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Thomas Baruchel
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thue-morse.v
172
thue-morse.v
@ -1113,169 +1113,27 @@ Lemma tm_step_next_range2_neighbor' : forall (n k : nat),
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- induction m.
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+ intros L. rewrite Nat.add_0_r in L. rewrite Nat.add_0_r in L.
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apply tm_step_next_range2_neighbor'. apply H. apply L.
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+
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+ intros L.
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rewrite Nat.add_succ_r in L. rewrite Nat.add_succ_r in L.
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assert (S k < 2^(S n + m)).
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assert (2^n < 2^(S n + m)).
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assert (n < S n + m). assert (n < S n). apply Nat.lt_succ_diag_r.
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generalize H0. apply Nat.lt_lt_add_r.
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generalize H0. apply Nat.pow_lt_mono_r. apply Nat.lt_1_2.
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generalize H0. generalize H. apply Nat.lt_trans.
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assert (H1 := H0). apply Nat.lt_succ_l in H1.
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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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rewrite <- Nat.add_succ_l.
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replace (2^(n + S m)) with (2^(n+m) + 2^(n+m)).
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rewrite Nat.add_assoc.
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rewrite Nat.add_assoc.
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(* rewrite Nat.add_succ_r. *)
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generalize L.
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apply tm_step_next_range2_neighbor'. rewrite <- Nat.add_succ_l.
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- assert (k < length (tm_step n)). rewrite tm_size_power2. apply I.
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apply nth_error_nth' with (d :=false) in H0.
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assert (S k < length (tm_step n)). rewrite tm_size_power2. apply H.
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apply nth_error_nth' with (d :=false) in H1.
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intros J. rewrite H0 in J. rewrite H1 in J.
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apply tm_step_next_range2' in H0. apply tm_step_next_range2' in H1.
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induction m.
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+ rewrite Nat.add_0_r. rewrite Nat.add_0_r.
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rewrite H0. rewrite H1. inversion J. rewrite H3. reflexivity.
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+
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assert (U: S k < 2^(S n+ m)).
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assert (N: k < 2^n). apply I.
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assert (L: 2^n < 2^(S n + m)).
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assert (M: n < S n + m). rewrite Nat.add_succ_comm.
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apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
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apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply M.
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generalize L. generalize H. apply Nat.lt_trans.
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assert (K := U). apply Nat.lt_succ_l in K.
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assert (nth_error (tm_step n) k = Some (nth k (tm_step n) false)).
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apply nth_error_nth'. rewrite tm_size_power2. apply I.
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assert (nth_error (tm_step n) (S k) = Some (nth (S k) (tm_step n) false)).
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apply nth_error_nth'. rewrite tm_size_power2. apply H.
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rewrite <- H2 in J. rewrite <- H3 in J.
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rewrite <- tm_step_next_range2_neighbor' in L.
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assert (nth_error (tm_step n) k = nth_error (tm_step (S n + m)) k).
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generalize K. generalize I. apply tm_step_stable.
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generalize H1. generalize I. apply tm_step_stable.
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assert (nth_error (tm_step n) (S k) = nth_error (tm_step (S n + m)) (S k)).
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generalize U. generalize H. apply tm_step_stable.
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generalize H0. generalize H. apply tm_step_stable.
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rewrite <- H2 in L. rewrite <- H3 in L. apply L.
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rewrite H4 in J. rewrite H5 in J.
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assert (nth_error (tm_step (S n + m)) k = Some (nth k (tm_step (S n + m)) false)).
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generalize K. rewrite <- tm_size_power2. apply nth_error_nth'.
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assert (nth_error (tm_step (S n + m)) (S k) = Some (nth (S k) (tm_step (S n + m)) false)).
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generalize U. rewrite <- tm_size_power2. apply nth_error_nth'.
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rewrite H6 in J. rewrite H7 in J.
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rewrite Nat.add_succ_comm in J. inversion J.
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assert ( S k + 2^(n + S m) < 2^(S n + S m)
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S k < 2^n ->
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eqb (nth k (tm_step n) false) (nth (S k) (tm_step n) false)
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= eqb
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partir de H9 et appliquer :
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Lemma tm_step_next_range2_neighbor : forall (n k : nat),
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S k < 2^n ->
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eqb (nth k (tm_step n) false) (nth (S k) (tm_step n) false)
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= eqb
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(nth (k + 2^n) (tm_step (S n)) false) (nth (S k + 2^n) (tm_step (S n)) false).
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assert (nth_error (tm_step (S n + m)) (k + 2 ^ (n + m))
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= nth_error (tm_step (S n + m)) (k + 2 ^ (n + m))
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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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assert (U: S k < 2^(S n+ m)).
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assert (N: k < 2^n). apply I.
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assert (L: 2^n < 2^(S n + m)).
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assert (M: n < S n + m). rewrite Nat.add_succ_comm.
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apply Nat.lt_add_pos_r. apply Nat.lt_0_succ.
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apply Nat.pow_lt_mono_r. apply Nat.lt_1_2. apply M.
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generalize L. generalize H. apply Nat.lt_trans.
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assert (K := U). apply Nat.lt_succ_l in K.
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assert (k < length (tm_step (S n + m))).
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rewrite tm_size_power2. apply K.
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apply nth_error_nth' with (d :=false) in H2.
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assert (S k < length (tm_step (S n + m))).
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rewrite tm_size_power2. apply U.
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apply nth_error_nth' with (d :=false) in H3.
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apply tm_step_next_range2' in H2. apply tm_step_next_range2' in H3.
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rewrite Nat.add_succ_comm in H2. rewrite <- Nat.add_succ_l in H2.
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rewrite Nat.add_succ_comm in H3. rewrite <- Nat.add_succ_l in H3.
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rewrite H2. rewrite H3.
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assert (nth_error (tm_step (n + S m)) k
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= Some (nth k (tm_step (n + S m)) false)).
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rewrite <- Nat.add_succ_comm. generalize K. rewrite <- tm_size_power2.
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apply nth_error_nth'.
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assert (nth_error (tm_step (n + S m)) (S k)
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= Some (nth (S k) (tm_step (n + S m)) false)).
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rewrite <- Nat.add_succ_comm. generalize U. rewrite <- tm_size_power2.
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apply nth_error_nth'.
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Some (negb (nth (S k) (tm_step (n + S m)) false))
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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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nth_error_nth':
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forall [A : Type] (l : list A) [n : nat] (d : A),
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n < length l -> nth_error l n = Some (nth n l d)
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assert (V: S k < 2^(n+m)).
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assert (N: k < 2^n). apply I.
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assert (L: 2^n <= 2^(n + m)).
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assert (M: n <= n + m). apply Nat.le_add_r.
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apply Nat.pow_le_mono_r. easy. apply M.
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generalize L. generalize H. apply Nat.lt_le_trans.
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assert (L := V). apply Nat.lt_succ_l in L.
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assert (k < length (tm_step (n + S m))).
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rewrite tm_size_power2. apply L.
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apply nth_error_nth' with (d :=false) in H2.
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assert (S k < length (tm_step (n + S m))).
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rewrite tm_size_power2. apply V.
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apply nth_error_nth' with (d :=false) in H3.
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rewrite <- Nat.add_succ_comm in V.
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rewrite <- Nat.add_succ_comm in L.
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assert (k < length (tm_step (S n + S m))).
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rewrite tm_size_power2. apply K.
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apply nth_error_nth' with (d :=false) in H2.
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assert (S k < length (tm_step (S n + S m))).
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rewrite tm_size_power2. apply U.
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apply nth_error_nth' with (d :=false) in H3.
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rewrite H2 in H3.
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intros J. rewrite H0 in J. rewrite H1 in J. inversion J.
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induction m. rewrite Nat.add_0_r. rewrite Nat.add_0_r.
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rewrite Bool.eqb_true_iff
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apply H0.
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Qed.
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Lemma tm_step_next_range2_neighbor : forall (n k : nat),
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