a half-implemented verified dimeq
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module Quox.Syntax.DimEq
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import public Quox.Syntax.Var
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import public Quox.Syntax.Dim
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import public Quox.Syntax.Subst
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import public Quox.Context
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import Data.Maybe
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import Data.DPair
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%default total
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mutual
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||| consistent (0≠1) set of constraints between dimension variables
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public export
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data DimEq' : Nat -> Type where
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||| empty context
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Nil : DimEq' 0
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||| Ψ, 𝑖, 𝑖=ε
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Const : (eqs : DimEq' d) -> (e : DimConst) -> DimEq' (S d)
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||| Ψ, 𝑖, 𝑖=𝑗 (Ψ ⊢ 𝑗 and 𝑗 is unassigned)
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Var : (eqs : DimEq' d) -> (i : Var d) -> (0 un : Unassigned eqs i) ->
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DimEq' (S d)
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||| Ψ, 𝑖 (𝑖 unassigned)
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None : (eqs : DimEq' d) -> DimEq' (S d)
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%name DimEq' eqs
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public export
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data Unassigned : DimEq' d -> Var d -> Type where
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UZ : Unassigned (None eqs) VZ
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USK : Unassigned eqs i -> Unassigned (Const eqs e) (VS i)
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USV : Unassigned eqs i -> Unassigned (Var eqs j un) (VS i)
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USN : Unassigned eqs i -> Unassigned (None eqs ) (VS i)
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%name Unassigned un
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||| set of constraints that might be inconsistent
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public export
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data DimEq : Nat -> Type where
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||| 0=1
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ZeroIsOne : DimEq d
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||| 0≠1, plus other constraints
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C : (eqs : DimEq' d) -> DimEq d
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%name DimEq eqs
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||| contains a value iff the dim ctx is consistent
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public export
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data IfConsistent : DimEq d -> Type -> Type where
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Nothing : IfConsistent ZeroIsOne a
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Just : a -> IfConsistent (C eqs) a
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export
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Functor (IfConsistent eqs) where
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map f Nothing = Nothing
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map f (Just x) = Just (f x)
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export
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Foldable (IfConsistent eqs) where
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foldr f z Nothing = z
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foldr f z (Just x) = f x z
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export
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Traversable (IfConsistent eqs) where
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traverse f Nothing = pure Nothing
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traverse f (Just x) = Just <$> f x
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||| performs an action if the dim ctx is consistent
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public export
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ifConsistent : Applicative f => (eqs : DimEq d) -> f a -> f (IfConsistent eqs a)
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ifConsistent ZeroIsOne act = pure Nothing
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ifConsistent (C _) act = Just <$> act
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public export %inline
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weakD : Dim d -> Dim (S d)
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weakD p = p // SS SZ
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public export
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tail' : DimEq' (S d) -> DimEq' d
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tail' (Const eqs e) = eqs
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tail' (Var eqs i un) = eqs
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tail' (None eqs ) = eqs
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public export
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tail : DimEq (S d) -> DimEq d
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tail ZeroIsOne = ZeroIsOne
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tail (C eqs) = C $ tail' eqs
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public export
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head' : DimEq' (S d) -> Maybe (Dim d)
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head' (Const _ e) = Just $ K e
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head' (Var _ i _) = Just $ B i
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head' (None _) = Nothing
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export
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tailU : Unassigned eqs (VS i) -> Unassigned (tail' eqs) i
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tailU (USK un) = un
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tailU (USV un) = un
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tailU (USN un) = un
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||| make a dim ctx where each variable has a constant assignment
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public export
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fromGround' : Context' DimConst d -> DimEq' d
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fromGround' [<] = Nil
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fromGround' (ctx :< e) = Const (fromGround' ctx) e
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||| make a dim ctx where each variable has a constant assignment
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public export
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fromGround : Context' DimConst d -> DimEq d
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fromGround = C . fromGround'
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||| make a dim ctx where each variable is unassigned
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public export
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new' : (d : Nat) -> DimEq' d
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new' 0 = Nil
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new' (S d) = None (new' d)
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||| make a dim ctx where each variable is unassigned
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public export
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new : (d : Nat) -> DimEq d
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new d = C $ new' d
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||| if the dim is a variable, then it is unassigned
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public export
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data UnassignedDim : DimEq' d -> Dim d -> Type where
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UDK : UnassignedDim eqs (K e)
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UDB : Unassigned eqs i -> UnassignedDim eqs (B i)
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export
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weakUD : {eqs : DimEq' (S d)} -> {p : Dim d} ->
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UnassignedDim (tail' eqs) p -> UnassignedDim eqs (weakD p)
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weakUD UDK = UDK
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weakUD (UDB un) {eqs = Const eqs e} = UDB $ USK un
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weakUD (UDB un) {eqs = Var eqs _ _} = UDB $ USV un
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weakUD (UDB un) {eqs = None eqs} = UDB $ USN un
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||| get the constraint on a variable 𝑖. if it is equal to another var 𝑗,
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||| then 𝑗 is not further constrained
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public export
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getVarPrf : (eqs : DimEq' d) -> Var d -> Subset (Dim d) (UnassignedDim eqs)
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getVarPrf (Const eqs e) VZ = Element (K e) UDK
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getVarPrf (Var eqs i un) VZ = Element (B $ VS i) (UDB $ USV un)
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getVarPrf (None eqs) VZ = Element (B VZ) (UDB UZ)
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getVarPrf (Const eqs _) (VS i) = let p = getVarPrf eqs i in
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Element (weakD p.fst) (weakUD p.snd)
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getVarPrf (Var eqs _ _) (VS i) = let p = getVarPrf eqs i in
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Element (weakD p.fst) (weakUD p.snd)
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getVarPrf (None eqs) (VS i) = let p = getVarPrf eqs i in
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Element (weakD p.fst) (weakUD p.snd)
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public export
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getVar : (eqs : DimEq' d) -> Var d -> Dim d
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getVar eqs i = fst $ getVarPrf eqs i
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public export
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getPrf : (eqs : DimEq' d) -> Dim d -> Subset (Dim d) (UnassignedDim eqs)
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getPrf eqs (K e) = Element (K e) UDK
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getPrf eqs (B i) = getVarPrf eqs i
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public export
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get : DimEq' d -> Dim d -> Dim d
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get eqs p = fst $ getPrf eqs p
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-- version of `get` that only shifts once but is even more annoying to prove
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-- anything about
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private
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getShift' : Shift d out -> DimEq' d -> Var d -> Maybe (Dim out)
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getShift' by (Const eqs e) VZ = Just $ K e
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getShift' by (Var eqs i un) VZ = Just $ B $ i // ssDown by
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getShift' by (None eqs) VZ = Nothing
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getShift' by eqs (VS i) = getShift' (ssDown by) (tail' eqs) i
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private
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getShift0' : DimEq' d -> Var d -> Maybe (Dim d)
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getShift0' = getShift' SZ
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private
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get' : DimEq' d -> Dim d -> Dim d
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get' eqs (K e) = K e
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get' eqs (B i) = fromMaybe (B i) $ getShift0' eqs i
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%transform "DimEq.get" get = get'
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public export
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Equal' : DimEq' d -> Rel (Dim d)
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Equal' eqs p q = get eqs p = get eqs q
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||| whether two dimensions are equal under the current constraints
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public export
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data Equal : DimEq d -> Rel (Dim d) where
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Eq01 : Equal ZeroIsOne p q
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EqC : Equal' eqs p q -> Equal (C eqs) p q
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%name DimEq.Equal prf
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export
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decEqual : (eqs : DimEq d) -> Dec2 (Equal eqs)
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decEqual ZeroIsOne _ _ = Yes Eq01
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decEqual (C eqs) p q = case get eqs p `decEq` get eqs q of
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Yes y => Yes $ EqC y
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No n => No $ \case EqC p => n p
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export
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equal : (eqs : DimEq d) -> Dim d -> Dim d -> Bool
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equal eqs p q = isYes $ decEqual eqs p q
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export
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{eqs : DimEq d} -> Reflexive _ (Equal eqs) where
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reflexive = case eqs of
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ZeroIsOne => Eq01
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C eqs => EqC Refl
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export
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Symmetric _ (Equal eqs) where
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symmetric Eq01 = Eq01
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symmetric (EqC eq) = EqC $ sym eq
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export
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Transitive _ (Equal eqs) where
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transitive Eq01 Eq01 = Eq01
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transitive (EqC p) (EqC q) = EqC $ p `trans` q
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export {eqs : DimEq d} -> Equivalence _ (Equal eqs) where
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||| extend the context with a new variable, possibly constrained
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public export
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(:<) : DimEq' d -> Maybe (Dim d) -> DimEq' (S d)
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eqs :< Nothing = None eqs
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eqs :< Just (K e) = Const eqs e
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eqs :< Just (B i) with (getVarPrf eqs i)
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_ | Element (K e) _ = Const eqs e
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_ | Element (B j) un = Var eqs j $ let UDB un = un in un
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infixl 7 :<?
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||| extend the context with a new variable, possibly constrained
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public export
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(:<?) : DimEq d -> Maybe (Dim d) -> DimEq (S d)
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ZeroIsOne :<? p = ZeroIsOne
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C eqs :<? p = C $ eqs :< p
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public export
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checkConst : DimConst -> DimConst -> DimEq' d -> DimEq d
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checkConst e f eqs = case decEq e f of Yes _ => C eqs; No _ => ZeroIsOne
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public export
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setConst : Var d -> DimConst -> DimEq' d -> DimEq d
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setConst VZ e (Const eqs f) = checkConst e f $ eqs :< Just (K e)
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setConst VZ e (Var eqs i un) = setConst i e eqs :<? Just (K e)
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setConst VZ e (None eqs) = C $ Const eqs e
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setConst (VS i) e (Const eqs f) = setConst i e eqs :<? Just (K f)
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setConst (VS i) e (Var eqs j un) = setConst i e eqs :<? Just (B j)
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setConst (VS i) e (None eqs) = setConst i e eqs :<? Nothing
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public export
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setVar : Var d -> Var d -> DimEq' d -> DimEq d
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setVar VZ VZ eqs = C eqs
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setVar VZ (VS j) (Const eqs e) = setConst j e eqs :<? Just (K e)
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setVar VZ (VS j) (Var eqs k un) = setVar j k eqs :<? Just (B k)
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setVar VZ (VS j) (None eqs) = C eqs :<? Just (B j)
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setVar (VS i) VZ (Const eqs e) = setConst i e eqs :<? Just (K e)
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setVar (VS i) VZ (Var eqs k un) = setVar i k eqs :<? Just (B k)
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setVar (VS i) VZ (None eqs) = C eqs :<? Just (B i)
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setVar (VS i) (VS j) (Const eqs e) = setVar i j eqs :<? Just (K e)
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setVar (VS i) (VS j) (Var eqs k un) = setVar i j eqs :<? Just (B k)
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setVar (VS i) (VS j) (None eqs) = setVar i j eqs :<? Nothing
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public export
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set : Dim d -> Dim d -> DimEq d -> DimEq d
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set p q ZeroIsOne = ZeroIsOne
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set (K e) (K f) (C eqs) = checkConst e f eqs
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set (K e) (B j) (C eqs) = setConst j e eqs
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set (B i) (K f) (C eqs) = setConst i f eqs
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set (B i) (B j) (C eqs) = setVar i j eqs
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private
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splitV0 : (p : Dim (S d)) -> Either (p = B VZ) (q : Dim d ** p = weakD q)
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splitV0 (K e) = Right (K e ** Refl)
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splitV0 (B VZ) = Left Refl
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splitV0 (B (VS i)) = Right (B i ** Refl)
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export
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0 getSnoc : (eqs : DimEq' d) -> (u : Maybe (Dim d)) -> (i : Var d) ->
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getVar (eqs :< u) (VS i) = weakD (getVar eqs i)
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getSnoc eqs Nothing i = Refl
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getSnoc eqs (Just (K e)) i = Refl
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getSnoc eqs (Just (B j)) i with (getVarPrf eqs j)
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_ | Element (K _) _ = Refl
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_ | Element (B _) _ = Refl
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export
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0 snocStrengthen : (p, q : Dim d) ->
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Equal' (eqs :< u) (weakD p) (weakD q) -> Equal' eqs p q
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snocStrengthen (K e) (K e) Refl = Refl
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snocStrengthen (K e) (B i) prf =
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shiftInj (SS SZ) _ _ $
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rewrite sym $ getSnoc eqs u i in prf
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snocStrengthen (B i) (K e) prf =
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shiftInj (SS SZ) _ _ $
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rewrite sym $ getSnoc eqs u i in prf
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snocStrengthen (B i) (B j) prf =
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shiftInj (SS SZ) _ _ $
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rewrite sym $ getSnoc eqs u i in
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rewrite sym $ getSnoc eqs u j in prf
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export
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0 snocStable : (eqs : DimEq d) -> (u : Maybe (Dim d)) -> (p, q : Dim d) ->
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Equal eqs p q -> Equal (eqs :<? u) (weakD p) (weakD q)
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snocStable ZeroIsOne u p q Eq01 = Eq01
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snocStable (C eqs) u (K e) (K e) (EqC Refl) = reflexive
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snocStable (C eqs) u (K e) (B i) (EqC prf) = EqC $
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rewrite getSnoc eqs u i in rewrite sym prf in Refl
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snocStable (C eqs) u (B i) (K e) (EqC prf) = EqC $
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rewrite getSnoc eqs u i in rewrite prf in Refl
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snocStable (C eqs) u (B i) (B j) (EqC prf) = EqC $
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rewrite getSnoc eqs u i in
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rewrite getSnoc eqs u j in
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rewrite prf in Refl
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export
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0 checkConstStable : (eqs : DimEq' d) -> (e, f : DimConst) ->
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(p, q : Dim d) -> Equal' eqs p q ->
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Equal (checkConst e f eqs) p q
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checkConstStable eqs e f p q prf with (decEq e f)
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_ | Yes _ = EqC prf
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_ | No _ = Eq01
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export
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0 setConstStable : (eqs : DimEq' d) -> (i : Var d) -> (e : DimConst) ->
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(p, q : Dim d) -> Equal' eqs p q ->
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Equal (setConst i e eqs) p q
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setConstStable (Const eqs f) VZ e p q prf with (decEq e f)
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_ | Yes _ = EqC prf
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_ | No _ = Eq01
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setConstStable (Const eqs f) (VS i) e p q prf = ?setConstStable_rhs_5
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setConstStable (Var eqs j un) VZ e p q prf = ?setConstStable_rhs_6
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setConstStable (Var eqs j un) (VS i) e p q prf = ?setConstStable_rhs_7
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setConstStable (None eqs) VZ e p q prf = ?setConstStable_rhs_8
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setConstStable (None eqs) (VS i) e p q prf = ?setConstStable_rhs_9
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export
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0 setVarStable : (eqs : DimEq' d) -> (i, j : Var d) ->
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(p, q : Dim d) -> Equal' eqs p q ->
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Equal (setVar i j eqs) p q
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export
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0 setStable : (eqs : DimEq d) -> (u, v, p, q : Dim d) ->
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Equal eqs p q -> Equal (set u v eqs) p q
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setStable ZeroIsOne p q u v Eq01 = Eq01
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setStable (C eqs) (K e) (K f) p q (EqC prf) = checkConstStable eqs e f p q prf
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setStable (C eqs) (K e) (B i) p q (EqC prf) = setConstStable eqs i e p q prf
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setStable (C eqs) (B i) (K e) p q (EqC prf) = setConstStable eqs i e p q prf
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setStable (C eqs) (B i) (B j) p q (EqC prf) = setVarStable eqs i j p q prf
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