Question

我对closed_subspace_equiv

有以下定义

library(dplyr)
library(tidyr)
foo <- data.frame(comb=replicate(10, 
                                 paste(paste(sample(LETTERS, 4), collapse=''),
                                       sample(c('p', 'n'), 1), 
                                       sample(1:100, 1), 
                                       paste(sample(letters, 2), collapse=''), 
                                       sep='.')
                                 ),
                  num = sample(1:10, 10, replace=T))

我想要的是

Record Closed_Subspace (V:Normed_Space) := {
closed_subspace  :> V -> Prop;
addition_closure : forall (x y:V),(closed_subspace x) -> (closed_subspace y) -> (closed_subspace (add V x y));
smul_closure     : forall (x:V) (a:R),(closed_subspace x) -> (closed_subspace (scalar_mul V a x));
subspace_closure : forall (x:V), closure (closed_subspace) x <-> closed_subspace x}.

Definition closed_subspace_equiv {V : Normed_Space} (U:Closed_Subspace V) (x y:V) (p:U x)(q:U y) := exists z:V,(add V x z = y) /\ (U z).

我该怎么做？

对于上下文，这里是Normed_Space。

Definition closed_subspace_equiv {V : Normed_Space} (U:Closed_Subspace V) (x y:U) := exists z:U,(add V x z = y).

Answer 1

V参数可以移动到记录正文中，使用:>语法自动创建强制。

Record Closed_Subspace := {
  normed_space :> Normed_Space;
  closed_subspace  :> normed_space -> Prop;
  addition_closure : forall x y:normed_space, closed_subspace x -> closed_subspace y -> closed_subspace (add normed_space x y);
  smul_closure     : forall (x:normed_space) (a:R), closed_subspace x -> closed_subspace (scalar_mul normed_space a x);
  subspace_closure : forall x:normed_space, closure (closed_subspace) x <-> closed_subspace x
}.

现在你的第二个定义有效：

Definition closed_subspace_equiv (U:Closed_Subspace) (x y:U) :=
  exists z:U, add _ x z = y.

保留参数V的另一种方法是定义Closed_Subspace，如下所示：

Record Closed_Subspace (V:Normed_Space) : Type := {
  normed_space := V;
  closed_subspace  :> V -> Prop;
  addition_closure : forall x y:V, closed_subspace x -> closed_subspace y -> closed_subspace (add V x y);
  smul_closure     : forall (x:V) (a:R), closed_subspace x -> closed_subspace (scalar_mul V a x);
  subspace_closure : forall (x:V), closure (closed_subspace) x <-> closed_subspace x
}.

手动添加必要的强制：

Coercion normed_space : Closed_Subspace >-> Normed_Space.

如何用“子类型”编写coq定义

1 个答案: