Prove IEEE connection in Coq

coq/Commands.v

@@ -86,3 +86,19 @@ Fixpoint liveVars V (f:cmd V) :NatSet.t :=
   | Let _ _ e g => NatSet.union (usedVars e) (liveVars g)
   | Ret e => usedVars e
   end.
+
+Lemma bstep_eq_env f:
+  forall E1 E2 Gamma v m,
+  (forall x, E1 x = E2 x) ->
+  bstep f E1 Gamma v m ->
+  bstep f E2 Gamma v m.
+Proof.
+  induction f; intros * eq_envs bstep_E1;
+    inversion bstep_E1; subst; simpl in *.
+  -  eapply eval_eq_env in H7; eauto. eapply let_b; eauto.
+     eapply IHf. instantiate (1:=(updEnv n v0 E1)).
+     + intros; unfold updEnv.
+       destruct (x=? n); auto.
+     + auto.
+  - apply ret_b. eapply eval_eq_env; eauto.
+Qed.
\ No newline at end of file

coq/Expressions.v

@@ -432,6 +432,19 @@ Proof.
   econstructor; try auto.
 Qed.
 
+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.
+
 (*
 (**
 Analogous lemma for unary expressions.

coq/Infra/MachineType.v

@@ -146,17 +146,17 @@ Definition join (m1:mType) (m2:mType) :=
 (*   destruct m1, m2; simpl in *; cbv in *; try congruence; try auto. *)
 (* Qed. *)
 
-Definition maxExponent (m:mType) :Z :=
+Definition maxExponent (m:mType) :positive :=
   match m with
-  | M0 => 0
+  | M0 => 1
   | M16 => 15
   | M32 => 127
   | M64 => 1023
   end.
 
-Definition minExponentPos (m:mType) :Z :=
+Definition minExponentPos (m:mType) :positive :=
   match m with
-  | M0 => 0
+  | M0 => 1
   | M16 => 14
   | M32 => 126
   | M64 => 1022
@@ -169,16 +169,16 @@ Goldberg - Handbook of Floating-Point Arithmetic: (p.183)
 which simplifies to 2^(e_max) for base 2
 **)
 Definition maxValue (m:mType) :=
-  Qpower (2#1) (maxExponent m).
+ (Z.pow_pos 2 (maxExponent m)) # 1.
 
 Definition minValue (m:mType) :=
-  Qinv (Qpower (2#1) (minExponentPos m)).
+  Qinv ((Z.pow_pos 2 (minExponentPos m)) # 1).
 
 (** Goldberg - Handbook of Floating-Point Arithmetic: (p.183)
   𝛃^(e_min -p + 1) = 𝛃^(e_min -1) for base 2
 **)
 Definition minDenormalValue (m:mType) :=
-  Qinv (Qpower (2#1) (minExponentPos m - 1)).
+  Qinv (Z.pow_pos 2 (minExponentPos m - 1) # 1).
 
 Definition normal (v:Q) (m:mType) :=
   Qle_bool (minValue m) (Qabs v) && Qle_bool (Qabs v) (maxValue m).
@@ -190,7 +190,7 @@ Definition Normal (v:R) (m:mType) :=
   (Q2R (minValue m) <= (Rabs v) /\ (Rabs v) <= Q2R (maxValue m))%R.
 
 Definition Denormal (v:R) (m:mType) :=
-  ((Rabs v) <= Q2R (minValue m) /\ ~ (v = 0))%R.
+  ((Rabs v) < Q2R (minValue m) /\ ~ (v = 0))%R.
 (**
   Predicate that is true if and only if the given value v is a valid
   floating-point value according to the the type m.