This commit is contained in:
Thomas Baruchel 2023-01-29 21:54:58 +01:00
parent d741cad63f
commit 54cdff7f66

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@ -1535,18 +1535,6 @@ Proof.
assert (K: even (length a') = true).
rewrite <- Nat.negb_odd. rewrite n0. reflexivity.
(*
assert (L: even (length (hd ++ a')) = true).
rewrite Nat.le_succ_l in J. apply Nat.lt_succ_l in J.
apply Nat.lt_succ_l in J. apply Nat.lt_succ_l in J.
generalize J. generalize H. apply tm_step_square_pos.
assert (M: even (length hd) = true).
rewrite app_length in L. rewrite Nat.even_add_even in L.
assumption.
apply Nat.EvenT_Even. apply Nat.even_EvenT. assumption.
*)
assert (W: {hd=nil} + {~ hd=nil}). apply list_eq_dec. apply bool_dec.
destruct W. rewrite e in H. right.
destruct n. inversion H. rewrite app_nil_l in H.
@ -1634,3 +1622,31 @@ Proof.
simpl. rewrite Nat.add_0_r. reflexivity.
left. assumption.
Qed.
Theorem tm_step_square_prefix_not_nil :
forall (n : nat) (hd a tl : list bool),
tm_step n = hd ++ a ++ a ++ tl -> 0 < length a -> hd <> nil.
Proof.
intros n hd a tl. intros H I.
assert (W: {hd=nil} + {~ hd=nil}). apply list_eq_dec. apply bool_dec.
destruct W. rewrite e in H. rewrite app_nil_l in H.
generalize dependent a.
generalize dependent tl.
induction n.
- intros tl a. intros H I.
destruct a. inversion I. inversion H.
assert (length (nil : list bool) = length (nil : list bool)).
reflexivity. rewrite H2 in H0 at 2. rewrite app_length in H0.
rewrite Nat.add_comm in H0. inversion H0.
- intros tl a. intros H I.
assert (hd <> nil
\/ exists a' tl', tm_step (pred (S n)) = a' ++ a' ++ tl' /\ 0 < length a').
generalize I. generalize H. replace (a++a++tl) with (hd ++ a ++ a ++ tl).
apply tm_step_square_prefix_not_nil0. rewrite e. reflexivity.
destruct H0. assumption. destruct H0. destruct H0.
rewrite <- pred_Sn in H0.
destruct H0.
generalize H1. generalize H0. apply IHn.
- assumption.
Qed.