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 → {!!})
>>
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>