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|
module Thinned
import Control.Order
import Control.Relation
import public Thinning
%prefix_record_projections off
-- Definition ------------------------------------------------------------------
||| Data embedded in a wider context via a thinning.
public export
record Thinned (t : SnocList a -> Type) (sx : SnocList a) where
constructor Over
{0 support : SnocList a}
value : t support
thin : support `Thins` sx
%name Thinned v, u
||| An equivalence relation on thinned objects.
public export
data (<~>) : Thinned t sx -> Thinned t sx -> Type where
UpToThin :
{0 v : t sx} ->
{0 thin1, thin2 : sx `Thins` sy} ->
thin1 <~> thin2 ->
(<~>) {t} (v `Over` thin1) (v `Over` thin2)
%name Thinned.(<~>) prf
export (.supportPrf) :
{0 v, u : Thinned t sx} ->
v <~> u ->
v.support = u.support
(UpToThin prf) .supportPrf = Refl
export (.valuePrf) :
{0 v, u : Thinned t sx} ->
(e : v <~> u) ->
v.value = (rewrite e.supportPrf in u.value)
(UpToThin prf) .valuePrf = Refl
export (.thinPrf) :
{0 v, u : Thinned t sx} ->
(e : v <~> u) ->
v.thin <~> (rewrite e.supportPrf in u.thin)
(UpToThin prf) .thinPrf = prf
--- Properties
export
irrelevantEquiv :
{0 v, u : Thinned t sx} ->
(0 prf : v <~> u) ->
v <~> u
irrelevantEquiv {v = v `Over` thin1, u = u `Over` thin2} prf =
rewrite prf.supportPrf in
rewrite prf.valuePrf in
UpToThin (rewrite sym prf.supportPrf in irrelevantEquiv prf.thinPrf)
export
Reflexive (Thinned t sx) (<~>) where
reflexive {x = t `Over` thin} = UpToThin reflexive
export
Symmetric (Thinned t sx) (<~>) where
symmetric (UpToThin prf) = UpToThin (symmetric prf)
export
Transitive (Thinned t sx) (<~>) where
transitive (UpToThin prf1) (UpToThin prf2) = UpToThin (transitive prf1 prf2)
export
Equivalence (Thinned t sx) (<~>) where
export
Preorder (Thinned t sx) (<~>) where
-- Operations ------------------------------------------------------------------
||| Map the underlying value.
public export
map : (forall sx. t sx -> u sx) -> Thinned t sx -> Thinned u sx
map f (value `Over` thin) = f value `Over` thin
||| Weaken the surrounding context.
public export
wkn : Thinned t sx -> sx `Thins` sy -> Thinned t sy
wkn (value `Over` thin') thin = value `Over` thin . thin'
||| Shift the surrounding context.
public export
shift : Thinned t sx -> Thinned t (sx :< x)
shift (value `Over` thin) = value `Over` Drop thin
--- Properties
export
mapCong :
{0 t, u : SnocList a -> Type} ->
(0 f : forall sx. t sx -> u sx) ->
{0 v1, v2 : Thinned t sx} ->
v1 <~> v2 ->
map f v1 <~> map f v2
mapCong f (UpToThin prf) = UpToThin prf
export
shiftCong : {0 v, u : Thinned t sx} -> v <~> u -> shift v <~> shift u
shiftCong (UpToThin prf) = UpToThin (dropCong prf)
-- Pairs -----------------------------------------------------------------------
||| Product of two values ensuring the whole context is used.
public export
record Pair (t, u : SnocList a -> Type) (sx : SnocList a) where
constructor MakePair
fst : Thinned t sx
snd : Thinned u sx
0 cover : Covering fst.thin snd.thin
||| Smart constructor for thinned pairs.
export
MkPair : Thinned t sx -> Thinned u sx -> Thinned (Pair t u) sx
MkPair (v `Over` thin1) (u `Over` thin2) =
let cp = coprod thin1 thin2 in
MakePair (v `Over` cp.thin1') (u `Over` cp.thin2') cp.cover `Over` cp.factor
--- Properties
export
mkPairCong :
{0 v1, w1 : Thinned t sx} ->
{0 v2, w2 : Thinned u sx} ->
v1 <~> w1 ->
v2 <~> w2 ->
MkPair v1 v2 <~> MkPair w1 w2
mkPairCong (UpToThin prf1) (UpToThin prf2) =
?mkPairCong_rhs_1
|
