trying something else

coq/Expressions.v

@@ -180,6 +180,50 @@ Proof.
     + apply IHe.
 Qed.
 
+Lemma expEqBool_trans e f g:
+  expEqBool e f = true ->
+  expEqBool f g = true ->
+  expEqBool e g = true.
+Proof.
+  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; apply andb_true_iff in H1; destruct H1; apply andb_true_iff in H2; destruct H2; simpl.
+  - apply EquivEqBoolEq in H1.
+    apply EquivEqBoolEq in H.
+    apply beq_nat_true in H2.
+    apply beq_nat_true in H0.
+    subst.
+    rewrite <- beq_nat_refl,mTypeEqBool_refl.
+    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.
+
+    
+
+
 Fixpoint toRExp (e:exp Q) :=
   match e with
   |Var _ m v => Var R m v