Expressions.v 32 KB
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(**
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  Formalization of the base expression language for the daisy framework
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 **)
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Require Import Coq.Reals.Reals Coq.micromega.Psatz Coq.QArith.QArith
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        Coq.QArith.Qreals Coq.Structures.Orders Coq.Structures.OrdersFacts.
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Require Import Daisy.Infra.RealRationalProps Daisy.Infra.RationalSimps
               Daisy.Infra.Ltacs.
Require Export Daisy.Infra.Abbrevs Daisy.Infra.RealSimps Daisy.Infra.NatSet
        Daisy.IntervalArithQ Daisy.IntervalArith Daisy.Infra.MachineType.
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(**
  Expressions will use binary operators.
  Define them first
**)
Inductive binop : Type := Plus | Sub | Mult | Div.
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Definition binopEq (b1:binop) (b2:binop) :=
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  match b1, b2 with
  | Plus, Plus => true
  | Sub,  Sub  => true
  | Mult, Mult => true
  | Div,  Div  => true
  | _,_ => false
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  end.

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(**
  Next define an evaluation function for binary operators on reals.
  Errors are added on the expression evaluation level later.
 **)
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Definition evalBinop (o:binop) (v1:R) (v2:R) :=
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  match o with
  | Plus => Rplus v1 v2
  | Sub => Rminus v1 v2
  | Mult => Rmult v1 v2
  | Div => Rdiv v1 v2
  end.
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Lemma binopEq_refl b:
  binopEq b b = true.
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Proof.
  case b; auto.
Qed.

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Lemma binopEq_compat_eq b1 b2:
  binopEq b1 b2 = true <-> b1 = b2.
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Proof.
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  split; case b1; case b2; intros; simpl in *; congruence.
Qed.

Lemma binopEq_compat_eq_false b1 b2:
  binopEq b1 b2 = false <-> ~ (b1 = b2).
Proof.
  split; intros neq.
  - hnf; intros; subst. rewrite binopEq_refl in neq.
    congruence.
  - destruct b1; destruct b2; cbv; congruence.
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Qed.

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(**
   Expressions will use unary operators.
   Define them first
 **)
Inductive unop: Type := Neg | Inv.

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Definition unopEq (o1:unop) (o2:unop) :=
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  match o1, o2 with
  | Neg, Neg => true
  | Inv, Inv => true
  | _ , _ => false
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  end.

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Lemma unopEq_refl b:
  unopEq b b = true.
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Proof.
  case b; auto.
Qed.

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Lemma unopEq_sym u1 u2:
  unopEq u1 u2 = unopEq u2 u1.
Proof.
  destruct u1,u2; compute; auto.
Qed.

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Lemma unopEq_compat_eq b1 b2:
  unopEq b1 b2 = true <-> b1 = b2.
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Proof.
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  split; case b1; case b2; intros; simpl in *; congruence.
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Qed.

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(**
   Define evaluation for unary operators on reals.
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   Errors are added in the expression evaluation level later.
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 **)
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Definition evalUnop (o:unop) (v:R):=
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  match o with
  |Neg => (- v)%R
  |Inv => (/ v)%R
  end .

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(**
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  Define expressions parametric over some value type V.
  Will ease reasoning about different instantiations later.
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**)
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Inductive exp (V:Type): Type :=
  Var: nat -> exp V
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| Const: mType -> V -> exp V
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| Unop: unop -> exp V -> exp V
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| Binop: binop -> exp V -> exp V -> exp V
| Downcast: mType -> exp V -> exp V.
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(**
  Boolean equality function on expressions.
  Used in certificates to define the analysis result as function
**)
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Fixpoint expEq (e1:exp Q) (e2:exp Q) :=
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  match e1, e2 with
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  | Var _ v1, Var _ v2 => (v1 =? v2)
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  | Const m1 n1, Const m2 n2 =>
    (mTypeEq m1 m2) && (Qeq_bool n1 n2)
  | Unop o1 e11, Unop o2 e22 =>
    (unopEq o1 o2) && (expEq e11 e22)
  | Binop o1 e11 e12, Binop o2 e21 e22 =>
    (binopEq o1 o2) && (expEq e11 e21) && (expEq e12 e22)
  | Downcast m1 f1, Downcast m2 f2 =>
    (mTypeEq m1 m2) && (expEq f1 f2)
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  | _, _ => false
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  end.

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Lemma expEq_refl e:
  expEq e e = true.
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Proof.
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  induction e; try (apply andb_true_iff; split); simpl in *; auto .
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  - symmetry; apply beq_nat_refl.
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  - apply mTypeEq_refl.
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  - apply Qeq_bool_iff; lra.
  - case u; auto.
  - case b; auto.
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  - apply mTypeEq_refl.
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Qed.
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Lemma expEq_sym e e':
  expEq e e' = expEq e' e.
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Proof.
  revert e'.
  induction e; intros e'; destruct e'; simpl; try auto.
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  - apply Nat.eqb_sym.
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  - f_equal.
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    + apply mTypeEq_sym; auto.
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    + apply Qeq_bool_sym.
  - f_equal.
    + destruct u; auto.
    + apply IHe.
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  - f_equal.
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    + f_equal.
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      * destruct b; auto.
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      * apply IHe1.
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    + apply IHe2.
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  - f_equal.
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    + apply mTypeEq_sym; auto.
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    + apply IHe.
Qed.

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Lemma expEq_trans e f g:
  expEq e f = true ->
  expEq f g = true ->
  expEq e g = true.
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Proof.
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  revert e f g; induction e;
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    destruct f; intros g eq1 eq2;
      destruct g; simpl in *; try congruence;
        try rewrite Nat.eqb_eq in *;
        subst; try auto.
  - andb_to_prop eq1;
      andb_to_prop eq2.
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    rewrite mTypeEq_compat_eq in L, L0; subst.
    rewrite mTypeEq_refl; simpl.
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    rewrite Qeq_bool_iff in *; lra.
  - andb_to_prop eq1;
      andb_to_prop eq2.
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    rewrite unopEq_compat_eq in *; subst.
    rewrite unopEq_refl; simpl.
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    eapply IHe; eauto.
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  - andb_to_prop eq1;
      andb_to_prop eq2.
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    rewrite binopEq_compat_eq in *; subst.
    rewrite binopEq_refl; simpl.
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    apply andb_true_iff.
    split; [eapply IHe1; eauto | eapply IHe2; eauto].
  - andb_to_prop eq1;
      andb_to_prop eq2.
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    rewrite mTypeEq_compat_eq in *; subst.
    rewrite mTypeEq_refl; simpl.
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    eapply IHe; eauto.
Qed.

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Module ExpOrderedType (V_ordered:OrderedType) <: OrderedType.
  Module V_orderedFacts := OrderedTypeFacts (V_ordered).

  Definition V := V_ordered.t.

  Fixpoint expCompare (e1:exp V) (e2:exp V) :=
    match e1, e2 with
    |Var _ n1, Var _ n2 => Nat.compare n1 n2
    |Var _ n1, _ => Lt
    | Const m1 v1, Const m2 v2 =>
      if mTypeEq m1 m2
      then V_ordered.compare v1 v2
      else (if morePrecise m1 m2 then Lt else Gt)
    | Const _ _, Var _ _ => Gt
    | Const _ _, _ => Lt
    | Unop u1 e1, Unop u2 e2 =>
      if unopEq u1 u2
      then expCompare e1 e2
      else (if unopEq u1 Neg then Lt else Gt)
    | Unop _ _, Binop _ _ _ => Lt
    | Unop _ _, Downcast _ _ => Lt
    | Unop _ _, _ => Gt
    | Downcast m1 e1, Downcast m2 e2 =>
      if mTypeEq m1 m2
      then expCompare e1 e2
      else (if morePrecise m1 m2 then Lt else Gt)
    | Downcast _ _, Binop _ _ _ => Lt
    | Downcast _ _, _ => Gt
    | Binop b1 e11 e12, Binop b2 e21 e22 =>
      let res := match b1, b2 with
                 | Plus, Plus => Eq
                 | Plus, _ => Lt
                 | Sub, Sub => Eq
                 | Sub, Plus => Gt
                 | Sub, _ => Lt
                 | Mult, Mult => Eq
                 | Mult, Div => Lt
                 | Mult, _ => Gt
                 | Div, Div => Eq
                 | Div, _ => Gt
                 end
      in
      match res with
      | Eq =>
        match expCompare e11 e21 with
        | Eq => expCompare e12 e22
        | Lt => Lt
        | Gt => Gt
        end
      | _ => res
      end
    |_ , _ => Gt
    end.

  Lemma expCompare_refl e: expCompare e e = Eq.
  Proof.
    induction e; simpl.
    - apply Nat.compare_refl.
    - rewrite mTypeEq_refl. apply V_orderedFacts.compare_refl.
    - rewrite unopEq_refl; auto.
    - rewrite IHe1, IHe2. destruct b; auto.
    - rewrite mTypeEq_refl; auto.
  Qed.

  (* Lemma expCompare_eq_compat_eq e1: *)
  (*   forall e2, *)
  (*     expCompare e1 e2 = Eq <-> *)
  (*     eq e1 e2. *)
  (* Proof. *)
  (*   induction e1; destruct e2; split; intros * cmp_res; simpl in *; *)
  (*     subst; try congruence. *)
  (*   - rewrite Nat.compare_eq_iff in cmp_res; subst; auto. *)
  (*   - inversion cmp_res; subst; simpl. apply Nat.compare_refl. *)
  (*   - destruct (mTypeEq m m0 *)
  (*     apply V_orderedFacts.compare_eq in cmp_res. *)
  (*     f_equal. *)
  (*     hnf in cmp_res. *)

  Lemma expCompare_eq_trans e1 :
    forall e2 e3,
      expCompare e1 e2 = Eq ->
      expCompare e2 e3 = Eq ->
      expCompare e1 e3 = Eq.
  Proof.
    induction e1; intros * eq12 eq23;
      destruct e2; destruct e3; simpl in *; try congruence.
    - rewrite Nat.compare_eq_iff in *; subst; auto.
    - destruct (mTypeEq m m0) eqn:?;
               [ destruct (mTypeEq m0 m1) eqn:? |
                 destruct (morePrecise m m0) eqn:?; congruence];
        [ | destruct (morePrecise m0 m1) eqn:?; congruence].
      type_conv. rewrite mTypeEq_refl.
      rewrite V_orderedFacts.compare_eq_iff in *;
        eapply V_orderedFacts.eq_trans; eauto.
    - destruct (unopEq u u0) eqn:?;
               destruct (unopEq u0 u1) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try rewrite unopEq_refl;
        try congruence.
      + eapply IHe1; eauto.
      + destruct (unopEq u0 Neg); congruence.
      + destruct (unopEq u Neg); congruence.
      + destruct (unopEq u Neg); congruence.
    - destruct b; destruct b0; try congruence;
        destruct b1; try congruence;
          destruct (expCompare e1_1 e2_1) eqn:?;
                   destruct (expCompare e2_1 e3_1) eqn:?;
                   try congruence; try erewrite IHe1_1; eauto.
    - destruct (mTypeEq m m0) eqn:?;
               destruct (mTypeEq m0 m1) eqn:?;
               type_conv;
        try rewrite mTypeEq_refl.
      + eapply IHe1; eauto.
      + destruct (morePrecise m0 m1); congruence.
      + destruct (morePrecise m m1); congruence.
      + destruct (morePrecise m m0); congruence.
  Qed.

  Lemma expCompare_antisym e1:
    forall e2,
      expCompare e1 e2 = CompOpp (expCompare e2 e1).
  Proof.
    induction e1; destruct e2; simpl; try auto.
    - apply Nat.compare_antisym.
    - rewrite mTypeEq_sym.
      destruct (mTypeEq m0 m) eqn:?;
               type_conv; try congruence; try rewrite mTypeEq_refl.
      + apply V_orderedFacts.compare_antisym.
      + destruct (morePrecise m m0) eqn:?;
                 destruct (morePrecise m0 m) eqn:?;
                 try (split; auto; fail).
        * pose proof (morePrecise_antisym _ _ Heqb0 Heqb1); type_conv;
            congruence.
        * destruct m, m0; unfold morePrecise in *; cbv; congruence.
    - rewrite unopEq_sym.
      destruct (unopEq u0 u) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try rewrite unopEq_refl, IHe1; try (apply IHe1).
      destruct (unopEq u Neg) eqn:?; try rewrite unopEq_compat_eq in *;
        destruct (unopEq u0 Neg) eqn:?; try rewrite unopEq_compat_eq in *;
        subst; simpl in *; try congruence.
      destruct u, u0; simpl in *; congruence.
    - destruct b, b0; simpl; try (split; auto; fail);
      destruct (expCompare e1_1 e2_1) eqn:first_comp;
      rewrite IHe1_1 in *; simpl in *;
        rewrite CompOpp_iff in first_comp;
        rewrite first_comp; simpl; try auto.
    - rewrite mTypeEq_sym.
      destruct (mTypeEq m0 m) eqn:?;
               type_conv; try auto.
      + destruct (morePrecise m m0) eqn:?;
                 destruct (morePrecise m0 m) eqn:?;
                 try (split; auto; fail).
        * pose proof (morePrecise_antisym _ _ Heqb0 Heqb1); type_conv;
            congruence.
        * destruct m, m0; unfold morePrecise in *; cbv; congruence.
  Qed.

  Lemma expCompare_lt_eq_is_lt e1:
    forall e2 e3,
      expCompare e1 e2 = Lt -> expCompare e2 e3 = Eq -> expCompare e1 e3 = Lt.
  Proof.
    induction e1; intros * compare_lt compare_eq; destruct e2; simpl in *;
      destruct e3; try congruence.
    - rewrite Nat.compare_eq_iff in compare_eq; subst; auto.
    - destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq  m0 m1) eqn:?.
      + pose proof (V_orderedFacts.compare_compat). unfold Proper in H.
        apply V_orderedFacts.compare_eq_iff in compare_eq.
        specialize (H v v (V_orderedFacts.eq_refl v) v0 v1 compare_eq).
        type_conv; rewrite mTypeEq_refl, <- H; auto.
      + rewrite mTypeEq_compat_eq in Heqb; subst.
        rewrite Heqb0. destruct (morePrecise m0 m1) eqn:?; congruence.
      + rewrite mTypeEq_compat_eq in Heqb0; subst.
        rewrite Heqb; destruct (morePrecise m m1) eqn:?; congruence.
      + destruct (morePrecise m0 m1); congruence.
    - destruct (unopEq u u0) eqn:?; destruct (unopEq u0 u1) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try rewrite unopEq_refl; try auto; try congruence.
      + eapply IHe1; eauto.
      + destruct (unopEq u0 Neg); congruence.
      + destruct (unopEq u Neg); try congruence.
        destruct (unopEq u u1); congruence.
      + destruct (unopEq u0 Neg); congruence.
    - destruct b; destruct b0; try congruence;
        destruct b1; try congruence;
          destruct (expCompare e1_1 e2_1) eqn:?;
               destruct (expCompare e2_1 e3_1) eqn:?;
               try congruence;
          try (erewrite IHe1_1; eauto; fail "");
          try erewrite expCompare_eq_trans; eauto.
    - destruct (mTypeEq m m0) eqn:?;
               destruct (mTypeEq m0 m1) eqn:?.
      + type_conv; subst. rewrite mTypeEq_refl. eapply IHe1; eauto.
      + destruct (morePrecise m0 m1); congruence.
      + rewrite mTypeEq_compat_eq in Heqb0; subst.
        rewrite Heqb. destruct (morePrecise m m1) eqn:?;  congruence.
      + destruct (morePrecise m0 m1); congruence.
  Qed.

  Lemma expCompare_eq_lt_is_lt e1:
    forall e2 e3,
      expCompare e1 e2 = Eq -> expCompare e2 e3 = Lt -> expCompare e1 e3 = Lt.
  Proof.
    induction e1; intros * compare_eq compare_lt; destruct e2; simpl in *;
      destruct e3; try congruence.
    - rewrite Nat.compare_eq_iff in compare_eq; subst; auto.
    - destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq  m0 m1) eqn:?.
      + pose proof (V_orderedFacts.compare_compat). unfold Proper in H.
        apply V_orderedFacts.compare_eq_iff in compare_eq.
        specialize (H v v0 compare_eq v1 v1 (V_orderedFacts.eq_refl v1)).
        type_conv; rewrite mTypeEq_refl, H; auto.
      + rewrite mTypeEq_compat_eq in Heqb; subst.
        rewrite Heqb0. destruct (morePrecise m0 m1) eqn:?; congruence.
      + rewrite mTypeEq_compat_eq in Heqb0; subst.
        rewrite Heqb; destruct (morePrecise m m1) eqn:?; congruence.
      + destruct (morePrecise m m0); congruence.
    - destruct (unopEq u u0) eqn:?; destruct (unopEq u0 u1) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try rewrite unopEq_refl; try auto; try congruence.
      + eapply IHe1; eauto.
      + rewrite Heqb0. destruct (unopEq u0 Neg); congruence.
      + destruct (unopEq u Neg); congruence.
      + destruct (unopEq u Neg); congruence.
    - destruct b; destruct b0;
        destruct b1; try congruence;
          destruct (expCompare e1_1 e2_1) eqn:?;
                   destruct (expCompare e2_1 e3_1) eqn:?;
                   try congruence;
          try (erewrite IHe1_1; eauto; fail "");
          try erewrite expCompare_eq_trans; eauto.
    - destruct (mTypeEq m m0) eqn:?;
               destruct (mTypeEq m0 m1) eqn:?.
      + type_conv; subst. rewrite mTypeEq_refl. eapply IHe1; eauto.
      + rewrite mTypeEq_compat_eq in Heqb; subst.
        rewrite Heqb0.
        destruct (morePrecise m0 m1); congruence.
      + rewrite mTypeEq_compat_eq in Heqb0; subst.
        rewrite Heqb. destruct (morePrecise m m1) eqn:?;  congruence.
      + destruct (morePrecise m m0); congruence.
  Qed.

  Definition eq e1 e2 :=
    expCompare e1 e2 = Eq.

  Definition lt (e1:exp V) (e2: exp V):=
    expCompare e1 e2 = Lt.

  Instance lt_strorder : StrictOrder lt.
  Proof.
    split.
    - unfold Irreflexive.
      unfold Reflexive.
      intros x; unfold complement.
      intros lt_x.
      induction x; unfold lt in *; simpl in lt_x.
      + rewrite PeanoNat.Nat.compare_refl in lt_x. congruence.
      + rewrite mTypeEq_refl, V_orderedFacts.compare_refl in *;
          congruence.
      + rewrite unopEq_refl in *; simpl in *.
        apply IHx; auto.
      + destruct b;
          destruct (expCompare x1 x1) eqn:?; try congruence.
      + rewrite mTypeEq_refl in lt_x.
        apply IHx; auto.
    - unfold Transitive.
      intros e1. unfold lt.
      induction e1; intros * lt_e1_e2 lt_e2_e3;
        simpl in *; destruct y; destruct z;
          simpl in *; try auto; try congruence.
      + rewrite <- nat_compare_lt in *. omega.
      + destruct (mTypeEq m m0) eqn:?;
                 destruct (mTypeEq m0 m1) eqn:?.
        * type_conv;
            rewrite mTypeEq_refl, V_orderedFacts.compare_lt_iff in *;
            eapply V_orderedFacts.lt_trans; eauto.
        * rewrite mTypeEq_compat_eq in Heqb; subst.
          rewrite Heqb0. assumption.
        * rewrite mTypeEq_compat_eq in Heqb0; subst.
          rewrite Heqb; assumption.
        * destruct (mTypeEq m m1) eqn:?.
          { rewrite mTypeEq_compat_eq in Heqb1; subst.
            destruct (morePrecise m0 m1) eqn:?;
                     destruct (morePrecise m1 m0) eqn:?;
                     try congruence.
            pose proof (morePrecise_antisym _ _ Heqb1 Heqb2).
            type_conv; congruence. }
          { destruct (morePrecise m m0) eqn:?;
                     destruct (morePrecise m0 m1) eqn:?;
                     try congruence.
            erewrite morePrecise_trans; eauto. }
      + destruct (unopEq u u0) eqn:?;
                 destruct (unopEq u0 u1) eqn:?;
                 try rewrite unopEq_compat_eq in *; subst;
          [ destruct (expCompare e1 y) eqn:?; try congruence;
            rewrite unopEq_refl;
            eapply IHe1; eauto
          | destruct (unopEq u0 Neg) eqn:?; try congruence;
            rewrite unopEq_compat_eq in *; subst
          | |].
        * rewrite Heqb0; auto.
        * destruct (unopEq u Neg) eqn:?; try congruence; rewrite unopEq_compat_eq in *; subst.
          rewrite Heqb; auto.
        * destruct (unopEq u u1) eqn:?; try congruence.
          rewrite unopEq_compat_eq in Heqb1; subst.
          destruct (unopEq u1 Neg) eqn:?; try congruence;
            destruct (unopEq u0 Neg) eqn:?; try congruence;
            rewrite unopEq_compat_eq in *; subst.
          simpl in *; congruence.
      + destruct b; destruct b0; destruct b1; try congruence;
          destruct (expCompare e1_1 y1) eqn:?; try congruence;
          destruct (expCompare y1 z1) eqn:?; try congruence;
          try (erewrite expCompare_eq_trans; eauto; fail);
          try (erewrite expCompare_eq_lt_is_lt; eauto; fail);
          try (erewrite expCompare_lt_eq_is_lt; eauto; fail);
          try (erewrite IHe1_1; eauto).
      + destruct (mTypeEq m m0) eqn:?;
                 destruct (mTypeEq m0 m1) eqn:?;
                 [type_conv; subst; rewrite mTypeEq_refl | | | ].
        * eapply IHe1; eauto.
        * rewrite mTypeEq_compat_eq in Heqb; subst.
          rewrite Heqb0; destruct (morePrecise m0 m1); congruence.
        * rewrite mTypeEq_compat_eq in Heqb0; subst.
          rewrite Heqb. destruct (morePrecise m m1); congruence.
        * destruct (mTypeEq m m1) eqn:?.
          { rewrite mTypeEq_compat_eq in Heqb1; subst.
            destruct (morePrecise m1 m0) eqn:?; try congruence.
            destruct (morePrecise m0 m1) eqn:?; try congruence.
            pose proof (morePrecise_antisym _ _ Heqb1 Heqb2).
            type_conv; subst. congruence. }
          { destruct (morePrecise m m0) eqn:?; try congruence.
            destruct (morePrecise m0 m1) eqn:?; try congruence.
            pose proof (morePrecise_trans _ _ _  Heqb2 Heqb3).
            rewrite H; auto. }
  Defined.

  Lemma eq_compat: Proper (eq ==> eq ==> iff) eq.
  Proof.
    unfold Proper; hnf.
    intros e1; induction e1;
    intros e2 e1_eq_e2; hnf;
    intros e3 e4 e3_eq_e4;
    unfold lt, eq in *;
    destruct e2,e3,e4; simpl in *; try congruence; try (split; auto; fail).
    - repeat rewrite Nat.compare_eq_iff in *; subst. split; try auto.
    - destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq m1 m2) eqn:?;
               [type_conv | | |].
      + rewrite V_orderedFacts.compare_eq_iff in *.
        rewrite (V_orderedFacts.compare_compat e1_eq_e2 e3_eq_e4).
        split; auto.
      + destruct (morePrecise m1 m2); congruence.
      + destruct (morePrecise m m0); congruence.
      + destruct (morePrecise m m0); congruence.
    - destruct (unopEq u u0) eqn:?;
               destruct (unopEq u1 u2) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try (destruct (unopEq u Neg); congruence);
            try (destruct (unopEq u1 Neg); congruence).
      specialize (IHe1 e2 e1_eq_e2 e3 e4 e3_eq_e4).
      simpl in *. destruct (unopEq u0 u2); try rewrite IHe1; split; auto.
    - destruct b; destruct b0; destruct b1; destruct b2; try congruence;
        try (split; auto; fail);
        destruct (expCompare e1_1 e2_1) eqn:?;
                 destruct (expCompare e3_1 e4_1) eqn:?;
                 try congruence;
        destruct (expCompare e1_1 e3_1) eqn:?;
                 destruct (expCompare e2_1 e4_1) eqn:?;
                 try (split; congruence);
      try (specialize (IHe1_2 _ e1_eq_e2 _ _ e3_eq_e4); simpl in *; rewrite IHe1_2 in *; split; auto; fail);
      try (split; try congruence; intros);
      try (specialize (IHe1_1 _ Heqc _ _ Heqc0); simpl in *; rewrite IHe1_1 in *; congruence);
      try (specialize (IHe1_1 _ Heqc _ _ Heqc0); simpl in *; rewrite <- IHe1_1 in *; congruence).
    -  destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq m1 m2) eqn:?;
               [type_conv | | |].
       + specialize (IHe1 _ e1_eq_e2 _ _ e3_eq_e4); simpl in *.
         destruct (mTypeEq m0 m2); try congruence.
         split; auto.
       + destruct (morePrecise m1 m2); congruence.
       + destruct (morePrecise m m0); congruence.
       + destruct (morePrecise m m0); congruence.
  Qed.

  Instance lt_compat: Proper (eq ==> eq ==> iff) lt.
  Proof.
    unfold Proper; hnf.
    intros e1; induction e1;
    intros e2 e1_eq_e2; hnf;
    intros e3 e4 e3_eq_e4;
    unfold lt, eq in *;
    destruct e2,e3,e4; simpl in *; try congruence; try (split; auto; fail).
    - rewrite Nat.compare_eq_iff in *; subst. split; try auto.
    - destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq m1 m2) eqn:?;
               [type_conv | | |].
      + rewrite V_orderedFacts.compare_eq_iff in *.
        rewrite (V_orderedFacts.compare_compat e1_eq_e2 e3_eq_e4).
        split; auto.
      + destruct (morePrecise m1 m2); congruence.
      + destruct (morePrecise m m0); congruence.
      + destruct (morePrecise m m0); congruence.
    - destruct (unopEq u u0) eqn:?;
               destruct (unopEq u1 u2) eqn:?;
               try rewrite unopEq_compat_eq in *; subst;
        try (destruct (unopEq u Neg); congruence);
            try (destruct (unopEq u1 Neg); congruence).
      specialize (IHe1 e2 e1_eq_e2 e3 e4 e3_eq_e4).
      simpl in *. destruct (unopEq u0 u2); try rewrite IHe1; split; auto.
    - pose proof eq_compat as eq_comp. unfold Proper, eq in eq_comp.
      destruct b, b0, b1, b2; try congruence; try (split; auto; fail);
        destruct (expCompare e1_1 e2_1) eqn:?;
                 destruct (expCompare e3_1 e4_1) eqn:?;
                 try congruence;
        destruct (expCompare e1_1 e3_1) eqn:?;
                 destruct (expCompare e2_1 e4_1) eqn:?;
                 try (split; congruence);
        try (specialize (IHe1_2 _ e1_eq_e2 _ _ e3_eq_e4); simpl in *; rewrite IHe1_2 in *; split; auto; fail);
        try (split; try congruence; intros);
        try (specialize (IHe1_1 _ Heqc _ _ Heqc0); simpl in *; rewrite IHe1_1 in *; congruence);
        try (specialize (IHe1_1 _ Heqc _ _ Heqc0); simpl in *; rewrite <- IHe1_1 in *; congruence);
        try (rewrite (eq_comp _ _ Heqc _ _ Heqc0) in *; congruence);
        try (rewrite <- (eq_comp _ _ Heqc _ _ Heqc0) in *; congruence).
    -  destruct (mTypeEq m m0) eqn:?; destruct (mTypeEq m1 m2) eqn:?;
               [type_conv | | |].
       + specialize (IHe1 _ e1_eq_e2 _ _ e3_eq_e4); simpl in *.
         destruct (mTypeEq m0 m2); try congruence.
         split; auto.
       + destruct (morePrecise m1 m2); congruence.
       + destruct (morePrecise m m0); congruence.
       + destruct (morePrecise m m0); congruence.
  Defined.

  Lemma compare_spec : forall x y, CompSpec eq lt x y (expCompare x y).
  Proof.
    intros e1 e2.
    destruct (expCompare e1 e2) eqn:?.
    - apply CompEq.
      unfold eq; auto.
    - apply CompLt. unfold lt; auto.
    - apply CompGt. unfold lt.
      rewrite expCompare_antisym in Heqc.
      rewrite CompOpp_iff in Heqc.
      simpl in *; auto.
  Qed.

End ExpOrderedType.

(* MARKER *)

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Fixpoint toRExp (e:exp Q) :=
  match e with
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  |Var _ v => Var R v
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  |Const m n => Const m (Q2R n)
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  |Unop o e1 => Unop o (toRExp e1)
  |Binop o e1 e2 => Binop o (toRExp e1) (toRExp e2)
  |Downcast m e1 => Downcast m (toRExp e1)
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  end.
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Fixpoint toREval (e:exp R) :=
  match e with
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  | Var _ v => Var R v
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  | Const _ n => Const M0 n
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  | Unop o e1 => Unop o (toREval e1)
  | Binop o e1 e2 => Binop o (toREval e1) (toREval e2)
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  | Downcast _ e1 =>   Downcast M0 (toREval e1)
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  end.
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Definition toRMap (d:nat -> option mType) (n:nat) :=
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  match d n with
  | Some m => Some M0
  | None => None
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  end.
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Arguments toRMap _ _/.

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(**
  Define a perturbation function to ease writing of basic definitions
**)
Definition perturb (r:R) (e:R) :=
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  (r * (1 + e))%R.
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Hint Unfold perturb.

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(**
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Define expression evaluation relation parametric by an "error" epsilon.
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The result value expresses float computations according to the IEEE standard,
using a perturbation of the real valued computation by (1 + delta), where
|delta| <= machine epsilon.
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**)
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Inductive eval_exp (E:env) (Gamma: nat -> option mType) :(exp R) -> R -> mType -> Prop :=
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| Var_load m x v:
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    Gamma x = Some m ->
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    E x = Some v ->
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    eval_exp E Gamma (Var R x) v m
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| Const_dist m n delta:
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    Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
    eval_exp E Gamma (Const m n) (perturb n delta) m
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| Unop_neg m f1 v1:
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    eval_exp E Gamma f1 v1 m ->
    eval_exp E Gamma (Unop Neg f1) (evalUnop Neg v1) m
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| Unop_inv m f1 v1 delta:
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    Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
    eval_exp E Gamma  f1 v1 m ->
    (~ v1 = 0)%R  ->
    eval_exp E Gamma (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
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| Downcast_dist m m1 f1 v1 delta:
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    (* Downcast expression f1 (evaluating to machine type m1), to a machine type m, less precise than m1.*)
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    isMorePrecise m1 m = true ->
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    Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
    eval_exp E Gamma f1 v1 m1 ->
    eval_exp E Gamma (Downcast m f1) (perturb v1 delta) m
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| Binop_dist m1 m2 op f1 f2 v1 v2 delta:
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    Rle (Rabs delta) (Q2R (mTypeToQ (join m1 m2))) ->
    eval_exp E Gamma f1 v1 m1 ->
    eval_exp E Gamma f2 v2 m2 ->
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    ((op = Div) -> (~ v2 = 0)%R) ->
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    eval_exp E Gamma (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta)  (join m1 m2).

Hint Constructors eval_exp.

(**
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  Show some simpler (more general) rule lemmata
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**)
Lemma Const_dist' m n delta v m' E Gamma:
  Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
  v = perturb n delta ->
  m' = m ->
  eval_exp E Gamma (Const m n) v m'.
Proof.
  intros; subst; auto.
Qed.

Hint Resolve Const_dist'.

Lemma Unop_neg' m f1 v1 v m' E Gamma:
  eval_exp E Gamma f1 v1 m ->
  v = evalUnop Neg v1 ->
  m' = m ->
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  eval_exp E Gamma (Unop Neg f1) v m'.
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Proof.
  intros; subst; auto.
Qed.

Hint Resolve Unop_neg'.

Lemma Unop_inv' m f1 v1 delta v m' E Gamma:
  Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
  eval_exp E Gamma  f1 v1 m ->
  (~ v1 = 0)%R  ->
  v = perturb (evalUnop Inv v1) delta ->
  m' = m ->
  eval_exp E Gamma (Unop Inv f1) v m'.
Proof.
  intros; subst; auto.
Qed.

Hint Resolve Unop_inv'.

Lemma Downcast_dist' m m1 f1 v1 delta v m' E Gamma:
  isMorePrecise m1 m = true ->
  Rle (Rabs delta) (Q2R (mTypeToQ m)) ->
  eval_exp E Gamma f1 v1 m1 ->
  v = (perturb v1 delta) ->
  m' = m ->
  eval_exp E Gamma (Downcast m f1) v m'.
Proof.
  intros; subst; eauto.
Qed.

Hint Resolve Downcast_dist'.

Lemma Binop_dist' m1 m2 op f1 f2 v1 v2 delta v m' E Gamma:
  Rle (Rabs delta) (Q2R (mTypeToQ m')) ->
  eval_exp E Gamma f1 v1 m1 ->
  eval_exp E Gamma f2 v2 m2 ->
  ((op = Div) -> (~ v2 = 0)%R) ->
  v = perturb (evalBinop op v1 v2) delta ->
  m' = join m1 m2 ->
  eval_exp E Gamma (Binop op f1 f2) v m'.
Proof.
  intros; subst; auto.
Qed.

Hint Resolve Binop_dist'.
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(**
  Define the set of "used" variables of an expression to be the set of variables
  occuring in it
**)
Fixpoint usedVars (V:Type) (e:exp V) :NatSet.t :=
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  match e with
  | Var _ x => NatSet.singleton x
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  | Unop u e1 => usedVars e1
  | Binop b e1 e2 => NatSet.union (usedVars e1) (usedVars e2)
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  | Downcast _ e1 => usedVars e1
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  | _ => NatSet.empty
  end.
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Lemma toRMap_eval_M0 f v E Gamma m:
  eval_exp E (toRMap Gamma) (toREval f) v m -> m = M0.
Proof.
  revert v E Gamma m.
  induction f; intros * eval_f; inversion eval_f; subst;
  repeat
    match goal with
    | H: context[toRMap _ _] |- _ => unfold toRMap in H
    | H: context[match ?Gamma ?v with | _ => _ end ] |- _ => destruct (Gamma v) eqn:?
    | H: Some ?m1 = Some ?m2 |- _ => inversion H; try auto
    | H: None = Some ?m |- _ => inversion H
    end; try auto.
  - eapply IHf; eauto.
  - eapply IHf; eauto.
  - assert (m1 = M0)
      by (eapply IHf1; eauto).
    assert (m2 = M0)
      by (eapply IHf2; eauto);
      subst; auto.
Qed.

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(**
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  If |delta| <= 0 then perturb v delta is exactly v.
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**)
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Lemma delta_0_deterministic (v:R) (delta:R):
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  (Rabs delta <= 0)%R ->
  perturb v delta = v.
Proof.
  intros abs_0; apply Rabs_0_impl_eq in abs_0; subst.
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  unfold perturb. lra.
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Qed.

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(**
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Evaluation with 0 as machine epsilon is deterministic
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**)
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Lemma meps_0_deterministic (f:exp R) (E:env) Gamma:
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  forall v1 v2,
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  eval_exp E (toRMap Gamma) (toREval f) v1 M0 ->
  eval_exp E (toRMap Gamma) (toREval f) v2 M0 ->
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  v1 = v2.
Proof.
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  induction f;
    intros v1 v2 ev1 ev2.
  - inversion ev1; inversion ev2; subst.
    rewrite H1 in H6.
    inversion H6; auto.
  - inversion ev1; inversion ev2; subst.
    simpl in *.
    rewrite Q2R0_is_0 in *;
    repeat (rewrite delta_0_deterministic; try auto).
  - inversion ev1; inversion ev2; subst; try congruence.
    + rewrite (IHf v0 v3); eauto.
    + rewrite (IHf v0 v3); eauto.
      simpl in *.
      rewrite Q2R0_is_0 in *;
        repeat (rewrite delta_0_deterministic; try auto).
  - inversion ev1; inversion ev2; subst.
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    assert (m0 = M0) by (eapply toRMap_eval_M0; eauto).
    assert (m3 = M0) by (eapply toRMap_eval_M0; eauto).
    assert (m1 = M0) by (eapply toRMap_eval_M0; eauto).
    assert (m2 = M0) by (eapply toRMap_eval_M0; eauto).
    subst.
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    rewrite (IHf1 v0 v4); try auto.
    rewrite (IHf2 v3 v5); try auto.
    simpl in *.
    rewrite Q2R0_is_0 in *.
    repeat (rewrite delta_0_deterministic; try auto).
  - inversion ev1; inversion ev2; subst.
    apply M0_least_precision in H1;
      apply M0_least_precision in H7; subst.
    rewrite (IHf v0 v3); try auto.
    simpl in *.
    rewrite Q2R0_is_0 in *.
    repeat (rewrite delta_0_deterministic; try auto).
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Qed.

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(**
Helping lemma. Needed in soundness proof.
For each evaluation of using an arbitrary epsilon, we can replace it by
evaluating the subexpressions and then binding the result values to different
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variables in the Environment.
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 **)
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Lemma binary_unfolding b f1 f2 E v1 v2 m1 m2 Gamma delta:
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  (b = Div -> ~(v2 = 0 )%R) ->
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  (Rabs delta <= Q2R (mTypeToQ (join m1 m2)))%R ->
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  eval_exp E Gamma f1 v1 m1 ->
  eval_exp E Gamma f2 v2 m2 ->
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  eval_exp E Gamma (Binop b f1 f2) (perturb (evalBinop b v1 v2) delta) (join m1 m2) ->
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  eval_exp (updEnv 2 v2 (updEnv 1 v1 emptyEnv))
           (updDefVars 2 m2 (updDefVars 1 m1 Gamma))
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             (Binop b (Var R 1) (Var R 2)) (perturb (evalBinop b v1 v2) delta) (join m1 m2).
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Proof.
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  intros no_div_zero eval_f1 eval_f2 eval_float.
  econstructor; try auto.
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Qed.

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Lemma eval_eq_env e:
  forall E1 E2 Gamma v m,
    (forall x, E1 x = E2 x) ->
    eval_exp E1 Gamma e v m ->
    eval_exp E2 Gamma e v m.
Proof.
  induction e; intros;
    (match_pat (eval_exp _ _ _ _ _) (fun H => inversion H; subst; simpl in *));
    try eauto.
  eapply Var_load; auto.
  rewrite <- (H n); auto.
Qed.

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(*
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(**
Analogous lemma for unary expressions.
**)
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Lemma unary_unfolding (e:exp R) (eps:R) (E:env) (v:R):
  (eval_exp eps E (Unop Inv e) v <->
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   exists v1,
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     eval_exp eps E e v1 /\
     eval_exp eps (updEnv 1 v1 E) (Unop Inv (Var R 1)) v).
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Proof.
  split.
  - intros eval_un.
    inversion eval_un; subst.
    exists v1.
    repeat split; try auto.
    constructor; try auto.
    constructor; auto.
  - intros exists_val.
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    destruct exists_val as [v1 [eval_f1 eval_e_E]].
    inversion eval_e_E; subst.
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    inversion H1; subst.
    unfold updEnv in *; simpl in *.
    constructor; auto.
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    inversion H3; subst; auto.
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Qed. *)
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(*   Using the parametric expressions, define boolean expressions for conditionals *)
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(* **)
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(* Inductive bexp (V:Type) : Type := *)
(*   leq: exp V -> exp V -> bexp V *)
(* | less: exp V -> exp V -> bexp V. *)
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(**
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  Define evaluation of boolean expressions
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 **)
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(* Inductive bval (E:env): (bexp R) -> Prop -> Prop := *)
(*   leq_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R): *)
(*     eval_exp E f1 v1 -> *)
(*     eval_exp E f2 v2 -> *)
(*     bval E (leq f1 f2) (Rle v1 v2) *)
(* |less_eval (f1:exp R) (f2:exp R) (v1:R) (v2:R): *)
(*     eval_exp E f1 v1 -> *)
(*     eval_exp E f2 v2 -> *)
(*     bval E (less f1 f2) (Rlt v1 v2). *)
(* (** *)
(*  Simplify arithmetic later by making > >= only abbreviations *)
(* **) *)
(* Definition gr := fun (V:Type) (f1: exp V) (f2: exp V) => less f2 f1. *)
(* Definition greq := fun (V:Type) (f1:exp V) (f2: exp V) => leq f2 f1. *)