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This commit is contained in:
Thomas Baruchel 2023-10-30 08:18:20 +01:00
parent 951e3cb244
commit ab46174c21
1 changed files with 9 additions and 14 deletions
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23
src/subsequences.v
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@ -25,15 +25,14 @@ Definition subsequence3 (l s : list Type) :=
Theorem subsequence_nil_r : forall (l : list Type), subsequence l nil. Theorem subsequence_nil_r : forall (l : list Type), subsequence l nil.
Proof. Proof.
intro l. unfold subsequence. exists l. exists nil. rewrite app_nil_r. intro l. exists l. exists nil. rewrite app_nil_r. split; easy.
split; easy.
Qed. Qed.
Theorem subsequence_nil_cons_r : forall (l: list Type) (a:Type), Theorem subsequence_nil_cons_r : forall (l: list Type) (a:Type),
~ subsequence nil (a::l). ~ subsequence nil (a::l).
Proof. Proof.
intros l a. unfold subsequence. unfold not. intro H. intros l a. unfold not. intro H.
destruct H. destruct H. destruct H. destruct H. destruct H. destruct H.
destruct x. rewrite app_nil_l in H0. destruct x. rewrite app_nil_l in H0.
destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H. destruct x0. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
@ -44,7 +43,7 @@ Qed.
Theorem subsequence2_nil_r : forall (l : list Type), subsequence2 l nil. Theorem subsequence2_nil_r : forall (l : list Type), subsequence2 l nil.
Proof. Proof.
intro l. unfold subsequence2. intro l.
exists (repeat false (length l)). rewrite repeat_length. exists (repeat false (length l)). rewrite repeat_length.
split. easy. split. easy.
induction l. reflexivity. simpl. assumption. induction l. reflexivity. simpl. assumption.
@ -54,7 +53,7 @@ Qed.
Theorem subsequence2_nil_cons_r : forall (l: list Type) (a:Type), Theorem subsequence2_nil_cons_r : forall (l: list Type) (a:Type),
~ subsequence2 nil (a::l). ~ subsequence2 nil (a::l).
Proof. Proof.
intros l a. unfold subsequence2. unfold not. intro H. destruct H. intros l a. unfold not. intro H. destruct H.
destruct H. assert (x = nil). destruct x. reflexivity. destruct H. assert (x = nil). destruct x. reflexivity.
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
rewrite H1 in H0. symmetry in H0. apply nil_cons in H0. contradiction H0. rewrite H1 in H0. symmetry in H0. apply nil_cons in H0. contradiction H0.
@ -64,7 +63,7 @@ Qed.
Theorem subsequence2_cons_l : forall (l s: list Type) (a: Type), Theorem subsequence2_cons_l : forall (l s: list Type) (a: Type),
subsequence2 l s -> subsequence2 (a::l) s. subsequence2 l s -> subsequence2 (a::l) s.
Proof. Proof.
intros l s a. intro H. unfold subsequence2 in H. intros l s a. intro H.
destruct H. destruct H. exists (false::x). destruct H. destruct H. exists (false::x).
split. simpl. apply eq_S. assumption. split. simpl. apply eq_S. assumption.
simpl. assumption. simpl. assumption.
@ -75,14 +74,11 @@ Theorem subsequence2_cons_r : forall (l s: list Type) (a: Type),
subsequence2 l (a::s) -> subsequence2 l s. subsequence2 l (a::s) -> subsequence2 l s.
Proof. Proof.
intro l. induction l. intros. apply subsequence2_nil_cons_r in H. intro l. induction l. intros. apply subsequence2_nil_cons_r in H.
contradiction H. intros s a0 . intro H. unfold subsequence2 in H. contradiction H. intros s a0 . intro H. destruct H. destruct H. destruct x.
destruct H. destruct H. destruct x.
symmetry in H0. apply nil_cons in H0. contradiction H0. symmetry in H0. apply nil_cons in H0. contradiction H0.
destruct b. simpl in H0. inversion H0. destruct b. simpl in H0. inversion H0.
unfold subsequence2. exists (false::x). exists (false::x). split; try rewrite <- H; reflexivity.
split; try rewrite <- H; reflexivity. simpl in H0. apply subsequence2_cons_l. apply IHl with (a := a0).
simpl in H0.
apply subsequence2_cons_l. apply IHl with (a := a0).
unfold subsequence2. exists (x). split. inversion H. unfold subsequence2. exists (x). split. inversion H.
reflexivity. assumption. reflexivity. assumption.
Qed. Qed.
@ -93,8 +89,7 @@ Theorem subsequence2_cons_eq : forall (l1 l2: list Type) (a: Type),
Proof. Proof.
intros l s a. split. intro H. unfold subsequence2 in H. intros l s a. split. intro H. unfold subsequence2 in H.
destruct H. destruct H. destruct x. inversion H0. destruct H. destruct H. destruct x. inversion H0.
destruct b. destruct b. exists (x). split. inversion H. reflexivity. simpl in H0.
exists (x). split. inversion H. reflexivity. simpl in H0.
inversion H0. reflexivity. simpl in H0. inversion H. inversion H0. reflexivity. simpl in H0. inversion H.
assert (subsequence2 l (a::s)). exists x. split; assumption. assert (subsequence2 l (a::s)). exists x. split; assumption.
apply subsequence2_cons_r with (a := a). assumption. apply subsequence2_cons_r with (a := a). assumption.