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Thomas Baruchel
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3bc04f10ca
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@ -2549,14 +2549,127 @@ Proof.
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destruct m. left.
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apply tm_step_palindrome_mod8 with (n := n) (tl := tl); assumption.
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right. rewrite J. apply Nat.pow_le_mono_r. easy.
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rewrite Nat.double_S. rewrite Nat.double_S.
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right. rewrite J. rewrite Nat.double_S. rewrite Nat.double_S.
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apply Nat.pow_le_mono_r. easy.
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rewrite <- Nat.succ_le_mono. rewrite <- Nat.succ_le_mono.
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rewrite <- Nat.succ_le_mono. rewrite <- Nat.succ_le_mono.
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rewrite <- Nat.succ_le_mono. apply le_0_n.
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rewrite <- Nat.succ_le_mono.
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apply le_0_n.
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Qed.
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Theorem tm_step_palindromic_power2_odd :
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forall (m n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ (rev a) ++ tl
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-> 6 < length a
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-> length a = 2^(S (Nat.double m))
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-> length (hd ++ a) mod (2 ^ (S (Nat.double m))) = 0.
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Proof.
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intros m n hd a tl. intros H I J.
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assert (E: length (hd ++ a) mod (2 ^ (S (Nat.double m))) = 0
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\/ 2^5 <= length a).
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generalize J. generalize I. generalize H.
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apply tm_step_palindromic_power2_odd_beta.
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generalize dependent hd.
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generalize dependent a.
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generalize dependent tl.
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generalize dependent n.
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induction m.
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- intros n tl a I J hd H E. destruct E. assumption.
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apply tm_step_palindromic_power2_odd_alpha with (n := n) (tl := tl).
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assumption. assumption. assumption.
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rewrite J in H0. rewrite <- Nat.pow_le_mono_r_iff in H0.
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inversion H0. inversion H2.
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apply Nat.lt_1_2.
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- intros n tl a I J hd H E. assert (E' := E).
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destruct E as [E0 | E1]. assumption.
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rewrite tm_step_palindromic_power2_odd_alpha with (n := n) (tl := tl).
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rewrite <- pred_Sn.
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assert (W: (length a) mod 4 = 0).
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rewrite Nat.double_S in J. rewrite Nat.pow_succ_r in J.
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rewrite Nat.pow_succ_r in J.
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rewrite Nat.mul_assoc in J.
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replace (2*2) with 4 in J. rewrite J.
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rewrite <- Nat.mul_mod_idemp_l. reflexivity.
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easy. reflexivity. apply le_0_n. apply le_0_n.
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assert (
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hd = tm_morphism (tm_morphism (firstn (length hd / 4)
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(tm_step (pred (pred n)))))
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/\ a = tm_morphism (tm_morphism
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(firstn (length a / 4) (skipn (length hd / 4)
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(tm_step (pred (pred n))))))
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/\ tl = tm_morphism (tm_morphism
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(skipn (length hd / 4 + Nat.div2 (length a))
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(tm_step (pred (pred n)))))).
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generalize W. generalize I. generalize H.
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apply tm_step_palindromic_even_morphism2.
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destruct H0 as [K H0]. destruct H0 as [L M].
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assert (V: 3 < n). generalize I. generalize H.
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apply tm_step_palindromic_length_12_n.
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destruct n. inversion V. destruct n. inversion V. inversion H1.
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rewrite K in H. rewrite L in H. rewrite M in H.
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rewrite tm_morphism_twice_rev in H.
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rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
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rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
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rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
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rewrite <- tm_step_lemma in H. rewrite <- tm_step_lemma in H.
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rewrite <- tm_morphism_eq in H. rewrite <- tm_morphism_eq in H.
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rewrite Nat.pred_succ in H. rewrite Nat.pred_succ in H.
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pose (hd' := (firstn (length hd / 4) (tm_step n))).
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pose (a' := (firstn (length a / 4) (skipn (length hd / 4) (tm_step n)))).
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pose (tl' := (skipn (length hd / 4 + Nat.div2 (length a)) (tm_step n))).
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fold hd' in H. fold a' in H. fold tl' in H.
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rewrite Nat.pred_succ in K. rewrite Nat.pred_succ in K.
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rewrite Nat.pred_succ in L. rewrite Nat.pred_succ in L.
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fold hd' in K. fold a' in L.
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assert (N: length a = length a). reflexivity. rewrite L in N at 2.
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rewrite tm_morphism_length in N. rewrite tm_morphism_length in N.
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rewrite Nat.mul_assoc in N. replace (2*2) with 4 in N.
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assert (O: length hd = length hd). reflexivity. rewrite K in O at 2.
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rewrite tm_morphism_length in O. rewrite tm_morphism_length in O.
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rewrite Nat.mul_assoc in O. replace (2*2) with 4 in O.
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rewrite app_length. rewrite N. rewrite O. rewrite <- Nat.mul_add_distr_l.
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rewrite Nat.mul_comm. rewrite Nat.div_mul. rewrite <- app_length.
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assert (Y: length a' = 2 ^ (S (Nat.double m))).
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rewrite J in N. rewrite Nat.double_S in N.
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rewrite Nat.pow_succ_r in N. rewrite Nat.pow_succ_r in N.
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rewrite Nat.mul_assoc in N. replace (2*2) with 4 in N.
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rewrite Nat.mul_cancel_l in N. rewrite N. reflexivity. easy. reflexivity.
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apply le_0_n. apply le_0_n.
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assert (Y': 6 < length a').
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rewrite N in E1.
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rewrite Nat.pow_succ_r in E1. rewrite Nat.pow_succ_r in E1.
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rewrite Nat.mul_assoc in E1. replace (2*2) with 4 in E1.
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rewrite <- Nat.mul_le_mono_pos_l in E1.
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assert (6 < 2^3). simpl.
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rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
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rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
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rewrite <- Nat.succ_lt_mono. rewrite <- Nat.succ_lt_mono.
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apply Nat.lt_0_succ. generalize E1. generalize H0.
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apply Nat.lt_le_trans. apply Nat.lt_0_succ. reflexivity.
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apply le_0_n. apply le_0_n.
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apply IHm with (n := n) (tl := tl').
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assumption. assumption. assumption.
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generalize Y. generalize Y'. generalize H.
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apply tm_step_palindromic_power2_odd_beta.
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easy. reflexivity. reflexivity.
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assumption. assumption. assumption.
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Qed.
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(*
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Lemma tm_step_proper_palindrome_center :
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forall (m n k : nat) (hd a tl : list bool),
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