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This commit is contained in:
Thomas Baruchel
parent
afc01a49bf
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c1dd9ca8fb
156
src/thue_morse.v
156
src/thue_morse.v
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@ -1012,30 +1012,10 @@ Proof.
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Qed.
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Lemma tm_step_factor4_odd_prefix : forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ tl
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-> length a = 4 -> odd (length hd) = true
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-> exists (x : bool) (u v : list bool), a = u ++ [x;x] ++ v.
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Proof.
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intros n hd a tl. intros H I J.
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assert (K: 1 = (length hd) mod 2).
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apply Nat.mod_unique with (q := Nat.div2 (length hd)). apply Nat.lt_1_2.
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replace (1) with (Nat.b2n (Nat.odd (length hd))) at 2.
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apply Nat.div2_odd. rewrite J. reflexivity.
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assert (L: (length hd) mod (2*2)
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= (length hd) mod 2 + 2 * (((length hd) / 2) mod 2)).
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apply Nat.mod_mul_r; easy. rewrite <- K in L.
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replace (2*2) with (4) in L. rewrite <- Nat.bit0_mod in L.
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assert (M: (length hd) mod 4 = 1 \/ (length hd) mod 4 = 3). rewrite L.
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destruct (Nat.testbit (length hd / 2) 0) ; [right | left] ; reflexivity.
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assert (forall (n' : nat) (w z y : list bool),
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Lemma tm_step_factor4_1mod4 : forall (n' : nat) (w z y : list bool),
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tm_step n' = w ++ z ++ y
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-> length z = 4 -> length w mod 4 = 1
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-> exists (x : bool), firstn 2 z = [x;x]).
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-> exists (x : bool), firstn 2 z = [x;x].
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intros n' w z y. intros G0 G1 G2.
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assert (U: 4 * (length w / 4) <= length w).
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apply Nat.mul_div_le. easy.
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@ -1070,11 +1050,11 @@ Proof.
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rewrite nth_error_app1 in O. rewrite nth_error_app2 in O.
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rewrite Nat.sub_succ_l in O. rewrite Nat.sub_diag in O.
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rewrite nth_error_app1 in O. destruct O.
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assert (nth_error z 0 = nth_error z 1). apply H0. reflexivity.
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rewrite nth_error_nth' with (d := false) in H2.
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rewrite nth_error_nth' with (d := false) in H2.
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exists (nth 0 z false). inversion H2.
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rewrite H4 at 2.
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assert (nth_error z 0 = nth_error z 1). apply H. reflexivity.
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rewrite nth_error_nth' with (d := false) in H1.
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rewrite nth_error_nth' with (d := false) in H1.
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exists (nth 0 z false). inversion H1.
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rewrite H3 at 2.
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destruct z. inversion G1. destruct z. inversion G1. reflexivity.
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rewrite G1. rewrite <- Nat.succ_lt_mono. apply Nat.lt_0_succ.
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rewrite G1. apply Nat.lt_0_succ.
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@ -1085,53 +1065,87 @@ Proof.
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reflexivity. easy. assumption.
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rewrite Nat.add_succ_r. rewrite <- Nat.div_mod_eq. reflexivity.
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reflexivity. apply Nat.lt_1_2. reflexivity.
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Qed.
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Lemma tm_step_factor4_3mod4 : forall (n' : nat) (w z y : list bool),
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tm_step n' = w ++ z ++ y
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-> length z = 4 -> length w mod 4 = 3
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-> exists (x : bool), skipn 2 z = [x;x].
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Proof.
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intros n hd a tl. intros H I M.
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assert ((length (hd ++ (firstn 2 a))) mod 4 = 1).
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rewrite app_length. rewrite firstn_length. rewrite I.
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rewrite <- Nat.add_mod_idemp_l. rewrite M. reflexivity. easy.
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assert (tm_step (S n) = tm_step n ++ map negb (tm_step n)).
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apply tm_build. rewrite H in H1.
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assert (length ((skipn 2 a)
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++ (firstn 2 (tl ++ (map negb (hd ++ a ++ tl))))) = 4).
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rewrite app_length. rewrite skipn_length. rewrite I.
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rewrite firstn_length. rewrite app_length. rewrite <- H.
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rewrite map_length.
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assert (4 <= length tl + length (tm_step n)).
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assert (Q: length a <= length (tm_step n)). rewrite H.
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rewrite app_length. rewrite app_length.
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rewrite Nat.add_assoc. rewrite Nat.add_shuffle0.
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apply Nat.le_add_l. rewrite I in Q.
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rewrite <- Nat.add_0_l at 1. apply Nat.add_le_mono.
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apply Nat.le_0_l. assumption. rewrite Nat.min_l. reflexivity.
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assert (2 <= 4). apply le_n_S. apply le_n_S. apply Nat.le_0_l.
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generalize H2. generalize H3. apply Nat.le_trans.
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replace ((hd ++ a ++ tl) ++ map negb (hd ++ a ++ tl))
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with ((hd ++ firstn 2 a)
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++ (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
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++ (skipn 2 (tl ++ map negb (hd ++ a ++ tl)))) in H1.
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assert (exists (x : bool),
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firstn 2 (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
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= [x;x]).
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generalize H0. generalize H2. generalize H1. apply tm_step_factor4_1mod4.
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destruct H3. exists x.
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rewrite <- H3. rewrite firstn_app. rewrite skipn_length. rewrite I.
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replace (2 - (4 - 2)) with 0. rewrite firstn_O.
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rewrite app_nil_r.
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destruct a. inversion I. destruct a. inversion I.
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destruct a. inversion I. destruct a. inversion I.
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destruct a. reflexivity. inversion I.
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reflexivity.
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rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
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apply app_inv_head_iff.
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rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
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rewrite firstn_skipn.
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rewrite <- firstn_skipn with (n := 2) (l := a) at 4.
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rewrite <- app_assoc. reflexivity.
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Admitted.
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Lemma tm_step_factor4_odd_prefix : forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ tl
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-> length a = 4 -> odd (length hd) = true
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-> exists (x : bool) (u v : list bool), a = u ++ [x;x] ++ v.
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Proof.
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intros n hd a tl. intros H I J.
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assert (K: 1 = (length hd) mod 2).
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apply Nat.mod_unique with (q := Nat.div2 (length hd)). apply Nat.lt_1_2.
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replace (1) with (Nat.b2n (Nat.odd (length hd))) at 2.
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apply Nat.div2_odd. rewrite J. reflexivity.
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assert (L: (length hd) mod (2*2)
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= (length hd) mod 2 + 2 * (((length hd) / 2) mod 2)).
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apply Nat.mod_mul_r; easy. rewrite <- K in L.
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replace (2*2) with (4) in L. rewrite <- Nat.bit0_mod in L.
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assert (M: (length hd) mod 4 = 1 \/ (length hd) mod 4 = 3). rewrite L.
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destruct (Nat.testbit (length hd / 2) 0) ; [right | left] ; reflexivity.
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destruct M as [M | M].
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- assert (exists (x : bool), firstn 2 a = [x;x]).
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generalize M. generalize I. generalize H. apply H0.
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destruct H1. exists x. exists nil. exists (skipn 2 a).
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rewrite app_nil_l. rewrite <- H1. symmetry. apply firstn_skipn.
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- assert ((length (hd ++ (firstn 2 a))) mod 4 = 1).
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rewrite app_length. rewrite firstn_length. rewrite I.
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rewrite <- Nat.add_mod_idemp_l. rewrite M. reflexivity. easy.
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assert (tm_step (S n) = tm_step n ++ map negb (tm_step n)).
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apply tm_build. rewrite H in H2.
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assert (length ((skipn 2 a)
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++ (firstn 2 (tl ++ (map negb (hd ++ a ++ tl))))) = 4).
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rewrite app_length. rewrite skipn_length. rewrite I.
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rewrite firstn_length. rewrite app_length. rewrite <- H.
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rewrite map_length.
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assert (4 <= length tl + length (tm_step n)).
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assert (Q: length a <= length (tm_step n)). rewrite H.
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rewrite app_length. rewrite app_length.
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rewrite Nat.add_assoc. rewrite Nat.add_shuffle0.
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apply Nat.le_add_l. rewrite I in Q.
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rewrite <- Nat.add_0_l at 1. apply Nat.add_le_mono.
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apply Nat.le_0_l. assumption. rewrite Nat.min_l. reflexivity.
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assert (2 <= 4). apply le_n_S. apply le_n_S. apply Nat.le_0_l.
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generalize H3. generalize H4. apply Nat.le_trans.
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replace ((hd ++ a ++ tl) ++ map negb (hd ++ a ++ tl))
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with ((hd ++ firstn 2 a)
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++ (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
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++ (skipn 2 (tl ++ map negb (hd ++ a ++ tl)))) in H2.
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assert (exists (x : bool),
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firstn 2 (skipn 2 a ++ firstn 2 (tl ++ map negb (hd ++ a ++ tl)))
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= [x;x]).
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generalize H1. generalize H3. generalize H2. apply H0.
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destruct H4. exists x. exists (firstn 2 a). exists nil.
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rewrite <- H4. rewrite firstn_app. rewrite skipn_length. rewrite I.
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replace (2 - (4 - 2)) with 0. rewrite firstn_O.
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rewrite app_nil_r. rewrite app_nil_r.
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destruct a. inversion I. destruct a. inversion I.
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destruct a. inversion I. destruct a. inversion I.
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destruct a. reflexivity. inversion I.
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reflexivity.
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rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
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apply app_inv_head_iff.
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rewrite <- app_assoc. symmetry. rewrite <- app_assoc.
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rewrite firstn_skipn.
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rewrite <- firstn_skipn with (n := 2) (l := a) at 4.
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rewrite <- app_assoc. reflexivity.
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generalize M. generalize I. generalize H. apply tm_step_factor4_1mod4.
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destruct H0. exists x. exists nil. exists (skipn 2 a).
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rewrite app_nil_l. rewrite <- H0. symmetry. apply firstn_skipn.
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- assert (exists (x : bool), skipn 2 a = [x;x]).
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generalize M. generalize I. generalize H. apply tm_step_factor4_3mod4.
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destruct H0. exists x. exists (firstn 2 a). exists nil.
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rewrite app_nil_r. rewrite <- H0. symmetry. apply firstn_skipn.
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- reflexivity.
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Qed.
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@ -280,6 +280,78 @@ Proof.
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Qed.
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Lemma tm_step_palindromic_negb_center :
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forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ map negb (rev a) ++ tl
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-> 1 < length a
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-> even (length (hd ++ a)) = true.
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Proof.
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intros n hd a tl. intros H I.
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assert (J: a = firstn (length a - 2) a ++ skipn (length a - 2) a).
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symmetry. apply firstn_skipn. rewrite J in H.
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rewrite rev_app_distr in H. rewrite map_app in H.
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rewrite <- app_assoc in H. rewrite <- app_assoc in H.
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rewrite app_assoc in H.
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replace ( skipn (length a - 2) a ++
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map negb (rev (skipn (length a - 2) a)) ++
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map negb (rev (firstn (length a - 2) a)) ++ tl )
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with ( ( skipn (length a - 2) a ++
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map negb (rev (skipn (length a - 2) a)) ) ++
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map negb (rev (firstn (length a - 2) a)) ++ tl ) in H.
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assert (length (skipn (length a - 2) a) = 2). rewrite skipn_length.
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replace (length a) with (2 + (length a - 2)) at 1. rewrite Nat.add_sub.
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reflexivity. rewrite Nat.add_sub_assoc. rewrite Nat.add_sub_swap.
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rewrite Nat.sub_diag. reflexivity. easy. rewrite Nat.le_succ_l.
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assumption.
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assert (length
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(skipn (length a - 2) a ++ map negb (rev (skipn (length a - 2) a))) = 4).
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rewrite app_length. rewrite map_length. rewrite rev_length.
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rewrite H0. reflexivity.
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assert (K: {even (length (hd ++ firstn (length a - 2) a))=true}
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+ {~ even (length (hd ++ firstn (length a - 2) a))=true}).
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apply bool_dec. destruct K.
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rewrite J. rewrite app_assoc. rewrite app_length. rewrite Nat.even_add.
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rewrite e. rewrite H0. reflexivity. apply not_true_is_false in n0.
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assert (exists (x : bool) (u v : list bool),
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(skipn (length a - 2) a ++ map negb (rev (skipn (length a - 2) a)))
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= u ++ [x;x] ++ v).
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assert (odd (length (hd ++ firstn (length a - 2) a)) = true).
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rewrite <- Nat.negb_even. rewrite n0. reflexivity. generalize H2.
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generalize H1. generalize H. apply tm_step_factor4_odd_prefix.
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destruct H2. destruct H2. destruct H2. rewrite H2 in H1.
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destruct x0.
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assert (skipn (length a - 2) a = [x;x]).
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assert (length (skipn (length a - 2) a) = length ([x;x])). rewrite H0.
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reflexivity. generalize H3. generalize H2. rewrite app_nil_l.
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apply app_eq_length_head. rewrite H3 in H.
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TODO contradiction en H
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Lemma tm_step_factor4_odd_prefix : forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ tl
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-> length a = 4 -> odd (length hd) = true
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-> exists (x : bool) (u v : list bool), a = u ++ [x;x] ++ v.
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pose (hd' := hd ++ firstn (length a - 2) a).
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pose (tl' := map negb (rev (firstn (length a - 2) a)) ++ tl).
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pose (a' := skipn (length a - 2) a).
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fold hd' in H. fold tl' in H. fold a' in H.
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replace (a' ++ map negb (rev a') ++ tl')
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with ((a' ++ map negb (rev a')) ++ tl') in H.
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assert (length pat = 4).
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(**
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In this notebook I call "proper palindrome" in the Thue-Morse sequence,
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any palindromic subsequence such that the middle (of the palindromic
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@ -355,6 +427,8 @@ Proof.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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generalize H0. generalize H. apply tm_step_palindromic_even_center.
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assert (H' := H).
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(* proof that 0 < n *)
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destruct n.
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assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
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@ -421,7 +495,7 @@ Proof.
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(skipn (Nat.div2 (length hd)) (tm_step n)))).
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fold hd' in H. fold a' in H. fold tl' in H.
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assert (length a' = 2^S (Nat.double m)). unfold a'.
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assert (I': length a' = 2^S (Nat.double m)). unfold a'.
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rewrite firstn_length_le. rewrite I. rewrite Nat.double_S.
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rewrite Nat.pow_succ_r. rewrite Nat.div2_double. reflexivity.
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apply le_0_n. rewrite skipn_length.
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@ -429,10 +503,29 @@ Proof.
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rewrite Nat.mul_comm. rewrite <- Nat.add_0_r at 1.
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replace 0 with (Nat.b2n (Nat.odd (length a))) at 2.
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rewrite <- Nat.div2_odd. rewrite Nat.mul_sub_distr_r.
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replace (length (tm_step n) * 2) with (length (tm_step (S n))).
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replace (Nat.div2 (length hd) * 2) with (length hd).
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rewrite H'. rewrite app_length. rewrite Nat.add_sub_swap.
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rewrite Nat.sub_diag. simpl. rewrite app_length.
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rewrite <- Nat.add_0_r at 1. rewrite <- Nat.add_le_mono_l.
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apply Nat.le_0_l. apply Nat.le_refl.
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rewrite Nat.mul_comm. symmetry. rewrite <- Nat.add_0_r at 1.
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replace 0 with (Nat.b2n (Nat.odd (length hd))) at 2.
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rewrite <- Nat.div2_odd. reflexivity.
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rewrite <- Nat.negb_even. rewrite L. reflexivity.
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rewrite tm_build. rewrite app_length. rewrite Nat.mul_comm.
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simpl. rewrite Nat.add_0_r. rewrite map_length. reflexivity.
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rewrite <- Nat.negb_even. rewrite K. reflexivity.
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apply Nat.lt_0_2.
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assert (K': even (length a') = true). rewrite I'.
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rewrite Nat.pow_succ_r. rewrite Nat.even_mul. reflexivity.
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apply le_0_n.
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re
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(* montrer que length (hd'++a') est pair à l'aide de
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H : tm_step n = hd' ++ a' ++ map negb (rev a') ++ tl'
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*)
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@ -807,6 +900,8 @@ Proof.
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*)
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(* TODO: reprendre tous les lemmes hd ++ a ++ rev a ++ tl
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et trouver un équivalent pour hd ++ a ++ map negb (rev a) ++ tl *)
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(*
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TODO: les palindromes de longueur 16 sont centrés autour
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