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thue-morse.v
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thue-morse.v
@ -615,19 +615,16 @@ Proof.
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Qed.
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Lemma tm_step_succ_double_index : forall (n k : nat),
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2 * k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step n) (S (2*k)).
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k < 2^n -> nth_error (tm_step n) k <> nth_error (tm_step (S n)) (S (2*k)).
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Proof.
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intros n k. intro H. rewrite tm_step_double_index.
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replace (nth_error (tm_step (S n)) (2*k)) with (nth_error (tm_step n) (2*k)).
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rewrite Nat.mul_comm. replace (2) with (2^1).
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replace (S (k*2^1)) with (k*2^1 + 2^0).
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apply tm_step_flip_low_bit. apply Nat.lt_0_1.
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simpl. rewrite Nat.mul_comm. assumption.
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rewrite Nat.add_1_r. reflexivity. easy.
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apply tm_step_stable. assumption.
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assert (2^n < 2^(S n)). apply Nat.pow_lt_mono_r.
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apply Nat.lt_1_2. apply Nat.lt_succ_diag_r.
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generalize H0. generalize H. apply Nat.lt_trans.
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simpl. rewrite Nat.add_0_r. replace (k*2) with (k+k).
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apply Nat.add_lt_mono; assumption.
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rewrite Nat.mul_comm. simpl. rewrite Nat.add_0_r. reflexivity.
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simpl. rewrite Nat.add_1_r. reflexivity. reflexivity.
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Qed.
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