How to define w types in agda?

How to define w types in agda?

[du yu]

[-- Attachment #1.1: Type: text/plain, Size: 560 bytes --]

I have the following initial thoughts but can't work out to define zero as 
a w type

data Zro : Set  where

data One : Set where
  O1 : One

data Two : Set where
  O2 : Two
  I2 : Two
-- w types 

rec2 : (x y : Set) -> Two -> Set
rec2 x _ O2 = x
rec2 _ y I2 = y


data W (A : Set) (B : A -> Set) : Set where -- well founded trees
  w : (s : A) -> B s -> W A B
  sup : (a : A) -> ((B a) -> ((x : A) -> W A B )) -> W A B

natw : Set
natw = W Two (rec2 Zro One) -- nat type as w type

zero_w : natw
zero_w = sup O2 (λ x y → {!!})


[-- Attachment #1.2: Type: text/html, Size: 896 bytes --]

Re: [HoTT] How to define w types in agda?

[Favonia]

[-- Attachment #1: Type: text/plain, Size: 1504 bytes --]

Hello,

The first argument to the function is of type Zro, and you can use the
absurd pattern to define such a function. See
http://agda.readthedocs.io/en/v2.5.3/language/function-definitions.html#absurd-patterns
.

However, there is a bigger issue---your W type seems wrong. At least I do
not see why it is equivalent to the standard presentation. See
https://github.com/agda/agda-stdlib/blob/master/src/Data/W.agda for a
correct implementation in the standard library.

Hope this helps.

Best,
Favonia

On Thu, Jan 25, 2018 at 11:26 AM du yu <doof...@gmail.com> wrote:

> I have the following initial thoughts but can't work out to define zero as
> a w type
>
> data Zro : Set  where
>
> data One : Set where
>   O1 : One
>
> data Two : Set where
>   O2 : Two
>   I2 : Two
> -- w types
>
> rec2 : (x y : Set) -> Two -> Set
> rec2 x _ O2 = x
> rec2 _ y I2 = y
>
>
> data W (A : Set) (B : A -> Set) : Set where -- well founded trees
>   w : (s : A) -> B s -> W A B
>   sup : (a : A) -> ((B a) -> ((x : A) -> W A B )) -> W A B
>
> natw : Set
> natw = W Two (rec2 Zro One) -- nat type as w type
>
> zero_w : natw
> zero_w = sup O2 (λ x y → {!!})
>
> --
> You received this message because you are subscribed to the Google Groups
> "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to HomotopyTypeThe...@googlegroups.com.
> For more options, visit https://groups.google.com/d/optout.
>

[-- Attachment #2: Type: text/html, Size: 2502 bytes --]

Re: [HoTT] How to define w types in agda?

[du yu]

[-- Attachment #1.1: Type: text/plain, Size: 1866 bytes --]

Wow thanks! I don't know it is already defined in stdlib (google search 
gives none!). As a beginner for dependent types , the data type definition 
is quite confusing sometimes!

On Friday, January 26, 2018 at 10:38:39 AM UTC+8, Favonia wrote:
>
> Hello,
>
> The first argument to the function is of type Zro, and you can use the 
> absurd pattern to define such a function. See 
> http://agda.readthedocs.io/en/v2.5.3/language/function-definitions.html#absurd-patterns
> .
>
> However, there is a bigger issue---your W type seems wrong. At least I do 
> not see why it is equivalent to the standard presentation. See 
> https://github.com/agda/agda-stdlib/blob/master/src/Data/W.agda for a 
> correct implementation in the standard library.
>
> Hope this helps.
>
> Best,
> Favonia
>
> On Thu, Jan 25, 2018 at 11:26 AM du yu <doo...@gmail.com <javascript:>> 
> wrote:
>
>> I have the following initial thoughts but can't work out to define zero 
>> as a w type
>>
>> data Zro : Set  where
>>
>> data One : Set where
>>   O1 : One
>>
>> data Two : Set where
>>   O2 : Two
>>   I2 : Two
>> -- w types 
>>
>> rec2 : (x y : Set) -> Two -> Set
>> rec2 x _ O2 = x
>> rec2 _ y I2 = y
>>
>>
>> data W (A : Set) (B : A -> Set) : Set where -- well founded trees
>>   w : (s : A) -> B s -> W A B
>>   sup : (a : A) -> ((B a) -> ((x : A) -> W A B )) -> W A B
>>
>> natw : Set
>> natw = W Two (rec2 Zro One) -- nat type as w type
>>
>> zero_w : natw
>> zero_w = sup O2 (λ x y → {!!})
>>
>> -- 
>> You received this message because you are subscribed to the Google Groups 
>> "Homotopy Type Theory" group.
>> To unsubscribe from this group and stop receiving emails from it, send an 
>> email to HomotopyTypeThe...@googlegroups.com <javascript:>.
>> For more options, visit https://groups.google.com/d/optout.
>>
>

[-- Attachment #1.2: Type: text/html, Size: 4414 bytes --]