Expressions.v 12 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 Coq.QArith.Qreals.
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Require Import Daisy.Infra.RealRationalProps Daisy.Infra.RationalSimps.
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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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(** TODO: simplify pattern matching **)
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Definition binopEqBool (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 binopEqBool_refl b:
  binopEqBool b b = true.
Proof.
  case b; auto.
Qed.

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

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Definition unopEqBool (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 unopEqBool_refl b:
  unopEqBool b b = true.
Proof.
  case b; auto.
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 :=
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  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 expEqBool (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 => andb (mTypeEqBool m1 m2) (Qeq_bool n1 n2)
  | Unop o1 e11, Unop o2 e22 => andb (unopEqBool o1 o2) (expEqBool e11 e22)
  | Binop o1 e11 e12, Binop o2 e21 e22 => andb (binopEqBool o1 o2) (andb (expEqBool e11 e21) (expEqBool e12 e22))
  | Downcast m1 f1, Downcast m2 f2 => andb (mTypeEqBool m1 m2) (expEqBool f1 f2)
  | _, _ => false
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  end.

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Lemma expEqBool_refl e:
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  expEqBool e e = true.
Proof.
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  induction e; try (apply andb_true_iff; split); simpl in *; auto; try (apply EquivEqBoolEq; auto). 
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  - symmetry; apply beq_nat_refl.
  - apply Qeq_bool_iff; lra.
  - case u; auto.
  - case b; auto.
  - apply andb_true_iff; split.
    apply IHe1. apply IHe2.
Qed.
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Lemma beq_nat_sym a b:
  beq_nat a b = beq_nat b a.
Proof.
  case_eq (a =? b); intros.
  - apply beq_nat_true in H.
    rewrite H.
    apply beq_nat_refl. 
  - apply beq_nat_false in H.
    case_eq (b =? a); intros.
    + apply beq_nat_true in H0.
      rewrite H0 in H.
      auto.
    + auto.
Qed.      

Lemma expEqBool_sym e e':
  expEqBool e e' = expEqBool e' e.
Proof.
  revert e'.
  induction e; intros e'; destruct e'; simpl; try auto.
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  - apply beq_nat_sym.
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  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply Qeq_bool_sym.
  - f_equal.
    + destruct u; auto.
    + apply IHe.
  - f_equal.      
    + destruct b; auto.
    + f_equal.
      * apply IHe1.
      * apply IHe2.
  - f_equal.
    + apply mTypeEqBool_sym; auto.
    + apply IHe.
Qed.

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Lemma expEqBool_trans e f g:
  expEqBool e f = true ->
  expEqBool f g = true ->
  expEqBool e g = true.
Proof.
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  revert e f g; induction e; destruct f; intros; simpl in H; inversion H; rewrite H; clear H; destruct g; simpl in H0; inversion H0; rewrite H0; clear H0; try (apply andb_true_iff in H1; destruct H1; apply andb_true_iff in H2; destruct H2; simpl).
  - apply beq_nat_true in H2.
    apply beq_nat_true in H1.
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    subst.
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    unfold expEqBool.
    rewrite <- beq_nat_refl. 
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    auto.
  - apply EquivEqBoolEq in H1.
    apply EquivEqBoolEq in H.
    subst.
    rewrite mTypeEqBool_refl; simpl.
    apply Qeq_bool_iff in H2.
    apply Qeq_bool_iff in H0.
    apply Qeq_bool_iff.
    lra.
  - assert (u = u0) by (destruct u; destruct u0; inversion H1; auto).
    assert (u0 = u1) by (destruct u0; destruct u1; inversion H; auto).
    subst.
    assert (unopEqBool u1 u1 = true) by (destruct u1; auto).
    apply andb_true_iff; split; try auto.
    eapply IHe; eauto.
  - apply andb_true_iff; split.
    + destruct b; destruct b0; destruct b1; auto.
    + apply andb_true_iff in H2; destruct H2.
      apply andb_true_iff in H0; destruct H0.
      apply andb_true_iff; split.
      eapply IHe1; eauto.
      eapply IHe2; eauto.
  - apply EquivEqBoolEq in H1.
    apply EquivEqBoolEq in H.
    subst.
    rewrite mTypeEqBool_refl; simpl.
    eapply IHe; eauto.
Qed.

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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 =>  (toREval e1)
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  end.
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Definition toREvalEnv (E:env) : env :=
  fun (n:nat) =>
    let s := (E n) in
    match s with
    | None => None
    | Some (r, _) => Some (r, M0)
    end.


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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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(**
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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) (defVars: nat -> option mType) :(exp R) -> R -> mType -> Prop :=
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| Var_load m x v:
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    defVars x = Some m ->
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    E x = Some (v, m) ->
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    eval_exp E defVars (Var R x) v m
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| Const_dist m n delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
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    eval_exp E defVars (Const m n) (perturb n delta) m
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| Unop_neg m f1 v1:
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    eval_exp E defVars f1 v1 m ->
    eval_exp E defVars (Unop Neg f1) (evalUnop Neg v1) m
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| Unop_inv m f1 v1 delta:
    Rle (Rabs delta) (Q2R (meps m)) ->
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    eval_exp E defVars  f1 v1 m ->
    eval_exp E defVars (Unop Inv f1) (perturb (evalUnop Inv v1) delta) m
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| Binop_dist m1 m2 op f1 f2 v1 v2 delta:
    Rle (Rabs delta) (Q2R (meps (computeJoin m1 m2))) ->
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    eval_exp E defVars f1 v1 m1 ->
    eval_exp E defVars f2 v2 m2 ->
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    ((op = Div) -> (~ v2 = 0)%R) ->
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    eval_exp E defVars (Binop op f1 f2) (perturb (evalBinop op v1 v2) delta)  (computeJoin m1 m2)
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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 ->
    Rle (Rabs delta) (Q2R (meps m)) ->
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    eval_exp E defVars f1 v1 m1 ->
    eval_exp E defVars (Downcast m f1) (perturb v1 delta) m.
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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
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  | 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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(**
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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.
  unfold perturb.
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  lra.
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Qed.

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Lemma general_meps_0_deterministic (f:exp R) (E:env) defVars:
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  forall v1 v2 m1,
    m1 = M0 ->
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    eval_exp E defVars (toREval f) v1 m1 ->
    eval_exp E defVars (toREval f) v2 M0 ->
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    v1 = v2.
Proof.
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  induction f; intros * m10_eq eval_v1 eval_v2.
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  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
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    rewrite H6 in H1; inversion H1; subst; auto.
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  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
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    + apply Ropp_eq_compat. apply (IHf v0 v3 M0); auto.     
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    + inversion H4.
    + inversion H5.
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    + rewrite (IHf v0 v3 M0); auto.
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  - inversion eval_v1; inversion eval_v2; subst; auto;
      try repeat (repeat rewrite delta_0_deterministic; simpl in *; rewrite Q2R0_is_0 in *; subst; auto); simpl.
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    destruct m0; destruct m2; inversion H5.
    destruct m3; destruct m4; inversion H11.
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    simpl in *.
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    rewrite (IHf1 v0 v4 M0); auto.
    rewrite (IHf2 v5 v3 M0); auto.
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    rewrite Q2R0_is_0 in H2,H12.
    rewrite delta_0_deterministic; auto.
    rewrite delta_0_deterministic; auto.
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  - simpl toREval in eval_v1.
    simpl toREval in eval_v2.
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    apply (IHf v1 v2 m1); auto.
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Qed.

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(* Lemma rnd_0_deterministic f E m v: *)
(*   eval_exp E (toREval (Downcast m f)) v M0 <-> *)
(*   eval_exp E (toREval f) v M0. *)
(* Proof. *)
(*   split; intros. *)
(*   - simpl in H. auto. *)
(*   - simpl; auto. *)
(* 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) defVars:
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  forall v1 v2,
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  eval_exp E defVars (toREval f) v1 M0 ->
  eval_exp E defVars (toREval f) v2 M0 ->
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  v1 = v2.
Proof.
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  intros v1 v2 ev1 ev2.
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  assert (M0 = M0) by auto.
  apply (general_meps_0_deterministic f H ev1 ev2). 
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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 m E vF defVars:
  eval_exp E defVars (Binop b f1 f2) vF m ->
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  exists vF1 vF2 m1 m2,
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    m = computeJoin m1 m2 /\
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    eval_exp E defVars f1 vF1 m1 /\
    eval_exp E defVars f2 vF2 m2 /\
    eval_exp (updEnv 2 m2 vF2 (updEnv 1 m1 vF1 emptyEnv))
             (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n)
             (Binop b (Var R 1) (Var R 2)) vF m.
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Proof.
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  intros eval_float.
  inversion eval_float; subst.
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  exists v1 ; exists v2; exists m1; exists m2; repeat split; try auto.
  eapply Binop_dist; eauto.
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  - pose proof (isMorePrecise_refl m1).
    eapply Var_load; eauto.
  - pose proof (isMorePrecise_refl m2).
    eapply Var_load; eauto.
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Qed.

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(* Analogous lemma for unary expressions. *)
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Lemma unary_unfolding (e:exp R) (m:mType) (E:env) (v:R) defVars:
  (eval_exp E defVars (Unop Inv e) v m ->
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   exists v1 m1,
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     eval_exp E defVars e v1 m1 /\
     eval_exp (updEnv 1 m1 v1 E) (fun n => if (n =? 1) then Some m1 else defVars n) (Unop Inv (Var R 1)) v m).
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Proof.
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  intros eval_un.
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    inversion eval_un; subst.
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    exists v1; exists m.
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    repeat split; try auto.
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    econstructor; try auto.
    pose proof (isMorePrecise_refl m).
    econstructor; eauto.
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Qed.
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(**
  Using the parametric expressions, define boolean expressions for conditionals
**)
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. *)