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complete basics

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This commit is contained in:
Michael Zhang 2020-06-04 00:02:38 -05:00
parent 9e8c105bc2
commit 3432fc448c
Signed by: michael
GPG key ID: BDA47A31A3C8EE6B
1 changed files with 97 additions and 36 deletions
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Basics.v
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@ -282,17 +282,20 @@ Proof. simpl. reflexivity. Qed.
model of the [orb] tests above.) The function should return [true] model of the [orb] tests above.) The function should return [true]
if either or both of its inputs are [false]. *) if either or both of its inputs are [false]. *)
Definition nandb (b1:bool) (b2:bool) : bool Definition nandb (b1:bool) (b2:bool) : bool :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match b1, b2 with
| true, true => false
| _, _ => true
end.
Example test_nandb1: (nandb true false) = true. Example test_nandb1: (nandb true false) = true.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_nandb2: (nandb false false) = true. Example test_nandb2: (nandb false false) = true.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_nandb3: (nandb false true) = true. Example test_nandb3: (nandb false true) = true.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_nandb4: (nandb true true) = false. Example test_nandb4: (nandb true true) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
(** [] *) (** [] *)
(** **** Exercise: 1 star, standard (andb3) (** **** Exercise: 1 star, standard (andb3)
@ -301,17 +304,20 @@ Example test_nandb4: (nandb true true) = false.
return [true] when all of its inputs are [true], and [false] return [true] when all of its inputs are [true], and [false]
otherwise. *) otherwise. *)
Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match b1, b2, b3 with
| true, true, true => true
| _, _, _ => false
end.
Example test_andb31: (andb3 true true true) = true. Example test_andb31: (andb3 true true true) = true.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_andb32: (andb3 false true true) = false. Example test_andb32: (andb3 false true true) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_andb33: (andb3 true false true) = false. Example test_andb33: (andb3 true false true) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_andb34: (andb3 true true false) = false. Example test_andb34: (andb3 true true false) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
(** [] *) (** [] *)
(* ================================================================= *) (* ================================================================= *)
@ -698,13 +704,16 @@ Fixpoint exp (base power : nat) : nat :=
Translate this into Coq. *) Translate this into Coq. *)
Fixpoint factorial (n:nat) : nat Fixpoint factorial (n:nat) : nat :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match n with
| 0 => 1
| S n => (S n) * factorial (n)
end.
Example test_factorial1: (factorial 3) = 6. Example test_factorial1: (factorial 3) = 6.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_factorial2: (factorial 5) = (mult 10 12). Example test_factorial2: (factorial 5) = (mult 10 12).
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
(** [] *) (** [] *)
(** Again, we can make numerical expressions easier to read and write (** Again, we can make numerical expressions easier to read and write
@ -791,17 +800,17 @@ Proof. simpl. reflexivity. Qed.
function. (It can be done with just one previously defined function. (It can be done with just one previously defined
function, but you can use two if you need to.) *) function, but you can use two if you need to.) *)
Definition ltb (n m : nat) : bool Definition ltb (n m : nat) : bool :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. 1 <=? m - n.
Notation "x <? y" := (ltb x y) (at level 70) : nat_scope. Notation "x <? y" := (ltb x y) (at level 70) : nat_scope.
Example test_ltb1: (ltb 2 2) = false. Example test_ltb1: (ltb 2 2) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_ltb2: (ltb 2 4) = true. Example test_ltb2: (ltb 2 4) = true.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
Example test_ltb3: (ltb 4 2) = false. Example test_ltb3: (ltb 4 2) = false.
(* FILL IN HERE *) Admitted. Proof. reflexivity. Qed.
(** [] *) (** [] *)
(* ################################################################# *) (* ################################################################# *)
@ -953,7 +962,12 @@ Proof.
Theorem plus_id_exercise : forall n m o : nat, Theorem plus_id_exercise : forall n m o : nat,
n = m -> m = o -> n + m = m + o. n = m -> m = o -> n + m = m + o.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros n m o.
intros H J.
rewrite -> H.
rewrite -> J.
reflexivity.
Qed.
(** [] *) (** [] *)
(** The [Admitted] command tells Coq that we want to skip trying (** The [Admitted] command tells Coq that we want to skip trying
@ -985,7 +999,12 @@ Theorem mult_S_1 : forall n m : nat,
m = S n -> m = S n ->
m * (1 + n) = m * m. m * (1 + n) = m * m.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros n m.
intros H.
simpl.
rewrite <- H.
reflexivity.
Qed.
(* (N.b. This proof can actually be completed with tactics other than (* (N.b. This proof can actually be completed with tactics other than
[rewrite], but please do use [rewrite] for the sake of the exercise.) [rewrite], but please do use [rewrite] for the sake of the exercise.)
@ -1213,15 +1232,21 @@ Qed.
Theorem andb_true_elim2 : forall b c : bool, Theorem andb_true_elim2 : forall b c : bool,
andb b c = true -> c = true. andb b c = true -> c = true.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros [] [].
(** [] *) - reflexivity.
- intros H. destruct H. reflexivity.
- reflexivity.
- intros H. destruct H. reflexivity.
Qed.
(** **** Exercise: 1 star, standard (zero_nbeq_plus_1) *) (** **** Exercise: 1 star, standard (zero_nbeq_plus_1) *)
Theorem zero_nbeq_plus_1 : forall n : nat, Theorem zero_nbeq_plus_1 : forall n : nat,
0 =? (n + 1) = false. 0 =? (n + 1) = false.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros [].
(** [] *) - reflexivity.
- reflexivity.
Qed.
(* ================================================================= *) (* ================================================================= *)
(** ** More on Notation (Optional) *) (** ** More on Notation (Optional) *)
@ -1327,9 +1352,11 @@ Theorem identity_fn_applied_twice :
(forall (x : bool), f x = x) -> (forall (x : bool), f x = x) ->
forall (b : bool), f (f b) = b. forall (b : bool), f (f b) = b.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros f x b.
rewrite -> x.
(** [] *) rewrite -> x.
reflexivity.
Qed.
(** **** Exercise: 1 star, standard (negation_fn_applied_twice) (** **** Exercise: 1 star, standard (negation_fn_applied_twice)
@ -1360,7 +1387,12 @@ Theorem andb_eq_orb :
(andb b c = orb b c) -> (andb b c = orb b c) ->
b = c. b = c.
Proof. Proof.
(* FILL IN HERE *) Admitted. intros [] [].
- reflexivity.
- simpl. intros H. rewrite -> H. reflexivity.
- simpl. intros H. rewrite -> H. reflexivity.
- reflexivity.
Qed.
(** [] *) (** [] *)
@ -1399,11 +1431,19 @@ Inductive bin : Type :=
for binary numbers, and a function [bin_to_nat] to convert for binary numbers, and a function [bin_to_nat] to convert
binary numbers to unary numbers. *) binary numbers to unary numbers. *)
Fixpoint incr (m:bin) : bin Fixpoint incr (m:bin) : bin :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match m with
| Z => B Z
| A n => B n
| B n => A (incr n)
end.
Fixpoint bin_to_nat (m:bin) : nat Fixpoint bin_to_nat (m:bin) : nat :=
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted. match m with
| Z => 0
| A n => 2 * (bin_to_nat n)
| B n => 1 + 2 * (bin_to_nat n)
end.
(** (b) Write five unit tests [test_bin_incr1], [test_bin_incr2], etc. (** (b) Write five unit tests [test_bin_incr1], [test_bin_incr2], etc.
for your increment and binary-to-unary functions. (A "unit for your increment and binary-to-unary functions. (A "unit
@ -1413,10 +1453,31 @@ Fixpoint bin_to_nat (m:bin) : nat
then converting it to unary should yield the same result as then converting it to unary should yield the same result as
first converting it to unary and then incrementing. *) first converting it to unary and then incrementing. *)
(* FILL IN HERE *) Example test_bin_incr1 : incr Z = B Z.
Proof. reflexivity. Qed.
Example test_bin_incr2 : incr (B Z) = A (B Z).
Proof. reflexivity. Qed.
Example test_bin_incr3 : incr (A (B Z)) = B (B Z).
Proof. reflexivity. Qed.
Example test_bin_incr4 : incr (B (B Z)) = A (A (B Z)).
Proof. reflexivity. Qed.
Example test_bin_incr5 : incr (A (A (B Z))) = B (A (B Z)).
Proof. reflexivity. Qed.
Example test_bin_to_nat0 : bin_to_nat Z = 0.
Proof. reflexivity. Qed.
Example test_bin_to_nat1 : bin_to_nat (B Z) = 1.
Proof. reflexivity. Qed.
Example test_bin_to_nat2 : bin_to_nat (A (B Z)) = 2.
Proof. reflexivity. Qed.
Example test_bin_to_nat3 : bin_to_nat (B (B Z)) = 3.
Proof. reflexivity. Qed.
Example test_bin_to_nat4 : bin_to_nat (A (A (B Z))) = 4.
Proof. reflexivity. Qed.
(* Do not modify the following line: *) (* Do not modify the following line: *)
Definition manual_grade_for_binary : option (nat*string) := None. Definition manual_grade_for_binary : option (nat*string) := None.
(** [] *) (** [] *)
(* Wed Jan 9 12:02:44 EST 2019 *) (* Wed Jan 9 12:02:44 EST 2019 *)