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thue-morse.v
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thue-morse.v
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@ -904,7 +904,7 @@ Proof.
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Qed.
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Qed.
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Theorem tm_step_stable : forall (n m k : nat),
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Theorem tm_step_stable : forall (n m k : nat),
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k < 2^n -> k < 2^m -> nth_error(tm_step n) k = nth_error (tm_step m) k.
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k < 2^n -> k < 2^m -> nth_error (tm_step n) k = nth_error (tm_step m) k.
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Proof.
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Proof.
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intros n m k. intros.
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intros n m k. intros.
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assert (I: n < m /\ max n m = m \/ m <= n /\ max n m = n).
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assert (I: n < m /\ max n m = m \/ m <= n /\ max n m = n).
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@ -1089,11 +1089,26 @@ Lemma tm_step_cancel_high_bits :
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<-> odd n = true.
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<-> odd n = true.
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Proof.
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Proof.
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intros k b n m. intros H I.
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intros k b n m. intros H I.
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assert (J: nth_error (tm_step (S n)) (2^n-1) = nth_error (tm_step (S n)) (2^n)
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<-> odd n = true).
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rewrite tm_step_single_bit_index.
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assert (nth_error (tm_step n) (2^n - 1) = nth_error (tm_step (S n)) (2^n-1)).
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apply tm_step_stable.
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assert (2^n - 1 < 2^n).
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apply Nat.sub_lt. replace (1) with (2^0) at 1. apply Nat.pow_le_mono_r.
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easy. apply le_0_n. simpl. reflexivity. apply Nat.lt_0_1.
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rewrite Nat.succ_lt_mono. simpl.
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(*
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(*
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assert (K: nth_error (tm_step n) a = Some (odd n)). rewrite I.
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assert (K: nth_error (tm_step n) a = Some (odd n)). rewrite I.
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apply tm_step_repunit.
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apply tm_step_repunit.
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*)
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*)
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Lemma tm_step_next_range :
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forall (n k : nat) (b : bool),
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nth_error (tm_step n) k = Some b -> nth_error (tm_step (S n)) k = Some b.
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Theorem tm_step_stable : forall (n m k : nat),
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k < 2^n -> k < 2^m -> nth_error(tm_step n) k = nth_error (tm_step m) k.
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