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Thomas Baruchel 2023-10-31 09:03:37 +01:00
parent d62a5752ae
commit 23a6870a6a
1 changed files with 20 additions and 20 deletions
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src/subsequences.v
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@ -142,7 +142,7 @@ Qed.
Theorem subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X), Theorem subsequence_cons_eq {X: Type} : forall (l1 l2: list X) (a: X),
subsequence (a::l1) (a::l2) <-> subsequence l1 l2. subsequence (a::l1) (a::l2) <-> subsequence l1 l2.
Proof. Proof.
intros l s a. split. intro H. intros l s a. split; intro H.
destruct H. destruct H. destruct H. destruct H. destruct H. destruct H.
destruct x. destruct x0. destruct x. destruct x0.
apply PeanoNat.Nat.neq_succ_0 in H. contradiction H. apply PeanoNat.Nat.neq_succ_0 in H. contradiction H.
@ -154,7 +154,7 @@ Proof.
inversion H0. rewrite <- app_assoc. apply app_inv_head_iff. inversion H0. rewrite <- app_assoc. apply app_inv_head_iff.
rewrite app_comm_cons. reflexivity. rewrite app_comm_cons. reflexivity.
intro H. destruct H. destruct H. destruct H. destruct H. destruct H. destruct H.
exists nil. exists (x::x0). simpl. split. rewrite H. reflexivity. exists nil. exists (x::x0). simpl. split. rewrite H. reflexivity.
rewrite H0. reflexivity. rewrite H0. reflexivity.
Qed. Qed.
@ -163,14 +163,14 @@ Qed.
Theorem subsequence2_cons_eq {X: Type}: forall (l1 l2: list X) (a: X), Theorem subsequence2_cons_eq {X: Type}: forall (l1 l2: list X) (a: X),
subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2. subsequence2 (a::l1) (a::l2) <-> subsequence2 l1 l2.
Proof. Proof.
intros l s a. split. intro H. unfold subsequence2 in H. intros l s a. split; intro H; destruct H; destruct H.
destruct H. destruct H. destruct x. inversion H0. destruct x. inversion H0.
destruct b. exists (x). split. inversion H. rewrite H2. reflexivity. simpl in H0. destruct b. exists (x). split. inversion H. rewrite H2. 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.
intro H. destruct H. destruct H. exists (true :: x). exists (true :: x).
split. simpl. apply eq_S. assumption. simpl. rewrite H0. reflexivity. split. simpl. apply eq_S. assumption. rewrite H0. reflexivity.
Qed. Qed.
@ -314,6 +314,19 @@ Proof.
Qed. Qed.
Theorem subsequence_dec {X: Type}:
(forall x y : X, {x = y} + {x <> y})
-> forall (l s : list X), { subsequence l s } + { ~ subsequence l s }.
Proof.
intro H. intros l s.
assert ({ subsequence2 l s } + { ~ subsequence2 l s }).
apply subsequence2_dec. assumption. destruct H0.
rewrite <- subsequence_eq_def_2 in s0. left. assumption.
assumption. rewrite <- subsequence_eq_def_2 in n. right. assumption.
assumption.
Qed.
Theorem subsequence_eq_def_3 {X: Type} : Theorem subsequence_eq_def_3 {X: Type} :
(forall (x y : X), {x = y} + {x <> y}) (forall (x y : X), {x = y} + {x <> y})
-> (forall l s : list X, subsequence l s <-> subsequence3 l s). -> (forall l s : list X, subsequence l s <-> subsequence3 l s).
@ -349,19 +362,6 @@ Proof.
Qed. Qed.
Theorem subsequence_dec {X: Type}:
(forall x y : X, {x = y} + {x <> y})
-> forall (l s : list X), { subsequence l s } + { ~ subsequence l s }.
Proof.
intro H. intros l s.
assert ({ subsequence2 l s } + { ~ subsequence2 l s }).
apply subsequence2_dec. assumption. destruct H0.
rewrite <- subsequence_eq_def_2 in s0. left. assumption.
assumption. rewrite <- subsequence_eq_def_2 in n. right. assumption.
assumption.
Qed.
@ -381,7 +381,7 @@ Qed.
Example test3: subsequence [1;2;3;4;5] [1;3;5]. Example test3: subsequence [1;2;3;4;5] [1;3;5].
Proof. Proof.
rewrite subsequence_eq_def. rewrite subsequence_eq_def_2.
exists [true; false; true; false; true]. exists [true; false; true; false; true].
split; reflexivity. split; reflexivity.
apply Nat.eq_dec. apply Nat.eq_dec.