diff --git a/thue-morse.v b/thue-morse.v
index 2e6eca0..5ec2df5 100644
--- a/thue-morse.v
+++ b/thue-morse.v
@@ -1209,23 +1209,23 @@ Proof.
   apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
   assumption.
 
-  assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases. destruct G.
+  assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases.
+
+  destruct G as [G1|G2].
 
   (* k = 0 *)
-  rewrite H0. rewrite Nat.mul_0_l. rewrite Nat.add_0_l.
+  rewrite G1. rewrite Nat.mul_0_l. rewrite Nat.add_0_l.
   replace (nth_error (tm_step n) 0) with (Some false).
   assert (S: n < S j \/ S j <= n). apply Nat.lt_ge_cases.
-  destruct S. rewrite Nat.lt_succ_r in H1.
-  apply Nat.pow_le_mono_r with (a := 2) in H1.
-  rewrite <- tm_size_power2 in H1. rewrite <- nth_error_None in H1.
-  rewrite H1. easy. easy.
-  rewrite Nat.le_succ_l in H1. apply Nat.pow_lt_mono_r with (a := 2) in H1.
+  destruct S. rewrite Nat.lt_succ_r in H0.
+  apply Nat.pow_le_mono_r with (a := 2) in H0.
+  rewrite <- tm_size_power2 in H0. rewrite <- nth_error_None in H0.
+  rewrite H0. easy. easy.
+  rewrite Nat.le_succ_l in H0. apply Nat.pow_lt_mono_r with (a := 2) in H0.
   rewrite tm_step_stable with (m := m). rewrite N. easy. assumption. assumption.
   apply Nat.lt_1_2. rewrite tm_step_head_1. reflexivity.
 
   (* k < 0 *)
-  rename H0 into G.
-
   assert (O: k*2^m + 2^j < 2^n). apply lt_split_bits; assumption.
   assert (P: 2^n <= 2^(m+n)). apply Nat.pow_le_mono_r. easy. apply Nat.le_add_l.