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https://github.com/pikapikapikaori/coq
Coq class projects.
https://github.com/pikapikapikaori/coq
coq
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Coq class projects.
- Host: GitHub
- URL: https://github.com/pikapikapikaori/coq
- Owner: pikapikapikaori
- License: gpl-3.0
- Archived: true
- Created: 2021-10-16T12:37:56.000Z (over 3 years ago)
- Default Branch: main
- Last Pushed: 2021-12-18T12:52:40.000Z (about 3 years ago)
- Last Synced: 2024-09-27T06:41:04.034Z (4 months ago)
- Topics: coq
- Language: Coq
- Homepage:
- Size: 79.1 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Coq
## Introduction
Coq class projects.
## Homework
Coq homework
### Homework 3
定义名为Z2的类型，其含zero和one，定义其上模2加法，用+表示，并证明下式。
```coq
Example testAdd1: zero + zero = zero.
Example testAdd2: zero + one = one.
Example testAdd3: one + zero = one.
Example testAdd4: one + one = zero.
```
### Homework 4
1. 定义函数gtb，使得gtb n m 返回布尔值 true当且仅当 n > m。至少用两种方法定义这个函数。
2. 证明下面的性质。
```coq
Theorem gtb_test : forall n : nat, gtb (n+1) 0 = true.
```
3. 定义斐波那契数列。
### Homework 5
1. 证明下面的性质。
```coq
Theorem plus_n_Sm : forall n m : nat,
S (n + m) = n + (S m).
Proof.
(* FILL IN HERE *) Admitted.
Theorem add_comm : forall n m : nat,
n + m = m + n.
Proof.
(* FILL IN HERE *) Admitted.
Theorem add_assoc : forall n m p : nat,
n + (m + p) = (n + m) + p.
Proof.
(* FILL IN HERE *) Admitted.
```
2. 给定自然数d与自然数列表L，定义函数`(hd (d L)`，使其返回列表第一个和最后一个自然数，例如：
```coq
Example test_ht1 : ht 0 [] = (0,0).
Proof. reflexivity. Qed.
Example test_ht2 : ht 0 [3] = (3,3).
Proof. reflexivity. Qed.
Example test_ht3 : ht 0 [1;2;3;4] = (1,4).
Proof. reflexivity. Qed.
```
### Homework 6
1. 补充完整下面的定义，判断一个集合是否为另一个集合的子集。
```coq
Fixpoint subset (s1 : bag) (s2 : bag) : bool
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example test_subset1: subset [1;2] [2;1;4;1] = true.
(* FILL IN HERE *) Admitted.
Example test_subset2: subset [1;2;2] [2;1;4;1] = false.
(* FILL IN HERE *) Admitted.
```
2. 证明如下性质
```coq
Theorem app_assoc4 : ∀ l1 l2 l3 l4 : natlist,
l1 ++ (l2 ++ (l3 ++ l4)) = ((l1 ++ l2) ++ l3) ++ l4.
Proof.
(* FILL IN HERE *) Admitted.
```
3. Before doing the next exercise, make sure you've filled in the definition of remove_one in the chapter Lists.
```coq
Theorem remove_does_not_increase_count: ∀ (s : bag),
(count 0 (remove_one 0 s)) <=? (count 0 s) = true.
Proof.
(* FILL IN HERE *) Admitted.
```
### Homework 7
1. 定义函数maxPair，找出输入的自然数的最大偶数和奇数，组成二元组输出，没有用0代替。
```coq
Example test_maxPair1: maxPair [1;2;5;4;8;10;3] = (5, 10).
Proof. reflexivity. Qed.
Example test_maxPair2: maxPair [2;4] = (0, 4).
Proof. reflexivity. Qed.
```
2. 把上面的函数maxPair改造成maxPair'，返回类型为option nat * option nat。当要找的数字不存在时用None而不是0替代。
```coq
Example test_maxPair‘1: maxPair’ [1;2;5;4;8;10;3] = (Some 5, Some 10).
Proof. reflexivity. Qed.
Example test_maxPair‘2: maxPair’ [2;4] = (None, Some 4).
Proof. reflexivity. Qed.
```
3.
```coq
Theorem rev_app_distr: ∀ X (l1 l2 : list X),
rev (l1 ++ l2) = rev l2 ++ rev l1.
Proof.
(* FILL IN HERE *) Admitted.
```
4.
```coq
Theorem rev_involutive : ∀ X : Type, ∀ l : list X,
rev (rev l) = l.
Proof.
(* FILL IN HERE *) Admitted.
```
### Homework 8
sf-exercise chapter 1-5.
### Homework 9
1.
```coq
Theorem or_commut : forall P Q : Prop, P \/ Q -> Q \/ P.
Proof. (* FILL IN HERE *) Admitted.
```
2.
```coq
Theorem not_both_true_and_false : ∀ P : Prop,
¬ (P ∧ ¬P).
Proof.
(* FILL IN HERE *) Admitted.
```
3.
```coq
Theorem or_distributes_over_and : ∀ P Q R : Prop,
P ∨ (Q ∧ R) -> (P ∨ Q) ∧ (P ∨ R).
Proof.
(* FILL IN HERE *) Admitted.
```
4.
```coq
Theorem or_distributes_over_and' : ∀ P Q R : Prop,
(P ∨ Q) ∧ (P ∨ R) -> P ∨ (Q ∧ R).
Proof.
(* FILL IN HERE *) Admitted.
```
### Homework 10
1.
```coq
Theorem In_app_iff : ∀ A l l' (a:A),
In a (l++l') ↔ In a l ∨ In a l'.
Proof.
intros A l. induction l as [|a' l' IH].
(* FILL IN HERE *) Admitted.
```
2.
Recall that functions returning propositions can be seen as properties of their arguments. For instance, if P has type nat → Prop, then P n states that property P holds of n.
Drawing inspiration from In, write a recursive function All stating that some property P holds of all elements of a list l. To make sure your definition is correct, prove the All_In lemma below. (Of course, your definition should notjust restate the left-hand side of All_In.)
```coq
Fixpoint All { T : Type} ( P : T → Prop) ( l : list T) : Prop
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Theorem All_In :
∀ T ( P : T → Prop) ( l : list T),
( ∀ x, In x l → P x ) ↔
All P l.
Proof.
(* FILL IN HERE *) Admitted.
```
### Homework 11
1. Inductively define a relation CE such that (CE m n) holds iff m and n are two consecutive even numbers with m smaller than n.
```coq
Example test_CE : CE 4 6.
Proof. (* Fill in here *) Admitted.
```
2.
```coq
Theorem CE_SS : forall n m, CE (S (S n)) (S (S m)) -> CE n m.
Proof. (* Fill in here *) Admitted.
```
3.
```coq
Theorem ev_sum : ∀ n m, ev n → ev m → ev (n + m).
Proof.
(* FILL IN HERE *) Admitted.
```
4.
```coq
Lemma le_trans : ∀ m n o, m ≤ n → n ≤ o → m ≤ o.
Proof.
(* FILL IN HERE *) Admitted.
```
### Homework 12
1. Prove the following theorem. Hint: You may find the theorem eqb_string_false_iff in the textbook useful.
```coq
Theorem t_update_neq : ∀ (A : Type) (m : total_map A) x1 x2 v,
x1 ≠ x2 →
(x1 !-> v ; m) x2 = m x2.
Proof.
(* FILL IN HERE *) Admitted.
```
2. Prove the following lemma. Hint: You may use functional_extensionality from the Coq library.
```coq
Lemma t_update_shadow : ∀ (A : Type) (m : total_map A) x v1 v2,
(x !-> v2 ; x !-> v1 ; m) = (x !-> v2 ; m).
Proof.
(* FILL IN HERE *) Admitted.
```
3. Construct a proof object for the following proposition.
```coq
Definition conj_disj : forall P Q R, P /\ (Q \/ R) -> (P /\ Q) \/ (P /\ R)
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *) . Admitted.
```
### Homework 13
1. Complete this proof without using the induction tactic.
```coq
Theorem plus_one_r' : ∀ n: nat,
n + 1 = S n.
Proof.
(* FILL IN HERE *) Admitted.
```
2. Show that the empty_relation defined in (an exercise in) IndProp is a partial function.
```coq
(* FILL IN HERE *)
```
3.
```coq
Theorem le_Sn_n : ∀ n ,
¬ ( S n ≤ n ) .
Proof .
(* FILL IN HERE *) Admitted .
```
4.
```coq
Theorem le_antisymmetric :
antisymmetric le .
Proof .
(* FILL IN HERE *) Admitted .
```
### Homework 14
1. Since the optimize_0plus transformation doesn't change the value of aexps, we should be able to apply it to all the aexps that appear in a bexp without changing the bexp's value. Write a function that performs this transformation on bexps and prove it is sound. Use the tacticals we've learned to make the proof as elegant as possible.
```coq
Fixpoint optimize_0plus_b ( b : bexp) : bexp
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Theorem optimize_0plus_b_sound : ∀ b,
beval ( optimize_0plus_b b) = beval b.
Proof.
(* FILL IN HERE *) Admitted.
```
## Quiz.
Class quizes.
### Quiz 1
编写二次方函数。
### Quiz 2
```coq
Definition bag := natlist.
Fixpoint count ( v : nat) ( s : bag) : nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
All these proofs can be done just by reflexivity.
Example test_count1: count 1 [1 ;2 ;3 ;1 ;4 ;1 ] = 3.
(* FILL IN HERE *) Admitted.
Example test_count2: count 6 [1 ;2 ;3 ;1 ;4 ;1 ] = 0.
(* FILL IN HERE *) Admitted.
```
### Quiz 3
Define type ListOfNatlists whose elements are natlists.
### Quiz 4
## sf-exercise
My solution for https://softwarefoundations.cis.upenn.edu