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qoc_jisuanji_tactics.html
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qoc_jisuanji_tactics.html
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<!--
This sample contains two very simple introductory exercises of
interactive proofs in Coq, one with natural numbers and one with lists.
It can be used as a gentle landing page, which requires no knowledge
of math-comp.
-->
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<p>
+ Friday, March 29, 2019, 23:00:00
<br/> OOO1337777 , COQ , 鸡算计 , mathematics , 数学 - tactic , suffice - 战术 , 满足
<br/> https://bilibili.com/video/av47713482
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<p>
Alt+↑/↓ – move through proof; Alt+→ or Alt+⏎ – go to cursor. <br/>
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from the standard library, and also show types of lemmas in the right pane.
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“Alt +↑/↓” : 通过证明。 “Alt +→”或“Alt +⏎” : 转到光标。
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From Qoc Require Import Jisuanji .
Import ssreflect .
(**MEMO:
短 :: 为了从“A”到“E”的目标，搜索/猜测某些“S”从“A”到“S”然后从“S”变为“E”可能更明智/更具策略性“。
Short :: for the goal of going from "A" to "E" , it may be more sensible/tactical to search/guess some "S" such to go from "A" to "S" then to go from "S" to "E" .
It may be more-sensible to refine/start/open the top of some deduction/proof of some end/goal [A |- E] by sequencing this deduction into two more-sensible parts/halves [A |- S] and [A |- S -> E] , where oneself shall search/guess/devine the mediator [S] by using the contextual sense of the end/goal .
In the most general text , [E] is the END/goal . And [S] is some SEARCHED/guessed/devined weaker SUB-end/goal which suffices ( expressed as [ S -> E ] ) for the end/goal [E] .
In the less general text , [W] is some searched/guessed/devined WEAKENER which defines this particular sub-end/goal [ S := (W -> E) ] ( the end UNDER [W] ) , which suffices ( [ (W -> E) -> E ] ) for the end/goal [E] . Alternatively , oneself may transpose this precedent result and view the sufficiency expression as [ W -> E ] and the sub-end/goal as [ S := ( (W -> E) -> E ) ] ( the RELATIVE [W] , which is some style of double negation of [W] above the end [E] not the bottom [False] )
Everythere , the identifier [p] is some outer-argument/parameter and [x] is some (varying) inner-argument . And the sense for the identifier [x] is not binary , possible senses are : [simultaneous] identifier , [parametric]/singly/pointwise identifier , [none] identifier .
-----
+ simultaneous have s_ , s_x : x / S p x
1(instance): x |- S p x
2(generalized): x ; s_ : (forall x , S p x) ; s_x : S x |- E p x
* have s : S
1(instance): |- S
2(generalized): s : S |- E
+ simultaneous suffices s_ : x / S p x
1(generalized): x ; s_ : (forall x , S p x) |- E p x
2(instance): x |- S p x
* suffices s : S
1(generalized): s : S |- E
2(instance): |- S
+ parametric have_relatively w : x / W p x
1(instance): x |- (forall x , W p x -> E p x) -> E p x
2(generalized): x ; w : W p x |- E p x
* have_relatively w : W
1(instance): |- (W -> E) -> E
2(generalized): w : W |- E
* have_under m : W
1(instance): |- W -> E
2(generalized): m : (W -> E) |- E
+ parametric suffices_relatively w : x / W p x
1(generalized): x ; w : W p x |- E p x
2(instance): x |- (forall x , W p x -> E p x) -> E p x
* suffices_relatively w : W
1(generalized): w : W |- E
2(instance): |- (W -> E) -> E
* suffices_under m : W
1(generalized): m : (W -> E) |- E
2(instance): |- W -> E
-----
+ simultaneous have s_ , s_x : x / S p x
:= gen have s_ , s_x : x / S p x
* have s : S
:= have s : S
+ simultaneous suffices s_ : x / S p x
:= wlog suff s_ : x / S p x
* suffices s : S
:= suff s : S
+ parametric have_relatively w : x / W p x
:= wlog w : x / W a x
* have_relatively w : W
:= ??
* have_under m : W
:= have suff m : W
+ parametric suffices_relatively w : x / W p x
:= ??
* suffices_relatively w : W
:= suff have w : W
* suffices_under m : W
:= ??
*)
Module Tactics .
Module Generalization .
Parameter A : forall (p : bool) , Type .
Parameter E : forall (p : bool) (x : bool) , Type .
Parameter S : forall (p : bool) (x : bool) , Type .
Parameter W : forall (p : bool) (x : bool) , Type .
Section section1 .
Variable p : bool .
Lemma lemma1 : A p -> forall x : bool , E p x .
Proof.
intros a x .
simultaneous have s_ , s_x : x / S p x . Show 2 . Undo .
(** Tóngshí jùyǒu *)
同时 具有 s_ , s_x : x / S p x . Show 2 . Undo .
(** 1(instance): x |- S p x
2(generalized): x ; s_ : (forall x , S p x) ; s_x : S x |- E p x *)
have s : S p x . Show 2. Undo.
(** Jùyǒu *)
具有 s : S p x . Show 2. Undo.
(** 1(instance): |- S
2(generalized): s : S |- E *)
Abort .
Lemma lemma1 : A p -> forall x : bool , E p x .
Proof.
intros a x .
simultaneous suffices s_ : x / S p x . Show 2 . Undo .
(** Tóngshí mǎnzú *)
同时 满足 s_ : x / S p x . Show 2 . Undo .
(** 1(generalized): x ; s_ : (forall x , S p x) |- E p x
2(instance): x |- S p x *)
suffices s : S p x . Show 2 . Undo .
(** mǎnzú *)
满足 s : S p x . Show 2 . Undo .
(** 1(generalized): s : S |- E
2(instance): |- S *)
Abort .
Lemma lemma1 : A p -> forall x : bool , E p x .
Proof.
intros a x .
parametric have_relatively w : x / W p x . Show 2 . Undo .
(** Cān jùyǒu_xiāngduì *)
参 具有_相对 w : x / W p x . Show 2 . Undo .
(** 1(instance): x |- (forall x , W p x -> E p x) -> E p x
2(generalized): x ; w : W p x |- E p x *)
have_relatively w : W p x . Show 2 . Undo .
(** Cān jùyǒu_xiāngduì *)
具有_相对 w : W p x . Show 2 . Undo .
(** 1(instance): |- (W -> E) -> E
2(generalized): w : W |- E *)
have_under m : W p x . Show 2 . Undo .
(** Jùyǒu_xià *)
具有_下 m : W p x . Show 2 . Undo .
(** 1(instance): |- W -> E
2(generalized): m : (W -> E) |- E *)
Abort .
Lemma lemma1 : A p -> forall x : bool , E p x .
Proof.
intros a x .
parametric suffices_relatively w : x / W p x . Show 2 . Undo .
(** Cān mǎnzú_xiāngduì *)
参 满足_相对 w : x / W p x . Show 2 . Undo .
(** 1(generalized): x ; w : W p x |- E p x
2(instance): x |- (forall x , W p x -> E p x) -> E p x *)
suffices_relatively w : W p x . Show 2 . Undo .
(** Mǎnzú_xiāngduì *)
满足_相对 w : W p x . Show 2 . Undo .
(** 1(generalized): w : W |- E
2(instance): |- (W -> E) -> E *)
suffices_under m : W p x . Show 2 . Undo .
(** Mǎnzú_xià *)
满足_下 m : W p x . Show 2 . Undo .
(** 1(generalized): m : (W -> E) |- E
2(instance): |- W -> E *)
Abort.
End section1.
Section Generalization_example .
Lemma quo_rem_unicity ( d : nat) :
forall ( q1 q2 r1 r2 : nat ) ,
q1*d + r1 = q2*d + r2 ->
r1 < d -> r2 < d ->
(pair q1 r1) = (pair q2 r2) .
Proof .
intros q1 q2 r1 r2 .
wlog: q1 q2 r1 r2 / q1 <= q2 . Show 2 . Undo .
parametric have_relatively w : q1 q2 r1 r2 / q1 <= q2 . Show 2 . Undo .
(** Cān jùyǒu_xiāngduì *)
参 具有_相对 w : q1 q2 r1 r2 / q1 <= q2 . Show 2 . Undo .
(** 1(instance): (用 q3 q4 r3 r4 : nat, q3 <= q4 ->
q3 * d + r3 = q4 * d + r4 -> r3 < d -> r4 < d -> (q3, r3) = (q4, r4))
-> q1 * d + r1 = q2 * d + r2 -> r1 < d -> r2 < d -> (q1, r1) = (q2, r2)
2(generalized): w : q1 <= q2 |- q1 * d + r1 = q2 * d + r2 -> r1 < d -> r2 < d -> (q1, r1) = (q2, r2) *)
Abort .
End Generalization_example .
End Generalization .
Definition tactics_move_apply_exact_elim_case
: forall n : nat , nat .
Proof .
move => m . Undo .
(** Yídòng *)
移动 => m .
apply : m . Undo .
(** Yìngyòng *)
应用 : m . Undo .
exact : m . Undo .
(** Quèqiè *)
确切 : m . Undo .
elim : m . Undo .
(** Xiāochú *)
消除 : m . Undo .
case : m . Undo .
(** Lìzi *)
例子 : m . Undo .
Abort.
(**TODO: solve this *)
Lemma tactics_abstract (n m : nat) : True.
have [:Sm] @plus_n_Sm : nat .
{ apply: (plus n).
abstract: Sm.
{ exact: (S m).
}
}
Restart .
(** Jùyǒu *)
具有 [:Sm] @plus_n_Sm : nat .
{ (** Yìngyòng *)
应用: (plus n).
(** Chōuxiàng *)
抽象: Sm.
{ (** Quèqiè *)
确切: (S m).
}
}
Abort.
Definition tactics_rewrite : forall n m : nat , n = m -> m = n .
Proof .
intros n m H.
rewrite H . Undo .
(** Gǎixiě *)
改写 H . Undo .
Abort.
Lemma tactic_pose : True.
pose f x y := x + y . Undo .
(** Bǎi *)
摆 f x y := x + y. Undo .
pose fix f (x y : nat) {struct x} : nat :=
if x is S p then S (f p y) else 0 .
Undo .
(** Bǎi gùdìng *)
摆 固定 f (x y : nat) {struct x} : nat :=
if x is S p then S (f p y) else 0 .
Undo .
Abort .
Lemma tactic_set (x y z : nat) : x + y = z.
set t := _ x. Undo .
(** Bǎiliè *)
摆列 t := _ x . Undo .
Abort .
Lemma tactic_lock n m : (S n) + m = match (S n) with
S p => m + (S p)
| 0 => m + 0
end.
rewrite {1}[S]lock .
rewrite -lock. Undo.
unlock .
move: (S n) . Restart .
(** Gǎixiě *)
改写 {1}[S]lock .
(** Kāisuǒ *)
开锁 .
(** Yídòng *)
移动: (S n).
Abort.
Definition add :=
( 固定 add (n m : nat) {构 n} : nat := 匹配 n 与
| 0 => m
| S p => S (add p m)
结束 ) .
(** Bù jiǎnhuà *)
Definition addn := 不简化 add .
Lemma notation_nosimpl : forall n m,
( if (addn (S n) (S m))
is (S p ) then p
else 0 )
= (addn (S m) n) .
intros n m .
simpl . fold addn .
move: (S m).
Abort.
Lemma tactic_congr x y :
x + (y * (y + x - x)) = (x * 1) + (y + 0) * y.
congr plus . Undo .
(** Cítóu *)
词头 plus . Undo .
congr ( _ + (_ * _)) . Undo .
(** Cítóu *)
词头 ( _ + (_ * _)) . Undo .
Abort .
Inductive test : nat -> Prop :=
| C1 : forall n , n = 1 -> test n
| C2 : forall n , n = 2 -> test n
| C3 : forall n , n = 3 -> test n
| C4 : forall n , n = 4 -> test n
| C5 : forall n , n = 5 -> test n .
Lemma tactic_last n (t : test n) : True.
case : t ;
last 2 [ move => k3
| move => k4 ] . Undo .
(** Lìzi *)
例子 : t ;
(** Hòu *)
后 2 [ (** Yídòng *) 移动 => k3
| (** Yídòng *) 移动 => k4 ] . Undo .
case : t ;
first last . Undo .
(** Lìzi *)
例子 : t ;
(** Qián hòu *)
前 后 .
Abort.
End Tactics .
Print Tactics .
</textarea>
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<script src="../ui-js/jscoq-loader.js" type="text/javascript"></script>
<script type="text/javascript">
var jscoq_ids = ['workspace'];
var jscoq_opts = {
prelude: true,
base_path: '../',
init_pkgs: ['init', 'qoc'],
all_pkgs: ['init', 'qoc', 'coq-base', 'coq-collections', 'coq-arith', 'coq-reals', 'math-comp'],
implicit_libs: true,
editor: { mode: { 'company-coq': true }, keyMap: 'default' }
};
/* Global reference */
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loadJsCoq(jscoq_opts.base_path)
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