Update

thue-morse.v

@@ -1192,11 +1192,43 @@ Qed.
 
 Theorem tm_step_flip_low_bit :
   forall (n m k j : nat),
-  0 < k -> j < m -> k * 2^m < 2^n
+  j < m -> k * 2^m < 2^n
   -> nth_error (tm_step n) (k * 2^m) <> nth_error (tm_step n) (k * 2^m + 2^j).
 Proof.
-  intros n m k j. intros G H I.
+  intros n m k j. intros H I.
 
+  assert (L: nth_error (tm_step m) 0 = Some false).
+  rewrite tm_step_head_1. simpl. reflexivity.
+
+  assert (M: 2^j < 2^m).
+  apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption.
+
+  assert (N: nth_error (tm_step m) (2^j) = Some true).
+  replace (nth_error (tm_step m) (2^j)) with (nth_error (tm_step (S j)) (2^j)).
+  rewrite tm_step_single_bit_index. reflexivity.
+  apply tm_step_stable. rewrite <- Nat.mul_1_l at 1.
+  rewrite Nat.pow_succ_r. rewrite <- Nat.mul_lt_mono_pos_r.
+  apply Nat.lt_1_2.
+  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
+  apply Nat.le_0_l. assumption.
+
+  assert (G: k = 0 \/ 0 < k). apply Nat.eq_0_gt_0_cases.
+  destruct G.
+
+  (* k = 0 *)
+  rewrite H0. 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.
+  rewrite tm_step_stable with (m := m). rewrite N. easy. assumption. assumption.
+  apply Nat.lt_1_2. rewrite tm_step_head_1. simpl. reflexivity.
+
+  (* k < 0 *)
+  rename H0 into G.
   assert ( nth_error (tm_step m) 0
          = nth_error (tm_step m) (2^j)
        <-> nth_error (tm_step (m+n)) (k * 2^m + 0)
@@ -1217,8 +1249,7 @@ Proof.
     with (nth_error (tm_step n) (k * 2 ^ m)) in H0.
   replace (nth_error (tm_step (m + n)) (k * 2 ^ m + 2^j))
     with (nth_error (tm_step n) (k * 2 ^ m + 2^j)) in H0.
-  replace (nth_error (tm_step m) 0) with (Some false) in H0.
-  replace (nth_error (tm_step m) (2^j)) with (Some true) in H0.
+  rewrite L in H0. rewrite N in H0.
 
   destruct (nth_error (tm_step n) (k * 2 ^ m)).
   destruct (nth_error (tm_step n) (k * 2 ^ m + 2^j)).
@@ -1232,15 +1263,6 @@ Proof.
   rewrite nth_error_nth' with (d := false). easy.
   rewrite tm_size_power2. apply lt_split_bits.
   assumption. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption. assumption.
-  replace (nth_error (tm_step m) (2^j)) with (nth_error (tm_step (S j)) (2^j)).
-  rewrite tm_step_single_bit_index. reflexivity.
-  apply tm_step_stable. rewrite <- Nat.mul_1_l at 1.
-  rewrite Nat.pow_succ_r. rewrite <- Nat.mul_lt_mono_pos_r.
-  apply Nat.lt_1_2.
-  rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
-  apply Nat.le_0_l. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
-  assumption.
-  rewrite tm_step_head_1. simpl. reflexivity.
 
   apply tm_step_stable. apply lt_split_bits.
   assumption. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. assumption. assumption.
@@ -1255,6 +1277,8 @@ Proof.
   generalize H1. generalize I. apply Nat.lt_le_trans.
 Qed.
 
+
+
 (* vérifier si les deux sont nécessaires *)
 Require Import BinPosDef.
 Require Import BinPos.