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thue-morse.v
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thue-morse.v
@ -859,7 +859,6 @@ Proof.
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rewrite nth_middle in L. apply L. rewrite H. reflexivity.
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rewrite nth_middle in L. apply L. rewrite H. reflexivity.
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Qed.
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Qed.
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Lemma tm_step_next_range :
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Lemma tm_step_next_range :
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forall (n : nat) (l1 l2 : list bool) (b : bool),
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forall (n : nat) (l1 l2 : list bool) (b : bool),
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tm_step n = l1 ++ b :: l2
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tm_step n = l1 ++ b :: l2
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@ -869,12 +868,25 @@ Proof.
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assert (nth_error (tm_step n) (length l1) = Some b).
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assert (nth_error (tm_step n) (length l1) = Some b).
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generalize H. apply list_concat_to_pos.
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generalize H. apply list_concat_to_pos.
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rewrite tm_build.
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rewrite tm_build.
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assert (I: length l1 < 2^n).
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assert (length l1 < length (tm_step n)).
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rewrite <- tm_size_power2. generalize H. apply list_app_length_lt.
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generalize H. apply list_app_length_lt.
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rewrite nth_error_app1. apply H0.
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rewrite nth_error_app1. apply H0. apply H1.
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rewrite tm_size_power2. apply I.
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Qed.
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Qed.
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Lemma tm_step_next_range' :
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forall (n k : nat) (l1 l2 : list bool) (b : bool),
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tm_step n = l1 ++ b :: l2
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-> nth_error (tm_step (n + k)) (length l1) = Some b.
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Proof.
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intros n k l1 l2 b. intros H.
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induction k.
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- rewrite Nat.add_0_r. generalize H. apply list_concat_to_pos.
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- rewrite Nat.add_succ_r.
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simpl.
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apply nth_error_split in IHk.
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Lemma tm_step_next_range2 :
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Lemma tm_step_next_range2 :
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forall (n : nat) (l1 l2 : list bool) (b : bool),
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forall (n : nat) (l1 l2 : list bool) (b : bool),
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tm_step n = l1 ++ b :: l2
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tm_step n = l1 ++ b :: l2
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