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module Core.Declarative
import Core.Environment
import Core.Term
import Core.Term.Substitution
import Core.Term.Thinned
import Core.Thinning
-- Definition ------------------------------------------------------------------
data EnvWf : Env n -> Type
data TypeWf : Env n -> Term n -> Type
data TypeConv : Env n -> Term n -> Term n -> Type
data TermWf : Env n -> Term n -> Term n -> Type
data TermConv : Env n -> Term n -> Term n -> Term n -> Type
data EnvWf where
Lin : EnvWf [<]
(:<) : EnvWf env -> TypeWf env (expand a) -> EnvWf (env :< a)
data TypeWf where
SetTyWf :
EnvWf env ->
---
TypeWf env Set
PiTyWf :
TypeWf env a ->
TypeWf (env :< pure a) b ->
---
TypeWf env (Pi a b)
LiftWf :
TermWf env a Set ->
---
TypeWf env a
data TypeConv where
ReflTy :
TypeWf env a ->
---
TypeConv env a a
SymTy :
TypeConv env a b ->
---
TypeConv env b a
TransTy :
TypeConv env a b ->
TypeConv env b c ->
---
TypeConv env a c
PiConv :
TypeWf env a ->
TypeConv env a c ->
TypeConv (env :< pure a) b d ->
---
TypeConv env (Pi a b) (Pi c d)
LiftConv :
TermConv env a b Set ->
---
TypeConv env a b
data TermWf where
PiTmWf :
TermWf env a Set ->
TermWf (env :< pure a) b Set ->
---
TermWf env (Pi a b) Set
VarWf :
EnvWf env ->
---
TermWf env (Var i) (expand $ index env i)
AbsWf :
TypeWf env a ->
TermWf (env :< pure a) t b ->
---
TermWf env (Abs t) (Pi a b)
AppWf :
TermWf env t (Pi a b) ->
TermWf env u a ->
---
TermWf env (App t u) (subst b $ Wkn (id _) :< pure u)
ConvWf :
TermWf env t a ->
TypeConv env a b ->
---
TermWf env t b
data TermConv where
ReflTm :
TermWf env t a ->
---
TermConv env t t a
SymTm :
TermConv env t u a ->
---
TermConv env u t a
TransTm :
TermConv env t u a ->
TermConv env u v a ->
---
TermConv env t v a
AppConv :
TermConv env f g (Pi a b) ->
TermConv env t u a ->
---
TermConv env (App f t) (App g u) (subst b $ Wkn (id _) :< pure t)
PiTmConv :
TypeWf env a ->
TermConv env a c Set ->
TermConv (env :< pure a) b d Set ->
---
TermConv env (Pi a b) (Pi c d) Set
PiEta :
TypeWf env a ->
TermWf env t (Pi a b) ->
TermWf env u (Pi a b) ->
TermConv (env :< pure a)
(App (wkn t $ drop $ id _) (Var FZ))
(App (wkn u $ drop $ id _) (Var FZ))
b ->
---
TermConv env t u (Pi a b)
PiBeta :
TypeWf env a ->
TermWf (env :< pure a) t b ->
TermWf env u a ->
---
TermConv env
(App (Abs t) u)
(subst t $ Wkn (id _) :< pure u)
(subst g $ Wkn (id _) :< pure u)
ConvConv :
TermConv env t u a ->
TypeConv env a b ->
---
TermConv env t u b
%name EnvWf envWf
%name TypeWf tyWf
%name TypeConv tyConv
%name TermWf tmWf
%name TermConv tmConv
-- Respects Environment Quotient -----------------------------------------------
export
envWfRespEq : EnvWf env1 -> env1 =~= env2 -> EnvWf env2
export
typeWfRespEq : TypeWf env1 a -> env1 =~= env2 -> TypeWf env2 a
export
typeConvRespEq : TypeConv env1 a b -> env1 =~= env2 -> TypeConv env2 a b
export
termWfRespEq : TermWf env1 t a -> env1 =~= env2 -> TermWf env2 t a
export
termConvRespEq : TermConv env1 t u a -> env1 =~= env2 -> TermConv env2 t u a
envWfRespEq envWf Refl = envWf
envWfRespEq (envWf :< tyWf) (prf :< prf') =
envWfRespEq envWf prf :<
rewrite sym prf' in typeWfRespEq tyWf prf
typeWfRespEq (SetTyWf envWf) prf = SetTyWf (envWfRespEq envWf prf)
typeWfRespEq (PiTyWf tyWf tyWf1) prf =
PiTyWf
(typeWfRespEq tyWf prf)
(typeWfRespEq tyWf1 $ prf :< Refl)
typeWfRespEq (LiftWf tmWf) prf = LiftWf (termWfRespEq tmWf prf)
typeConvRespEq (ReflTy tyWf) prf = ReflTy (typeWfRespEq tyWf prf)
typeConvRespEq (SymTy tyConv) prf = SymTy (typeConvRespEq tyConv prf)
typeConvRespEq (TransTy tyConv tyConv1) prf =
TransTy
(typeConvRespEq tyConv prf)
(typeConvRespEq tyConv1 prf)
typeConvRespEq (PiConv tyWf tyConv tyConv1) prf =
PiConv
(typeWfRespEq tyWf prf)
(typeConvRespEq tyConv prf)
(typeConvRespEq tyConv1 $ prf :< Refl)
typeConvRespEq (LiftConv tmConv) prf = LiftConv (termConvRespEq tmConv prf)
termWfRespEq (PiTmWf tmWf tmWf1) prf =
PiTmWf
(termWfRespEq tmWf prf)
(termWfRespEq tmWf1 $ prf :< Refl)
termWfRespEq (VarWf {i} envWf) prf =
rewrite indexCong prf i in
VarWf (envWfRespEq envWf prf)
termWfRespEq (AbsWf tyWf tmWf) prf =
AbsWf
(typeWfRespEq tyWf prf)
(termWfRespEq tmWf $ prf :< Refl)
termWfRespEq (AppWf tmWf tmWf1) prf =
AppWf
(termWfRespEq tmWf prf)
(termWfRespEq tmWf1 prf)
termWfRespEq (ConvWf tmWf tyConv) prf =
ConvWf
(termWfRespEq tmWf prf)
(typeConvRespEq tyConv prf)
termConvRespEq (ReflTm tmWf) prf = ReflTm (termWfRespEq tmWf prf)
termConvRespEq (SymTm tmConv) prf = SymTm (termConvRespEq tmConv prf)
termConvRespEq (TransTm tmConv tmConv1) prf =
TransTm
(termConvRespEq tmConv prf)
(termConvRespEq tmConv1 prf)
termConvRespEq (AppConv tmConv tmConv1) prf =
AppConv
(termConvRespEq tmConv prf)
(termConvRespEq tmConv1 prf)
termConvRespEq (PiTmConv tyWf tmConv tmConv1) prf =
PiTmConv
(typeWfRespEq tyWf prf)
(termConvRespEq tmConv prf)
(termConvRespEq tmConv1 $ prf :< Refl)
termConvRespEq (PiEta tyWf tmWf tmWf1 tmConv) prf =
PiEta
(typeWfRespEq tyWf prf)
(termWfRespEq tmWf prf)
(termWfRespEq tmWf1 prf)
(termConvRespEq tmConv $ prf :< Refl)
termConvRespEq (PiBeta tyWf tmWf tmWf1) prf =
PiBeta
(typeWfRespEq tyWf prf)
(termWfRespEq tmWf $ prf :< Refl)
(termWfRespEq tmWf1 prf)
termConvRespEq (ConvConv tmConv tyConv) prf =
ConvConv
(termConvRespEq tmConv prf)
(typeConvRespEq tyConv prf)
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