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qoc_jisuanji_ends.html
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qoc_jisuanji_ends.html
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<!--
This sample contains two very simple introductory exercises of
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It can be used as a gentle landing page, which requires no knowledge
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+ Friday, April 26, 2019, 22:22:00
<br/> OOO1337777 , COQ , 鸡算计 , mathematics , 数学 - ends , 鸡算计.在线 - 目的们 , http://鸡算计.在线
<br/> http://鸡算计.在线
<br/> https://www.bilibili.com/video/av50587460/
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From Qoc Require Import Jisuanji .
(**
短 : “鸡算计.在线”的目的是让每个人都能读写数学。在未来5年到10年，这可以用于国民教育。
Short ：the objective of the "鸡算计.在线" is to enable everyone to read and write mathematics . In the next 5 years to 10 years , this could be used in the national education .
Outline ::
* PART 1 : ENDS . 第1部分 : 目的们
* PART 2 : MORE TACTICS . 第2部分：更多战术们
*)
(** * PART 1 : ENDS . 第1部分 : 目的们 *)
(**
http://鸡算计.在线
http://xn--lzzm24a52n.xn--3ds443g/
短 : “鸡算计.在线”的目的是让每个人都能读写数学。在未来5年到10年，这可以用于国民教育。
Short ：the objective of the "鸡算计.在线" is to enable everyone to read and write mathematics . In the next 5 years to 10 years , this could be used in the national education .
*)
(** * PART 2 : MORE TACTICS . 第2部分：更多战术们 *)
(**cs3110 *)
(** 假设 *)
论点 p_implies_p : forall P : Prop,
P -> P.
证明.
移动外 P P_holds.
假设.
据证实.
(** 同一 , 对称 , transitivity *)
论点 forty_two : 41 + 1 = 42.
证明.
同一. 复原.
对称. 同一. 复原 2.
外传递. 确切 (eq_refl (3 + (38 + 1))).
同一.
退出.
(** 琐细 *)
论点 p_implies_p' : forall P : Prop,
P -> P .
证明.
琐细.
据证实.
(** 自动 *)
论点 modus_tollens: forall (P Q : Prop),
(P -> Q) -> ~Q -> ~P.
证明.
自动.
据证实.
(** 辨析 *)
论点 false_implies_anything : forall P : Prop,
0 = 1 -> P.
证明.
移动外 P zero_equals_one.
辨析.
据证实.
(** 确切 *)
论点 everything : 42 = 42.
证明.
确切 (eq_refl 42).
据证实.
(** 矛盾 *)
论点 law_of_矛盾 : forall (P Q : Prop),
P -> ~P -> Q.
证明.
移动外 P Q P_and_not_P.
矛盾.
据证实.
(** Transforming goals *)
(** 移动外 *)
论点 my移动外 : forall A B C D : Prop, D .
证明.
移动外 A B C .
退出.
(** 简化 *)
论点 switch_to_honors : 10 + 2 = 12.
证明.
计时了 简化.
计时了 同一.
退出.
(** 展开 *)
定义 plus_two (x : nat) : nat :=
x + 2.
论点 switch_to_honors_again :
plus_two 10 = 12.
证明.
展开 plus_two.
同一.
据证实.
(** 应用 *)
论点 modus_ponens : forall (P Q : Prop),
(P -> Q) -> P -> Q.
证明.
移动外 P Q P_implies_Q P_holds.
应用 P_implies_Q.
退出.
论点 modus_ponens'' : forall (P Q : Prop),
P -> (P -> Q) -> Q.
证明.
自动.
据证实.
论点 double_negation : forall (P : Prop),
P -> ~~P.
证明.
展开 not. 移动外个 P.
应用 modus_ponens''.
据证实.
(** rewrite *)
论点 add_comm : forall (x y : nat),
x + y = y + x.
证明.
移动外. 归纳 x 如 [ | x' IHx' ].
- 琐细.
- 简化. 改写 -> IHx'.
琐细.
据证实.
(** 逆温 *)
论点 succ_eq_implies_eq : forall (x y : nat),
S x = S y -> x = y.
证明.
移动外 x y succ_eq.
逆温 succ_eq.
琐细.
据证实.
(** 左 , 右 *)
论点 or_左 : forall (P Q : Prop),
P -> P \/ Q.
证明.
移动外 P Q P_holds.
左.
退出.
论点 or_右 : forall (P Q : Prop),
Q -> P \/ Q.
证明.
移动外 P Q Q_holds.
右.
退出.
(** 更换 *)
论点 one_x_one : forall (x : nat),
1 + x + 1 = 2 + x.
证明.
移动外个. 简化.
更换 (x + 1) 与 (S x).
退出.
(** 分裂 *)
论点 implies_and : forall (P Q R : Prop),
P -> (P -> Q) -> (P -> R) -> (Q /\ R).
证明.
移动外 P Q R P_holds.
移动外 P_implies_Q P_implies_R.
分裂. 复原. 构造函数.
- 应用 P_implies_Q . 假设 .
- 应用 P_implies_R . 假设 .
据证实.
论点 and_左 : forall (P Q : Prop),
(P /\ Q) -> P.
证明.
移动外 P Q P_and_Q.
解构 P_and_Q 如 [P_holds Q_holds].
假设.
据证实.
论点 or_comm : forall (P Q : Prop),
P \/ Q -> Q \/ P.
证明.
移动外 P Q P_or_Q.
解构 P_or_Q 如 [P_holds | Q_holds].
- 右. 复原. 构造函数 2. 假设.
- 左. 复原. 构造函数 1. 假设.
据证实.
归纳的 element :=
| grass : element
| fire : element
| water : element.
定义 weakness (e : element) : element :=
匹配 e 与
| grass => fire
| fire => water
| water => grass
结束.
论点 never_weak_to_self : forall (e : element),
weakness e <> e.
证明.
解构 e.
- 简化. 辨析.
- 简化. 辨析.
- 简化. 辨析.
退出.
论点 n_plus_n : forall (n : nat),
n + n = n * 2.
证明.
归纳 n 如 [| x IH].
- 同一.
- 简化. 改写 <- IH. 自动.
据证实.
(** 自动 *)
(** logical t自动 *)
论点 demorgan : forall (P Q : Prop),
~(P \/ Q) -> ~P /\ ~Q.
证明.
同义反复.
据证实.
从 Coq 要求 进口 Lia.
论点 dfoil : forall a ,
(a + 2) * (1 + 2) = a*1 + 2*1 + a*2 + 2*2.
证明.
移动外. lia.
据证实.
从 Coq 要求 进口 Arith.
(** algebraic rihg *)
论点 foil : forall a b c d,
(a + b) * (c + d) = a*c + b*c + a*d + b*d.
证明.
移动外. ring.
据证实.
(** ** alt
----------------------------------------------------------------------------- *)
Reset p_implies_p.
(** assumption *)
Lemma p_implies_p : forall P : Prop,
P -> P.
Proof.
intros P P_holds.
assumption.
Qed.
(** reflexivity , symmetry , transitivity *)
Lemma forty_two : 41 + 1 = 42.
Proof.
reflexivity. Undo.
symmetry. reflexivity. Undo 2.
etransitivity. exact (eq_refl (3 + (38 + 1))).
reflexivity.
Abort.
(** trivial *)
Lemma p_implies_p' : forall P : Prop,
P -> P .
Proof.
trivial.
Qed.
(** auto *)
Lemma modus_tollens: forall (P Q : Prop),
(P -> Q) -> ~Q -> ~P.
Proof.
auto.
Qed.
(** discriminate *)
Lemma false_implies_anything : forall P : Prop,
0 = 1 -> P.
Proof.
intros P zero_equals_one.
discriminate.
Qed.
(** exact *)
Lemma everything : 42 = 42.
Proof.
exact (eq_refl 42).
Qed.
(** contradiction *)
Lemma law_of_contradiction : forall (P Q : Prop),
P -> ~P -> Q.
Proof.
intros P Q P_and_not_P.
contradiction.
Qed.
(** Transforming goals *)
(** intros *)
Lemma myintros : forall A B C D : Prop, D .
Proof.
intros A B C .
Abort.
(** simpl *)
Lemma switch_to_honors : 10 + 2 = 12.
Proof.
Time simpl.
Time reflexivity.
Abort.
(** unfold *)
Definition plus_two (x : nat) : nat :=
x + 2.
Lemma switch_to_honors_again :
plus_two 10 = 12.
Proof.
unfold plus_two.
reflexivity.
Qed.
(** apply *)
Lemma modus_ponens : forall (P Q : Prop),
(P -> Q) -> P -> Q.
Proof.
intros P Q P_implies_Q P_holds.
apply P_implies_Q.
Abort.
Lemma modus_ponens'' : forall (P Q : Prop),
P -> (P -> Q) -> Q.
Proof.
auto.
Qed.
Lemma double_negation : forall (P : Prop),
P -> ~~P.
Proof.
unfold not. intro P.
apply modus_ponens''.
Qed.
(** rewrite *)
Lemma add_comm : forall (x y : nat),
x + y = y + x.
Proof.
intros. induction x as [ | x' IHx' ].
- trivial.
- simpl. rewrite -> IHx'.
trivial.
Qed.
(** inversion *)
Lemma succ_eq_implies_eq : forall (x y : nat),
S x = S y -> x = y.
Proof.
intros x y succ_eq.
inversion succ_eq.
trivial.
Qed.
(** left , right *)
Lemma or_left : forall (P Q : Prop),
P -> P \/ Q.
Proof.
intros P Q P_holds.
left.
Abort.
Lemma or_right : forall (P Q : Prop),
Q -> P \/ Q.
Proof.
intros P Q Q_holds.
right.
Abort.
(** replace *)
Lemma one_x_one : forall (x : nat),
1 + x + 1 = 2 + x.
Proof.
intro. simpl.
replace (x + 1) with (S x).
Abort.
(** split *)
Lemma implies_and : forall (P Q R : Prop),
P -> (P -> Q) -> (P -> R) -> (Q /\ R).
Proof.
intros P Q R P_holds.
intros P_implies_Q P_implies_R.
split. Undo. constructor.
- apply P_implies_Q . assumption .
- apply P_implies_R . assumption .
Qed.
Lemma and_left : forall (P Q : Prop),
(P /\ Q) -> P.
Proof.
intros P Q P_and_Q.
destruct P_and_Q as [P_holds Q_holds].
assumption.
Qed.
Lemma or_comm : forall (P Q : Prop),
P \/ Q -> Q \/ P.
Proof.
intros P Q P_or_Q.
destruct P_or_Q as [P_holds | Q_holds].
- right. Undo. constructor 2. assumption.
- left. Undo. constructor 1. assumption.
Qed.
Inductive element :=
| grass : element
| fire : element
| water : element.
Definition weakness (e : element) : element :=
match e with
| grass => fire
| fire => water
| water => grass
end.
Lemma never_weak_to_self : forall (e : element),
weakness e <> e.
Proof.
destruct e.
- simpl. discriminate.
- simpl. discriminate.
- simpl. discriminate.
Abort.
Lemma n_plus_n : forall (n : nat),
n + n = n * 2.
Proof.
induction n as [| x IH].
- reflexivity.
- simpl. rewrite <- IH. auto.
Qed.
(** auto *)
(** logical tauto *)
Lemma demorgan : forall (P Q : Prop),
~(P \/ Q) -> ~P /\ ~Q.
Proof.
tauto.
Qed.
From Coq Require Import Lia.
Lemma dfoil : forall a ,
(a + 2) * (1 + 2) = a*1 + 2*1 + a*2 + 2*2.
Proof.
intros. lia.
Qed.
From Coq Require Import Arith.
(** algebraic rihg *)
Lemma foil : forall a b c d,
(a + b) * (c + d) = a*c + b*c + a*d + b*d.
Proof.
intros. ring.
Qed.
</textarea>
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