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Thomas Baruchel
parent
76f7d464e0
commit
c05caff356
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@ -160,6 +160,81 @@ Proof.
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Qed.
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(*
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Lemma xxx :
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forall (n : nat) (hd a tl : list bool),
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tm_step n = hd ++ a ++ a ++ tl
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-> 0 < length a
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-> a = rev a \/ a = map negb (rev a).
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Proof.
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intros n hd a tl. intros H I.
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destruct n. assert (length (tm_step 0) = length (tm_step 0)). reflexivity.
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rewrite H in H0 at 2. rewrite app_length in H0. rewrite app_length in H0.
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rewrite app_length in H0. rewrite Nat.add_comm in H0.
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destruct a. inversion I. rewrite <- Nat.add_assoc in H0.
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rewrite <- Nat.add_assoc in H0. simpl in H0. rewrite Nat.add_succ_r in H0.
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apply Nat.succ_inj in H0. apply Nat.neq_0_succ in H0. contradiction.
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assert (exists k j, length a = S (Nat.double k) * 2^j).
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apply trailing_zeros; assumption. destruct H0. destruct H0.
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assert (x = 0 \/ x = 1). generalize H0. generalize H.
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apply tm_step_square_size.
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generalize dependent n. generalize dependent hd. generalize dependent a.
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generalize dependent tl. generalize dependent x.
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induction x0; intros x K tl a J I hd n H.
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- destruct K; rewrite H0 in I; destruct a.
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inversion J. destruct a. left. reflexivity. inversion I.
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inversion I. destruct a. inversion I. destruct a. inversion I.
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destruct a.
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assert ({b=b1} + {~ b=b1}). apply bool_dec. destruct H1.
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rewrite e. left. reflexivity.
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assert ({b0=b1} + {~ b0=b1}). apply bool_dec. destruct H1.
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rewrite e in H.
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replace (hd ++ [b; b1; b1] ++ [b; b1; b1] ++ tl)
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with ((hd ++ [b]) ++ [b1;b1] ++ [b] ++ [b1;b1] ++ tl) in H.
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apply tm_step_consecutive_identical' in H. inversion H.
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rewrite <- app_assoc. reflexivity.
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assert (b = b0). destruct b; destruct b0; destruct b1;
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reflexivity || contradiction n0 || contradiction n1; reflexivity.
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rewrite H1 in H.
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replace (hd ++ [b0; b0; b1] ++ [b0; b0; b1] ++ tl)
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with (hd ++ [b0;b0] ++ [b1] ++ [b0;b0] ++ ([b1] ++ tl)) in H.
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apply tm_step_consecutive_identical' in H. inversion H. reflexivity.
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inversion I.
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- assert (even (length a) = true).
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rewrite Nat.mul_comm in I. rewrite Nat.pow_succ_r in I. rewrite I.
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rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity.
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apply Nat.le_0_l.
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assert (even (length (hd ++ a)) = true).
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generalize J. generalize H. apply tm_step_square_pos.
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assert (even (length hd) = true).
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rewrite app_length in H1. rewrite Nat.even_add in H1.
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rewrite H0 in H1. destruct (Nat.even (length hd)). reflexivity.
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inversion H1.
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(se débarrasser du S x0 en I)
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generalize H. generalize n. generalize hd. generalize I. generalize J.
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apply IHx0.
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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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*)
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Lemma xxx :
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forall (j n : nat) (hd a tl : list bool),
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@ -192,13 +267,195 @@ Proof.
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rewrite Nat.pow_succ_r.
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rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
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rewrite Nat.even_mul. rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l.
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assert (0 < length a). destruct I; rewrite H1.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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apply Nat.mul_pos_pos. lia.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy.
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assert (even (length (hd ++ a)) = true). generalize H1. generalize H.
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apply tm_step_square_pos.
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assert (even (length hd) = true). rewrite app_length in H2.
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rewrite Nat.even_add in H2. rewrite H0 in H2.
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destruct (even (length hd)). reflexivity. inversion H2.
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assert (2 <= n).
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assert (length (tm_step n) = length (tm_step n)). reflexivity.
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rewrite H in H4 at 2. rewrite app_length in H4. rewrite Nat.add_comm in H4.
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rewrite app_length in H4. destruct I; rewrite H5 in H4;
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rewrite tm_size_power2 in H4; rewrite Nat.double_S in H4;
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symmetry in H4; apply Nat.eq_le_incl in H4.
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assert (2^(S (S (Nat.double j))) <= 2^n).
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assert (2^(S (S (Nat.double j)))
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<= 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
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rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
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generalize H6. apply Nat.le_trans. rewrite <- Nat.pow_le_mono_r_iff in H6.
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assert (2 <= S (S (Nat.double j))). lia. generalize H6. generalize H7.
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apply Nat.le_trans. apply Nat.lt_1_2.
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assert (3 * 2^(S (S (Nat.double j))) <= 2^n).
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assert (3 * 2^(S (S (Nat.double j)))
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<= 3 * 2 ^ (S (S (Nat.double j))) + length (a ++ tl) + length hd).
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rewrite <- Nat.add_assoc. apply Nat.le_add_r. generalize H4.
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generalize H6. apply Nat.le_trans.
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assert (2^(S (S (Nat.double j))) <= 3 * 2^(S (S (Nat.double j)))).
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rewrite <- Nat.mul_1_l at 1. apply Nat.mul_le_mono_pos_r.
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rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy. lia.
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assert (2^(S (S (Nat.double j))) <= 2^n).
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generalize H6. generalize H7. apply Nat.le_trans.
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rewrite <- Nat.pow_le_mono_r_iff in H8.
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assert (2 <= S (S (Nat.double j))). lia. generalize H8. generalize H9.
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apply Nat.le_trans. apply Nat.lt_1_2.
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destruct n. inversion H4. destruct n. inversion H4. inversion H6.
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assert(
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tm_step (S (S n)) = tm_morphism
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(firstn (Nat.div2 (length hd)) (tm_step (S n)) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step (S n))) ++
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))) ++
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skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n)))).
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generalize H0. generalize H0. generalize H3. generalize H.
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apply tm_step_morphism4.
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pose (hd' := firstn (Nat.div2 (length hd)) (tm_step (S n))).
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pose (a' := firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length hd)) (tm_step (S n)))).
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pose (tl' := skipn (Nat.div2 (length (hd ++ a ++ a))) (tm_step (S n))).
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fold hd' in H5. fold a' in H5. fold tl' in H5.
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assert (length hd' = Nat.div2 (length hd)). unfold hd'.
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rewrite firstn_length_le. reflexivity.
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rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
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rewrite tm_size_power2. rewrite <- Nat.pow_succ_r.
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rewrite <- tm_size_power2. rewrite <- Nat.Even_double.
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rewrite H. rewrite app_length. lia.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption. lia. lia.
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assert (length hd = length (tm_morphism hd')).
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rewrite tm_morphism_length. rewrite H6. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double. reflexivity.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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assert (length a' = Nat.div2 (length a)). unfold a'.
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rewrite firstn_length_le. reflexivity. rewrite skipn_length.
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rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
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rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
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rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
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rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double.
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rewrite H. rewrite app_length. rewrite app_length. lia.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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lia. lia.
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assert (length a = length (tm_morphism a')).
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rewrite tm_morphism_length. rewrite H8. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double. reflexivity.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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rewrite H in H5. rewrite tm_morphism_app in H5.
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assert (hd = tm_morphism hd'). generalize H7. generalize H5.
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apply app_eq_length_head. rewrite <- H10 in H5.
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apply app_inv_head in H5. rewrite tm_morphism_app in H5.
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assert (a = tm_morphism a'). generalize H9. generalize H5.
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apply app_eq_length_head. rewrite <- H11 in H5.
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apply app_inv_head in H5. rewrite tm_morphism_app in H5.
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assert (length a = length (tm_morphism (
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n)))))).
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rewrite tm_morphism_length. rewrite firstn_length_le.
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rewrite <- Nat.double_twice. rewrite <- Nat.Even_double.
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reflexivity.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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rewrite skipn_length.
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rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
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rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
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rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
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rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double.
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rewrite H. rewrite app_assoc.
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rewrite app_length. rewrite Nat.add_sub_swap. rewrite Nat.sub_diag.
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rewrite app_length. lia. apply Nat.le_refl.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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lia. lia.
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assert (a = tm_morphism (
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firstn (Nat.div2 (length a))
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(skipn (Nat.div2 (length (hd ++ a))) (tm_step (S n))))).
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generalize H12. generalize H5. apply app_eq_length_head.
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rewrite <- H13 in H5. apply app_inv_head in H5.
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assert (H' := H).
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rewrite H10 in H'. rewrite H11 in H'. rewrite H5 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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rewrite <- tm_morphism_app in H'.
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rewrite <- tm_step_lemma in H'. rewrite <- tm_morphism_eq in H'.
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assert (even (length a') = true). unfold a'.
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rewrite firstn_length_le.
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destruct I; rewrite H14; rewrite Nat.double_S.
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rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
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rewrite Nat.div2_double.
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rewrite Nat.even_mul. reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
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rewrite Nat.mul_comm. rewrite Nat.pow_succ_r. rewrite Nat.pow_succ_r.
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rewrite <- Nat.mul_assoc.
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rewrite Nat.div2_double.
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rewrite Nat.even_mul. rewrite Nat.even_mul.
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reflexivity. apply Nat.le_0_l. apply Nat.le_0_l.
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rewrite skipn_length.
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rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
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rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
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rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
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rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double.
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rewrite H. rewrite app_length. rewrite app_length. lia.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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lia. lia.
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assert (0 < length a'). unfold a'. rewrite firstn_length_le.
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destruct (length a). destruct I. inversion H1. inversion H1.
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destruct n0. inversion H0. replace (S (S n0)) with (n0 + 1*2).
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rewrite Nat.div2_div. rewrite Nat.div_add. rewrite Nat.add_1_r. lia.
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easy. lia.
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rewrite skipn_length.
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rewrite Nat.mul_le_mono_pos_l with (p := 2). rewrite <- Nat.double_twice.
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rewrite tm_size_power2. rewrite Nat.mul_sub_distr_l.
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rewrite <- Nat.pow_succ_r. rewrite <- tm_size_power2.
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rewrite <- Nat.Even_double. rewrite <- Nat.double_twice.
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rewrite <- Nat.Even_double.
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rewrite H. rewrite app_length. rewrite app_length. lia.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
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lia. lia.
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assert (even (length (hd' ++ a')) = true). generalize H15.
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generalize H'. apply tm_step_square_pos.
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assert (even (length hd') = true). rewrite app_length in H16.
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rewrite Nat.even_add in H16. rewrite H14 in H16.
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destruct (even (length hd')). reflexivity. inversion H16.
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assert (
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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 a'))
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(skipn (Nat.div2 (length (hd' ++ a'))) (tm_step n)) ++
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skipn (Nat.div2 (length (hd' ++ a' ++ a'))) (tm_step n))).
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generalize H14. generalize H14. generalize H17. generalize H'.
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apply tm_step_morphism4.
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