max4 a b c d) = max4 (Q2R a) (Q2R b) (Q2R c) (Q2R d).
Proof.
  unfold IntervalArithQ.max4, max4; repeat rewrite Q2R_max; auto.
Qed.

Corollary Q2R_min4 a b c d:
  Q2R (IntervalArithQ.min4 a b c d) = min4 (Q2R a) (Q2R b) (Q2R c) (Q2R d).
Proof.
  unfold IntervalArith.min4, min4; repeat rewrite Q2R_min; auto.
Qed.

Ltac env_assert absenv e name :=
  assert (exists iv err, absenv e = (iv,err)) as name by (destruct (absenv e); repeat eexists; auto).

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Lemma validBoundsDiv_uneq_zero e1 e2 absenv P ivlo_e2 ivhi_e2 err:
  absenv e2 = ((ivlo_e2,ivhi_e2), err) ->
  validIntervalbounds (Binop Div e1 e2) absenv P = true ->
  (ivhi_e2 < 0) \/ (0 < ivlo_e2).
Proof.
  intros absenv_eq validBounds.
  unfold validIntervalbounds in validBounds.
  env_assert absenv (Binop Div e1 e2) abs_div; destruct abs_div as [iv_div [err_div abs_div]].
  env_assert absenv e1 abs_e1; destruct abs_e1 as [iv_e1 [err_e1 abs_e1]].
  rewrite abs_div, abs_e1, absenv_eq in validBounds.
  apply Is_true_eq_left in validBounds.
  apply andb_prop_elim in validBounds.
  destruct validBounds as [_ validBounds]; apply andb_prop_elim in validBounds.
  destruct validBounds as [_ nodiv0].
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  apply Is_true_eq_true in nodiv0.
  apply le_neq_bool_to_lt_prop; auto.
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Qed.

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Theorem validIntervalbounds_sound (e:exp Q):
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  forall (absenv:analysisResult) (P:precond) cenv vR,
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    precondValidForExec P cenv ->
    validIntervalbounds e absenv P = true ->
    eval_exp 0%R cenv (toRExp e) vR ->
    (Q2R (fst (fst(absenv e))) <= vR <= Q2R (snd (fst (absenv e))))%R.
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Proof.
  induction e.
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  - intros absenv P cenv vR precond_valid valid_bounds eval_e.
    pose proof (ivbounds_approximatesPrecond_sound (Var Q n) absenv P valid_bounds) as env_approx_p.
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    unfold validIntervalbounds in valid_bounds.
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    destruct (absenv (Var Q n)); inversion valid_bounds.
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  - intros absenv P cenv vR precond_valid valid_bounds eval_e.
    pose proof (ivbounds_approximatesPrecond_sound (Param Q n) absenv P valid_bounds) as env_approx_p.
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    unfold validIntervalbounds in valid_bounds.
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    assert (exists intv err, absenv (Param Q n) = (intv,err))
      as absenv_n
        by (destruct (absenv (Param Q n)); repeat eexists; auto).
    destruct absenv_n as [intv [err absenv_n]].
    rewrite absenv_n in valid_bounds.
    unfold precondValidForExec in precond_valid.
    specialize (env_approx_p n).
    assert (exists ivlo ivhi, P n = (ivlo, ivhi)) as p_n
        by (destruct (P n); repeat eexists; auto).
    destruct p_n as [ivlo [ivhi p_n]].
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    unfold isSupersetIntv, freeVars in env_approx_p.
    assert (In n (n::nil)) by (unfold In; auto).
    specialize (env_approx_p H).
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    rewrite p_n, absenv_n in env_approx_p.
    specialize (precond_valid n); rewrite p_n in precond_valid.
    inversion eval_e; subst.
    rewrite absenv_n; simpl.
    rewrite perturb_0_val; auto.
    destruct intv as [abslo abshi]; simpl in *.
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    apply andb_prop_elim in env_approx_p.
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    destruct env_approx_p as [abslo_le_ivlo ivhi_le_abshi].
    destruct precond_valid as [ivlo_le_env env_le_ivhi].
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    apply Is_true_eq_true in abslo_le_ivlo; apply Is_true_eq_true in ivhi_le_abshi.
    unfold Qleb in abslo_le_ivlo, ivhi_le_abshi.
    apply Qle_bool_iff in abslo_le_ivlo; apply Qle_bool_iff in ivhi_le_abshi.
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    apply Qle_Rle in abslo_le_ivlo; apply Qle_Rle in ivhi_le_abshi.
    split.
    + eapply Rle_trans.
      apply abslo_le_ivlo.
      apply ivlo_le_env.
    + eapply Rle_trans.
      apply env_le_ivhi.
      apply ivhi_le_abshi.
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  - intros absenv P cenv vR valid_precond valid_bounds eval_e.
    pose proof (ivbounds_approximatesPrecond_sound (Const v) absenv P valid_bounds) as env_approx_p.
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    unfold validIntervalbounds in valid_bounds.
    destruct (absenv (Const v)) as [intv err].
    simpl.
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
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    destruct valid_bounds as [valid_lo valid_hi].
    inversion eval_e; subst.
    rewrite perturb_0_val; auto.
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    unfold contained; simpl.
    split.
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    + apply Is_true_eq_true in valid_lo.
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      unfold Qleb in *.
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      apply Qle_bool_iff in valid_lo.
      apply Qle_Rle in valid_lo; auto.
    + apply Is_true_eq_true in valid_hi.
      unfold Qleb in *.
      apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_hi; auto.
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  - intros absenv P cenv vR valid_precond valid_bounds eval_e;
      inversion eval_e; subst.
    pose proof (ivbounds_approximatesPrecond_sound (Binop b e1 e2) absenv P valid_bounds) as env_approx_p.
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    rewrite perturb_0_val in eval_e; auto.
    rewrite perturb_0_val; auto.
    simpl in valid_bounds.
    env_assert absenv (Binop b e1 e2) absenv_bin.
    env_assert absenv e1 absenv_e1.
    env_assert absenv e2 absenv_e2.
    destruct absenv_bin as [iv [err absenv_bin]]; rewrite absenv_bin in valid_bounds; rewrite absenv_bin.
    destruct absenv_e1 as [iv1 [err1 absenv_e1]]; rewrite absenv_e1 in valid_bounds.
    destruct absenv_e2 as [iv2 [err2 absenv_e2]]; rewrite absenv_e2 in valid_bounds.
    apply Is_true_eq_left in valid_bounds.
    apply andb_prop_elim in valid_bounds.
    destruct valid_bounds as [valid_rec valid_bin].
    apply andb_prop_elim in valid_rec.
    destruct valid_rec as [valid_e1 valid_e2].
    apply Is_true_eq_true in valid_e1; apply Is_true_eq_true in  valid_e2.
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    specialize (IHe1 absenv P cenv v1 valid_precond valid_e1 H4);
      specialize (IHe2 absenv P cenv v2 valid_precond valid_e2 H5).
    rewrite absenv_e1 in IHe1.
    rewrite absenv_e2 in IHe2.
    destruct b; simpl in *.
    + pose proof (additionIsValid (Q2R (fst iv1),Q2R (snd iv1)) (Q2R (fst iv2), Q2R (snd iv2))) as valid_add.
      unfold validIntervalAdd in valid_add.
      specialize (valid_add v1 v2 IHe1 IHe2).
      unfold contained in valid_add.
      unfold isSupersetIntv in valid_bin.
      apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
      destruct valid_add as [valid_add_lo valid_add_hi].
      split.
      { eapply Rle_trans. apply valid_lo.
        unfold ivlo. unfold addIntv.
        simpl in valid_add_lo.
        repeat rewrite <- Q2R_plus in valid_add_lo.
        rewrite <- Q2R_min4 in valid_add_lo.
        unfold absIvUpd; auto. }
      { eapply Rle_trans.
        Focus 2.
        apply valid_hi.
        unfold ivlo, addIntv.
        simpl in valid_add_hi.
        repeat rewrite <- Q2R_plus in valid_add_hi.
        rewrite <- Q2R_max4 in valid_add_hi.
        unfold absIvUpd; auto. }
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    + pose proof (subtractionIsValid (Q2R (fst iv1),Q2R (snd iv1)) (Q2R (fst iv2), Q2R (snd iv2))) as valid_sub.
      specialize (valid_sub v1 v2 IHe1 IHe2).
      unfold contained in valid_sub.
      unfold isSupersetIntv in valid_bin.
      apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
      destruct valid_sub as [valid_sub_lo valid_sub_hi].
      split.
      * eapply Rle_trans. apply valid_lo.
        unfold ivlo. unfold subtractIntv.
        simpl in valid_sub_lo.
        repeat rewrite <- Rsub_eq_Ropp_Rplus in valid_sub_lo.
        repeat rewrite <- Q2R_minus in valid_sub_lo.
        rewrite <- Q2R_min4 in valid_sub_lo.
        unfold absIvUpd; auto.
      * eapply Rle_trans.
        Focus 2.
        apply valid_hi.
        unfold ivlo, addIntv.
        simpl in valid_sub_hi.
        repeat rewrite <- Rsub_eq_Ropp_Rplus in valid_sub_hi.
        repeat rewrite <- Q2R_minus in valid_sub_hi.
        rewrite <- Q2R_max4 in valid_sub_hi.
        unfold absIvUpd; auto.
    + pose proof (interval_multiplication_valid (Q2R (fst iv1),Q2R (snd iv1)) (Q2R (fst iv2), Q2R (snd iv2))) as valid_mul.
      specialize (valid_mul v1 v2 IHe1 IHe2).
      unfold contained in valid_mul.
      unfold isSupersetIntv in valid_bin.
      apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
      destruct valid_mul as [valid_mul_lo valid_mul_hi].
      split.
      * eapply Rle_trans. apply valid_lo.
        unfold ivlo. unfold multIntv.
        simpl in valid_mul_lo.
        repeat rewrite <- Q2R_mult in valid_mul_lo.
        rewrite <- Q2R_min4 in valid_mul_lo.
        unfold absIvUpd; auto.
      * eapply Rle_trans.
        Focus 2.
        apply valid_hi.
        unfold ivlo, addIntv.
        simpl in valid_mul_hi.
        repeat rewrite <- Q2R_mult in valid_mul_hi.
        rewrite <- Q2R_max4 in valid_mul_hi.
        unfold absIvUpd; auto.
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    + pose proof (divisionIsValid v1 v2 (Q2R (fst iv1), Q2R (snd iv1)) (Q2R (fst iv2),Q2R (snd iv2))) as valid_div.
      unfold contained in valid_div.
      unfold isSupersetIntv in valid_bin.
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      apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_bin nodiv0].
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      apply andb_prop_elim in valid_bin; destruct valid_bin as [valid_lo valid_hi].
      apply Is_true_eq_true in valid_lo; apply Is_true_eq_true in valid_hi.
      apply Qle_bool_iff in valid_lo; apply Qle_bool_iff in valid_hi.
      apply Qle_Rle in valid_lo; apply Qle_Rle in valid_hi.
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      apply orb_prop_elim in nodiv0.
      assert (snd iv2 < 0 \/ 0 < fst iv2).
      * destruct nodiv0 as [lt_0 | lt_0];
          apply andb_prop_elim in lt_0; destruct lt_0 as [le_0 neq_0];
            apply Is_true_eq_true in le_0; apply Is_true_eq_true in neq_0;
              apply negb_true_iff in neq_0; apply Qeq_bool_neq in neq_0;
                rewrite Qle_bool_iff in le_0;
                rewrite Qle_lteq in le_0; destruct le_0 as [lt_0 | eq_0];
                  [ | exfalso; apply neq_0; auto | | exfalso; apply neq_0; symmetry; auto]; auto.
      * destruct valid_div as [valid_div_lo valid_div_hi]; simpl; try auto.
        { rewrite <- Q2R0_is_0.
          destruct H; [left | right]; apply Qlt_Rlt; auto. }
        { unfold divideInterval, IVlo, IVhi in valid_div_lo, valid_div_hi.
          simpl in *.
          assert (Q2R (fst iv2) <= (Q2R (snd iv2)))%R by lra.
          assert (~ snd iv2 == 0).
          - destruct H; try lra.
            hnf; intros ivhi2_0.
            apply Rle_Qle in H0.
            rewrite ivhi2_0 in H0.
            lra.
          - assert (~ fst iv2 == 0).
            + destruct H; try lra.
              hnf; intros ivlo2_0.
              apply Rle_Qle in H0.
              rewrite ivlo2_0 in H0.
              lra.
              + split.
                * eapply Rle_trans. apply valid_lo.
                  unfold ivlo. unfold multIntv.
                  simpl in valid_div_lo.
                  rewrite <- Q2R_inv in valid_div_lo; [ | auto].
                  rewrite <- Q2R_inv in valid_div_lo; [ | auto].
                  repeat rewrite <- Q2R_mult in valid_div_lo.
                  rewrite <- Q2R_min4 in valid_div_lo; auto.
                * eapply Rle_trans.
                  Focus 2.
                  apply valid_hi.
                  simpl in valid_div_hi.
                  rewrite <- Q2R_inv in valid_div_hi; [ | auto].
                  rewrite <- Q2R_inv in valid_div_hi; [ | auto].
                  repeat rewrite <- Q2R_mult in valid_div_hi.
                  rewrite <- Q2R_max4 in valid_div_hi; auto. }
Qed.