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Thomas Baruchel
parent
b8b836fb53
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166
src/thue_morse.v
166
src/thue_morse.v
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@ -1237,3 +1237,169 @@ Proof.
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apply app_inv_head_iff.
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rewrite app_assoc. rewrite firstn_skipn. reflexivity.
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Qed.
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(**
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Some more lemmas are added there as convenient tools for further
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investigations; they have been added quite late, after many
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following results were proved, but they are inserted here in
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case some proofs are re-written later.
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*)
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Lemma tm_step_morphism2 :
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forall (n : nat) (hd tl : list bool),
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tm_step (S n) = hd ++ tl
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-> even (length hd) = true
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-> tm_step (S n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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skipn (Nat.div2 (length hd)) (tm_step n)).
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Proof.
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intros n hd tl. intros H I.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (tl = tm_morphism (skipn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app3.
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rewrite H0 in H. rewrite H1 in H. rewrite <- tm_morphism_app in H.
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assumption.
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Qed.
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Lemma tm_step_morphism3 :
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forall (n : nat) (hd a tl : list bool),
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tm_step (S n) = hd ++ a ++ tl
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-> even (length hd) = true
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-> even (length a) = true
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-> tm_morphism (tm_step n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step n)) ++
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skipn (Nat.div2 (length hd + length a)) (tm_step n)).
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Proof.
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intros n hd a tl. intros H I J. rewrite <- tm_step_lemma in H.
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assert (even (length (hd ++ a)) = true). rewrite app_length.
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rewrite Nat.even_add. rewrite I. rewrite J. reflexivity.
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rewrite app_assoc in H.
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assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app2.
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assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc in H.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (a = tm_morphism (skipn (Nat.div2 (length hd))
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(firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize I. symmetry in H1. generalize H1. apply tm_morphism_app3.
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rewrite skipn_firstn_comm in H4.
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rewrite H3 in H. rewrite H4 in H. rewrite H2 in H.
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rewrite app_length in H.
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rewrite <- tm_morphism_app in H.
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rewrite <- tm_morphism_app in H.
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replace (Nat.div2 (length hd + length a))
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with ((length hd) / 2 + Nat.div2 (length a)) in H.
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rewrite <- Nat.div2_div in H. rewrite Nat.add_sub_swap in H.
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rewrite Nat.sub_diag in H. rewrite Nat.add_0_l in H.
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rewrite Nat.div2_div in H at 3. rewrite <- Nat.div_add in H.
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rewrite Nat.mul_comm in H.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))) in H.
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rewrite <- Nat.div2_odd in H. rewrite <- Nat.div2_div in H.
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assumption.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity. easy.
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apply Nat.le_refl.
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replace (length a) with (Nat.div2 (length a) * 2) at 2.
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rewrite <- Nat.div_add. rewrite <- Nat.div2_div. reflexivity.
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easy. rewrite Nat.mul_comm.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
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rewrite Nat.div2_odd. reflexivity.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity.
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Qed.
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Lemma tm_step_morphism4 :
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forall (n : nat) (hd a b tl : list bool),
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tm_step (S n) = hd ++ a ++ b ++ tl
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-> even (length hd) = true
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-> even (length a) = true
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-> even (length b) = true
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-> tm_step (S n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step n)) ++
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firstn (Nat.div2 (length b))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)) ++
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skipn (Nat.div2 (length (hd ++ a ++ b))) (tm_step n)).
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Proof.
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intros n hd a b tl. intros H I J K.
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assert (even (length (hd ++ a)) = true).
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rewrite app_length. rewrite Nat.even_add.
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rewrite I. rewrite J. reflexivity.
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rewrite app_assoc in H. rewrite <- tm_step_lemma in H.
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assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app2.
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assert (b ++ tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc in H. symmetry in H1. symmetry in H2.
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assert (even (length (hd ++ a ++ b)) = true).
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rewrite app_length. rewrite Nat.even_add. rewrite I.
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rewrite app_length. rewrite Nat.even_add. rewrite J. rewrite K.
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reflexivity.
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assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a ++ b)))
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(tm_step n))).
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replace (hd ++ a ++ b ++ tl) with ((hd ++ a ++ b) ++ tl) in H.
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generalize H3. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (a = tm_morphism (skipn (Nat.div2 (length hd))
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(firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize I. generalize H1. apply tm_morphism_app3.
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assert (b = tm_morphism (firstn (Nat.div2 (length b))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize K. generalize H2. apply tm_morphism_app2.
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rewrite skipn_firstn_comm in H6. rewrite app_length in H6.
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replace (Nat.div2 (length hd + length a))
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with ((length hd) / 2 + Nat.div2 (length a)) in H6.
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rewrite <- Nat.div2_div in H6. rewrite Nat.add_sub_swap in H6.
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rewrite Nat.sub_diag in H6. rewrite Nat.add_0_l in H6.
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rewrite H5 in H. rewrite H6 in H. rewrite H7 in H. rewrite H4 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_step_lemma in H.
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assumption.
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apply Nat.le_refl.
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rewrite <- Nat.div_add. rewrite <- Nat.div2_div.
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rewrite Nat.mul_comm.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
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rewrite <- Nat.div2_odd. reflexivity.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity.
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easy.
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Qed.
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@ -1377,173 +1377,6 @@ Qed.
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Lemma tm_step_morphism2 :
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forall (n : nat) (hd tl : list bool),
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tm_step (S n) = hd ++ tl
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-> even (length hd) = true
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-> tm_step (S n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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skipn (Nat.div2 (length hd)) (tm_step n)).
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Proof.
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intros n hd tl. intros H I.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (tl = tm_morphism (skipn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app3.
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rewrite H0 in H. rewrite H1 in H. rewrite <- tm_morphism_app in H.
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assumption.
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Qed.
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Lemma tm_step_morphism3 :
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forall (n : nat) (hd a tl : list bool),
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tm_step (S n) = hd ++ a ++ tl
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-> even (length hd) = true
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-> even (length a) = true
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-> tm_morphism (tm_step n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step n)) ++
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skipn (Nat.div2 (length hd + length a)) (tm_step n)).
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Proof.
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intros n hd a tl. intros H I J. rewrite <- tm_step_lemma in H.
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assert (even (length (hd ++ a)) = true). rewrite app_length.
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rewrite Nat.even_add. rewrite I. rewrite J. reflexivity.
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rewrite app_assoc in H.
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assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app2.
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assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc in H.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (a = tm_morphism (skipn (Nat.div2 (length hd))
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(firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize I. symmetry in H1. generalize H1. apply tm_morphism_app3.
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rewrite skipn_firstn_comm in H4.
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rewrite H3 in H. rewrite H4 in H. rewrite H2 in H.
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rewrite app_length in H.
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rewrite <- tm_morphism_app in H.
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rewrite <- tm_morphism_app in H.
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replace (Nat.div2 (length hd + length a))
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with ((length hd) / 2 + Nat.div2 (length a)) in H.
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rewrite <- Nat.div2_div in H. rewrite Nat.add_sub_swap in H.
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rewrite Nat.sub_diag in H. rewrite Nat.add_0_l in H.
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rewrite Nat.div2_div in H at 3. rewrite <- Nat.div_add in H.
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rewrite Nat.mul_comm in H.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))) in H.
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rewrite <- Nat.div2_odd in H. rewrite <- Nat.div2_div in H.
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assumption.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity. easy.
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apply Nat.le_refl.
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replace (length a) with (Nat.div2 (length a) * 2) at 2.
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rewrite <- Nat.div_add. rewrite <- Nat.div2_div. reflexivity.
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easy. rewrite Nat.mul_comm.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
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rewrite Nat.div2_odd. reflexivity.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity.
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Qed.
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Lemma tm_step_morphism4 :
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forall (n : nat) (hd a b tl : list bool),
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tm_step (S n) = hd ++ a ++ b ++ tl
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-> even (length hd) = true
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-> even (length a) = true
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-> even (length b) = true
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-> tm_step (S n) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step n) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step n)) ++
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firstn (Nat.div2 (length b))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)) ++
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skipn (Nat.div2 (length (hd ++ a ++ b))) (tm_step n)).
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Proof.
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intros n hd a b tl. intros H I J K.
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assert (even (length (hd ++ a)) = true).
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rewrite app_length. rewrite Nat.even_add.
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rewrite I. rewrite J. reflexivity.
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rewrite app_assoc in H. rewrite <- tm_step_lemma in H.
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assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app2.
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assert (b ++ tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc in H. symmetry in H1. symmetry in H2.
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assert (even (length (hd ++ a ++ b)) = true).
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rewrite app_length. rewrite Nat.even_add. rewrite I.
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rewrite app_length. rewrite Nat.even_add. rewrite J. rewrite K.
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reflexivity.
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assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a ++ b)))
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(tm_step n))).
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replace (hd ++ a ++ b ++ tl) with ((hd ++ a ++ b) ++ tl) in H.
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generalize H3. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
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assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
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assert (a = tm_morphism (skipn (Nat.div2 (length hd))
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(firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize I. generalize H1. apply tm_morphism_app3.
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assert (b = tm_morphism (firstn (Nat.div2 (length b))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
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generalize K. generalize H2. apply tm_morphism_app2.
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rewrite skipn_firstn_comm in H6. rewrite app_length in H6.
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replace (Nat.div2 (length hd + length a))
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with ((length hd) / 2 + Nat.div2 (length a)) in H6.
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rewrite <- Nat.div2_div in H6. rewrite Nat.add_sub_swap in H6.
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rewrite Nat.sub_diag in H6. rewrite Nat.add_0_l in H6.
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rewrite H5 in H. rewrite H6 in H. rewrite H7 in H. rewrite H4 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_step_lemma in H.
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assumption.
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apply Nat.le_refl.
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rewrite <- Nat.div_add. rewrite <- Nat.div2_div.
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rewrite Nat.mul_comm.
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replace (2 * Nat.div2 (length a))
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with (2 * Nat.div2 (length a) + Nat.b2n (Nat.odd (length a))).
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rewrite <- Nat.div2_odd. reflexivity.
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rewrite <- Nat.negb_even. rewrite J.
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rewrite <- Nat.add_0_r. reflexivity.
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easy.
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Qed.
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Lemma tm_step_palindromic_even_morphism1 :
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forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ (rev a) ++ tl
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@ -1558,9 +1391,11 @@ Lemma tm_step_palindromic_even_morphism1 :
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(rev
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(firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step (pred n))))) ++
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skipn (length a + Nat.div2 (length hd)) (tm_step (pred n))).
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skipn (Nat.div2 (length hd) + length a) (tm_step (pred n))).
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Proof.
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intros n hd a tl. intros H I J.
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intros n hd a tl. intros H Z J.
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(* proof that n <> 0 *)
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destruct n. assert (length (tm_step 0) <= length (tm_step 0)).
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apply Nat.le_refl.
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rewrite H in H0 at 1. rewrite app_length in H0. rewrite app_length in H0.
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@ -1570,7 +1405,104 @@ Proof.
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assert (K: 1 <= length a + (length tl + length hd)).
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rewrite Nat.le_succ_l. apply Nat.lt_lt_add_r. assumption.
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rewrite <- Nat.sub_0_le in K. rewrite K in H0. apply Nat.le_0_r in H0.
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rewrite H0 in I. inversion I.
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rewrite H0 in Z. inversion Z.
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assert (even (length (hd ++ a)) = true). generalize Z. generalize H.
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apply tm_step_palindromic_even_center.
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assert (I: even (length hd) = true). rewrite app_length in H0.
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rewrite Nat.even_add in H0. rewrite J in H0.
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destruct (even (length hd)). reflexivity. inversion H0.
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assert (K: even (length (rev a)) = true). rewrite rev_length. assumption.
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rewrite app_assoc in H. rewrite <- tm_step_lemma in H.
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assert (hd ++ a = tm_morphism (firstn (Nat.div2 (length (hd ++ a)))
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(tm_step n))).
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generalize H0. generalize H. apply tm_morphism_app2.
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rewrite <- app_assoc in H. symmetry in H1.
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assert (even (length (hd ++ a ++ (rev a))) = true).
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rewrite app_length. rewrite Nat.even_add. rewrite I.
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rewrite app_length. rewrite Nat.even_add. rewrite J. rewrite K.
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reflexivity.
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|
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assert (tl = tm_morphism (skipn (Nat.div2 (length (hd ++ a ++ (rev a))))
|
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(tm_step n))).
|
||||
replace (hd ++ a ++ (rev a) ++ tl) with ((hd ++ a ++ (rev a)) ++ tl) in H.
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generalize H2. generalize H. apply tm_morphism_app3.
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rewrite <- app_assoc. rewrite <- app_assoc. reflexivity.
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||||
|
||||
assert (hd = tm_morphism (firstn (Nat.div2 (length hd)) (tm_step n))).
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generalize I. generalize H. apply tm_morphism_app2.
|
||||
|
||||
assert (a = tm_morphism (skipn (Nat.div2 (length hd))
|
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(firstn (Nat.div2 (length (hd ++ a))) (tm_step n)))).
|
||||
generalize I. generalize H1. apply tm_morphism_app3.
|
||||
|
||||
rewrite skipn_firstn_comm in H5. rewrite app_length in H5.
|
||||
|
||||
replace (Nat.div2 (length hd + length a))
|
||||
with ((length hd) / 2 + Nat.div2 (length a)) in H5.
|
||||
rewrite <- Nat.div2_div in H5. rewrite Nat.add_sub_swap in H5.
|
||||
rewrite Nat.sub_diag in H5. rewrite Nat.add_0_l in H5.
|
||||
|
||||
rewrite H4 in H. rewrite H5 in H. rewrite H3 in H.
|
||||
rewrite tm_morphism_rev in H.
|
||||
rewrite <- tm_morphism_app in H. rewrite <- tm_morphism_app in H.
|
||||
rewrite <- tm_morphism_app in H. rewrite tm_step_lemma in H.
|
||||
|
||||
rewrite app_length in H. rewrite app_length in H. rewrite rev_length in H.
|
||||
replace (length a + length a) with ((length a)*2) in H.
|
||||
replace (Nat.div2 (length hd + length a * 2))
|
||||
with ((length hd + length a * 2) / 2) in H.
|
||||
rewrite Nat.div_add in H. rewrite <- Nat.div2_div in H.
|
||||
|
||||
assumption. easy.
|
||||
rewrite Nat.div2_div. reflexivity. rewrite Nat.mul_comm. simpl.
|
||||
rewrite Nat.add_0_r. reflexivity.
|
||||
|
||||
apply Nat.le_refl.
|
||||
rewrite <- Nat.div_add. rewrite <- Nat.div2_div.
|
||||
rewrite Nat.mul_comm.
|
||||
|
||||
rewrite Nat.div2_odd with (a := length a) at 2.
|
||||
rewrite <- Nat.negb_even. rewrite J. simpl.
|
||||
rewrite Nat.add_0_r. rewrite Nat.add_0_r. reflexivity. easy.
|
||||
Qed.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
assert (tm_step (S n) = tm_morphism
|
||||
(firstn (Nat.div2 (length hd)) (tm_step n) ++
|
||||
firstn (Nat.div2 (length a))
|
||||
(skipn (Nat.div2 (length hd)) (tm_step n)) ++
|
||||
firstn (Nat.div2 (length (rev a)))
|
||||
(skipn (Nat.div2 (length (hd ++ a))) (tm_step n)) ++
|
||||
skipn (Nat.div2 (length (hd ++ a ++ (rev a)))) (tm_step n))).
|
||||
assert (K : even (length (rev a)) = true).
|
||||
rewrite rev_length. assumption.
|
||||
generalize K. generalize J. generalize H1. generalize H.
|
||||
apply tm_step_morphism4. rewrite rev_length in H2.
|
||||
|
||||
rewrite <- pred_Sn.
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
assert (even (length (hd ++ a)) = true). generalize I. generalize H.
|
||||
apply tm_step_palindromic_even_center.
|
||||
|
|
|
|||
Loading…
Reference in New Issue