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module Data.Term.Property
import Data.Term
import Data.Vect.Properties.Index
import Data.Vect.Quantifiers
import Data.Vect.Quantifiers.Extra
import Syntax.PreorderReasoning
-- Definition ------------------------------------------------------------------
public export
record Property (sig : Signature) (k : Nat) where
constructor MkProp
Prop : forall n. (Fin k -> Term sig n) -> Type
cong : forall n. {f, g : Fin k -> Term sig n} -> f .=. g -> Prop f -> Prop g
%name Property p, q
-- Instances -------------------------------------------------------------------
public export
Unifies : Term sig k -> Term sig k -> Property sig k
Unifies t u = MkProp
{ Prop = \f => f <$> t = f <$> u
, cong = \cong, prf => Calc $
|~ _ <$> t
~~ _ <$> t ..<(subCong cong t)
~~ _ <$> u ...(prf)
~~ _ <$> u ...(subCong cong u)
}
public export
All : Vect n (Property sig k) -> Property sig k
All ps = MkProp (\f => All (\p => p.Prop f) ps) (\cong => mapRel (\p => p.cong cong))
-- Equivalence -----------------------------------------------------------------
public export 0
(<=>) : Property sig k -> Property sig k -> Type
p <=> q = forall n. (f : Fin k -> Term sig n) -> p.Prop f <=> q.Prop f
-- Properties
export
unifiesSym : Unifies t u <=> Unifies u t
unifiesSym f = MkEquivalence (\prf => sym prf) (\prf => sym prf)
export
unifiesOp :
{k : Nat} ->
{0 op : sig.Operator k} ->
{0 ts, us : Vect k (Term sig j)} ->
Unifies (Op op ts) (Op op us) <=> All (tabulate (\i => Unifies (index i ts) (index i us)))
unifiesOp f = MkEquivalence
{ leftToRight = \prf => tabulate (\i => Unifies (index i ts) (index i us)) (leftToRight prf)
, rightToLeft = \prf => irrelevantEq $ cong (Op op) $ rightToLeft prf
}
where
leftToRight :
(Unifies (Op op ts) (Op op us)).Prop f ->
(i : _) ->
f <$> index i ts = f <$> index i us
leftToRight prf i =
Calc $
|~ f <$> index i ts
~~ index i (map (f <$>) ts) ...(sym $ indexNaturality i (f <$>) ts)
~~ index i (map (f <$>) us) ...(cong (index i) $ opInjectiveTs' prf)
~~ f <$> index i us ...(indexNaturality i (f <$>) us)
rightToLeft :
{k : Nat} ->
{ts, us : Vect k (Term sig j)} ->
(All (tabulate (\i => Unifies (index i ts) (index i us)))).Prop f ->
map (f <$>) ts = map (f <$>) us
rightToLeft {ts = [], us = []} [] = Refl
rightToLeft {ts = t :: ts, us = u :: us} (prf :: prfs) = cong2 (::) prf (rightToLeft prfs)
-- Nothing ---------------------------------------------------------------------
public export 0
Nothing : Property sig k -> Type
Nothing p = forall n. (f : Fin k -> Term sig n) -> Not (p.Prop f)
-- Properties
export
nothingEquiv : p <=> q -> Nothing p -> Nothing q
nothingEquiv eq absurd f x = absurd f ((eq f).rightToLeft x)
-- Extensions ------------------------------------------------------------------
public export
Extension : Property sig k -> (Fin k -> Term sig n) -> Property sig n
Extension p f = MkProp
{ Prop = \g => p.Prop (g . f)
, cong = \prf => p.cong (\i => subCong prf (f i))
}
-- Properties
export
nothingExtends : Nothing p -> Nothing (Extension p f)
nothingExtends absurd g x = void $ absurd (g . f) x
export
extendUnit : (p : Property sig k) -> p <=> Extension p Var
extendUnit p f = MkEquivalence id id
export
extendAssoc :
(p : Property sig j) ->
(f : Fin j -> Term sig k) ->
(g : Fin k -> Term sig m) ->
Extension (Extension p f) g <=> Extension p (g . f)
extendAssoc p f g h =
MkEquivalence
(p.cong (\i => subAssoc h g (f i)))
(p.cong (\i => sym $ subAssoc h g (f i)))
export
extendUnify :
(t, u : Term sig j) ->
(f : Fin j -> Term sig k) ->
(g : Fin k -> Term sig m) ->
Extension (Unifies t u) (g . f) <=> Extension (Unifies (f <$> t) (f <$> u)) g
extendUnify t u f g h =
MkEquivalence
(\prf => Calc $
|~ (h . g) <$> (f <$> t)
~~ ((h . g) . f) <$> t ...(sym $ subAssoc (h . g) f t)
~~ (h . (g . f)) <$> t ...(subCong (\i => subAssoc h g (f i)) t)
~~ (h . (g . f)) <$> u ...(prf)
~~ ((h . g) . f) <$> u ...(sym $ subCong (\i => subAssoc h g (f i)) u)
~~ (h . g) <$> (f <$> u) ...(subAssoc (h . g) f u))
(\prf => Calc $
|~ (h . (g . f)) <$> t
~~ ((h . g) . f) <$> t ...(sym $ subCong (\i => subAssoc h g (f i)) t)
~~ (h . g) <$> (f <$> t) ...(subAssoc (h . g) f t)
~~ (h . g) <$> (f <$> u) ...(prf)
~~ ((h . g) . f) <$> u ...(sym $ subAssoc (h . g) f u)
~~ (h . (g . f)) <$> u ...(subCong (\i => subAssoc h g (f i)) u))
|
