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This commit is contained in:
Thomas Baruchel 2023-01-17 21:07:54 +01:00
parent 4f5b891ca2
commit 8f027411c3
1 changed files with 10 additions and 10 deletions
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src/thue_morse3.v
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@ -18,8 +18,14 @@ Lemma tm_step_palindromic_odd : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ tl
-> a = rev a
-> odd (length a) = true
-> length a <> 5.
-> length a < 4.
Proof.
(* end of the lemma *)
assert (LEMMA: forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ tl
-> a = rev a
-> odd (length a) = true
-> length a <> 5).
intros n hd a tl. intros H I J.
destruct a. easy. destruct a. easy.
destruct a. easy. destruct a. easy.
@ -87,14 +93,8 @@ Proof.
simpl. apply not_eq_S. apply not_eq_S. apply not_eq_S.
apply not_eq_S. apply not_eq_S. apply Nat.neq_succ_0.
Qed.
(* end of the lemma *)
Lemma tm_step_palindromic_odd' : forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ tl
-> a = rev a
-> odd (length a) = true
-> length a < 4.
Proof.
intros n hd a tl. intros H I J.
assert (length a <= 5 \/ 5 < length a). apply Nat.le_gt_cases.
@ -104,7 +104,7 @@ Proof.
apply Nat.lt_eq_cases in H0. destruct H0. assumption.
rewrite H0 in J. inversion J.
assert (length a <> 5). generalize J. generalize I. generalize H.
apply tm_step_palindromic_odd. rewrite H0 in H1. contradiction H1.
apply LEMMA. rewrite H0 in H1. contradiction H1.
reflexivity.
(* main part of the proof:
@ -201,6 +201,6 @@ Proof.
assert (odd (length v) = true). rewrite H4. reflexivity.
generalize H7. generalize H6. generalize H. rewrite H2.
rewrite <- app_assoc. rewrite <- app_assoc. rewrite app_assoc.
apply tm_step_palindromic_odd. rewrite H4 in H7. contradiction H7.
apply LEMMA. rewrite H4 in H7. contradiction H7.
reflexivity.
Qed.