[Agda] unsolved metas in Respects₂

Sergei Meshveliani mechvel at botik.ru
Sun Dec 15 21:56:35 CET 2013

```Please,
how to fix the below code?

It defines  StrictTotalOrder  on  A × B,  where the comparison is by the
first component (so that B is a dummy factor).

The  development Agda of November ~20 2013 MAlonzo
reports of
unsolved metas  in the line of  " = (foo1 , foo2)".

I have introduced several intermediate signatures -- to make sure in the
types, and wonder what else is needed to resolve this direct product
expression.

Thanks,

------
Sergei

-------------------------------------------------------------------------
open import Level           using (Level; _⊔_)
open import Function        using (flip; _on_)
open import Relation.Binary using
(Rel; Reflexive; Symmetric; Transitive; IsEquivalence;
module IsEquivalence; _Respects_; _Respects₂_; Trichotomous;
IsStrictTotalOrder; module IsStrictTotalOrder; StrictTotalOrder;
module StrictTotalOrder
)
open import Data.Product using (_×_; _,_; proj₁; proj₂)

stoForPairBy1 : ∀ {α α= α< β} → StrictTotalOrder α α= α< → Set β →
StrictTotalOrder (α ⊔ β) α= α<
stoForPairBy1 {α} {α=} {α<} {β} stoA B =
record{ Carrier            = A × B
; _≈_                = _=₁_
; _<_                = _<₁_
; isStrictTotalOrder = isSTO₁ }
where
open StrictTotalOrder stoA using ()
renaming (Carrier to A; _≈_ to _=a_; _<_ to _<a_;
isStrictTotalOrder to isSTO-A)
open IsStrictTotalOrder isSTO-A using ()
renaming (isEquivalence to aEquiv; trans to <a-trans;
compare to aCompare; <-resp-≈ to <a-cong)
open IsEquivalence aEquiv using ()
renaming (refl to aRefl; sym to aSym; trans to aTrans)

_=₁_ :  Rel (A × B) α=
_=₁_ =  _=a_ on proj₁

_<₁_ :  Rel (A × B) α<
_<₁_ =  _<a_ on proj₁

=₁refl : Reflexive _=₁_
=₁refl = aRefl

=₁sym : Symmetric _=₁_
=₁sym = aSym

=₁trans : Transitive _=₁_
=₁trans = aTrans

=₁equiv : IsEquivalence _=₁_
=₁equiv = record{ refl  = \ {x}         → =₁refl {x}
; sym   = \ {x} {y}     → =₁sym {x} {y}
; trans = \ {x} {y} {z} → =₁trans {x} {y} {z} }

<₁trans : Transitive _<₁_
<₁trans = <a-trans

compare₁ : Trichotomous _=₁_ _<₁_
compare₁ (a , _) (a' , _) =  aCompare a a'

<a-cong1 : {x : A} → (_<a_ x) Respects _=a_
<a-cong1 = proj₁ <a-cong

<a-cong2 : {y : A} → (flip _<a_ y) Respects _=a_
<a-cong2 = proj₂ <a-cong

cong-<₁1 : ∀ {p} → (_<₁_ p) Respects _=₁_
cong-<₁1 {p1 , _} {q1 , _} {q1' , _} q=q' p<q =  <a-cong1 q=q' p<q

cong-<₁2 : ∀ {q} → (flip _<₁_ q) Respects _=₁_
cong-<₁2 {q1 , _} {p1 , _} {p1' , _} p=p' p<q =  <a-cong2 p=p' p<q

cong-<₁ : _<₁_ Respects₂ _=₁_
-- {α ⊔ β} {α<} {α=} {A × B} _<₁_ _=₁_
cong-<₁ = pp
where
foo1 : ∀ {p} → (_<₁_ p) Respects _=₁_
foo1 {p} {q} {q'}= cong-<₁1 {p} {q} {q'}

foo2 : ∀ {q} → (flip _<₁_ q) Respects _=₁_
foo2 {q} {p} {p'} =  cong-<₁2 {q} {p} {p'}

pp : (∀ {p} → (_<₁_ p) Respects _=₁_) ×
(∀ {q} → (flip _<₁_ q) Respects _=₁_)
pp = (foo1 , foo2)

isSTO₁ : IsStrictTotalOrder _=₁_ _<₁_
isSTO₁ = record{ isEquivalence = =₁equiv
; trans    = \ {x} {y} {z} → <₁trans {x} {y} {z}
; compare  = compare₁
; <-resp-≈ = cong-<₁
}

```