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Poly.v

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This commit is contained in:
Michael Zhang 2020-06-04 05:32:54 -05:00
parent ec45376417
commit 027cc8dc14
Signed by: michael
GPG key ID: BDA47A31A3C8EE6B
1 changed files with 42 additions and 15 deletions
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Poly.v
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@ -406,18 +406,29 @@ Definition list123''' := [1; 2; 3].
Theorem app_nil_r : forall (X:Type), forall l:list X, Theorem app_nil_r : forall (X:Type), forall l:list X,
l ++ [] = l. l ++ [] = l.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros H L.
induction L.
- reflexivity.
- simpl. rewrite -> IHL. reflexivity.
Qed.
Theorem app_assoc : forall A (l m n:list A), Theorem app_assoc : forall A (l m n:list A),
l ++ m ++ n = (l ++ m) ++ n. l ++ m ++ n = (l ++ m) ++ n.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros A.
induction l.
- reflexivity.
- intros m n. simpl. rewrite <- IHl. reflexivity.
Qed.
Lemma app_length : forall (X:Type) (l1 l2 : list X), Lemma app_length : forall (X:Type) (l1 l2 : list X),
length (l1 ++ l2) = length l1 + length l2. length (l1 ++ l2) = length l1 + length l2.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros X.
(** [] *) induction l1.
- reflexivity.
- intros l2. simpl. rewrite <- IHl1. reflexivity.
Qed.
(** **** Exercise: 2 stars, standard, optional (more_poly_exercises) (** **** Exercise: 2 stars, standard, optional (more_poly_exercises)
@ -426,13 +437,25 @@ Proof.
Theorem rev_app_distr: forall X (l1 l2 : list X), Theorem rev_app_distr: forall X (l1 l2 : list X),
rev (l1 ++ l2) = rev l2 ++ rev l1. rev (l1 ++ l2) = rev l2 ++ rev l1.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros X.
induction l1. induction l2.
- reflexivity.
- simpl. rewrite -> app_nil_r. reflexivity.
- intros l2. simpl. rewrite -> IHl1. rewrite <- app_assoc. reflexivity.
Qed.
Theorem rev_involutive : forall X : Type, forall l : list X, Theorem rev_involutive : forall X : Type, forall l : list X,
rev (rev l) = l. rev (rev l) = l.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros X.
(** [] *) induction l.
- reflexivity.
- simpl.
rewrite -> rev_app_distr.
simpl.
rewrite -> IHl.
reflexivity.
Qed.
(* ================================================================= *) (* ================================================================= *)
(** ** Polymorphic Pairs *) (** ** Polymorphic Pairs *)
@ -515,13 +538,17 @@ Fixpoint combine {X Y : Type} (lx : list X) (ly : list Y)
given unit test. *) given unit test. *)
Fixpoint split {X Y : Type} (l : list (X*Y)) Fixpoint split {X Y : Type} (l : list (X*Y))
: (list X) * (list Y) : (list X) * (list Y) :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match l with
| nil => (nil, nil)
| (x, y) :: t => match split t with
| (a, b) => (x :: a, y :: b)
end
end.
Example test_split: Example test_split:
split [(1,false);(2,false)] = ([1;2],[false;false]). split [(1,false);(2,false)] = ([1;2],[false;false]).
Proof. Proof. reflexivity. Qed.
(* FILL IN HERE *) Admitted.
(** [] *) (** [] *)
(* ================================================================= *) (* ================================================================= *)
@ -696,16 +723,16 @@ Proof. reflexivity. Qed.
and returns a list of just those that are even and greater than and returns a list of just those that are even and greater than
7. *) 7. *)
Definition filter_even_gt7 (l : list nat) : list nat Definition filter_even_gt7 (l : list nat) : list nat :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. filter (fun n => andb (7 <? n) (evenb n)) l.
Example test_filter_even_gt7_1 : Example test_filter_even_gt7_1 :
filter_even_gt7 [1;2;6;9;10;3;12;8] = [10;12;8]. filter_even_gt7 [1;2;6;9;10;3;12;8] = [10;12;8].
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_filter_even_gt7_2 : Example test_filter_even_gt7_2 :
filter_even_gt7 [5;2;6;19;129] = []. filter_even_gt7 [5;2;6;19;129] = [].
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
(** [] *) (** [] *)
(** **** Exercise: 3 stars, standard (partition) (** **** Exercise: 3 stars, standard (partition)