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module Core.Declarative
import Core.Context
import Core.Environment
import Core.Name
import Core.Term
import Core.Term.Substitution
import Core.Thinning
import Core.Var
import Data.Nat
import Syntax.PreorderReasoning
public export
data EnvWf : Env sx -> Type
public export
data TypeWf : Env sx -> Term sx -> Type
public export
data TypeConv : Env sx -> Term sx -> Term sx -> Type
public export
data TermWf : Env sx -> Term sx -> Term sx -> Type
public export
data TermConv : Env sx -> Term sx -> Term sx -> Term sx -> Type
data EnvWf where
Lin :
---
EnvWf [<]
(:<) :
EnvWf env ->
TypeWf env (expand t) ->
---
EnvWf (Add env t)
data TypeWf where
SetType :
EnvWf env ->
---
TypeWf env Set
PiType :
TypeWf env f ->
TypeWf (env :< f) g ->
---
TypeWf env (Pi n f g)
LiftType :
TermWf env a Set ->
---
TypeWf env a
data TypeConv where
ReflType :
TypeWf env a ->
---
TypeConv env a a
SymType :
TypeConv env a b ->
---
TypeConv env b a
TransType :
TypeConv env a b ->
TypeConv env b c ->
---
TypeConv env a c
PiTypeConv :
TypeWf env f ->
TypeConv env f h ->
TypeConv (env :< f) g e ->
---
TypeConv env (Pi n f g) (Pi n h e)
LiftConv :
TermConv env a b Set ->
---
TypeConv env a b
data TermWf where
VarTerm :
EnvWf env ->
---
TermWf env (Var i) (index env i)
PiTerm :
TermWf env f Set ->
TermWf (env :< f) g Set ->
---
TermWf env (Pi n f g) Set
AbsTerm :
TypeWf env f ->
TermWf (env :< f) t g ->
---
TermWf env (Abs n t) (Pi n f g)
AppTerm :
TermWf env t (Pi n f g) ->
TermWf env u f ->
---
TermWf env (App t u) (subst g $ sub1 u)
ConvTerm :
TermWf env t a ->
TypeConv env a b ->
---
TermWf env t b
data TermConv where
ReflTerm :
TermWf env t a ->
---
TermConv env t t a
SymTerm :
TermConv env t u a ->
---
TermConv env u t a
TransTerm :
TermConv env t u a ->
TermConv env u v a ->
---
TermConv env t v a
ConvTermConv :
TermConv env t u a ->
TypeConv env a b ->
---
TermConv env t u b
PiTermConv :
TypeWf env f ->
TermConv env f h Set ->
TermConv (env :< f) g e Set ->
---
TermConv env (Pi n f g) (Pi n h e) Set
PiBeta :
TypeWf env f ->
TermWf (env :< f) t g ->
TermWf env u f ->
---
TermConv env (App (Abs n t) u) (subst t $ sub1 u) (subst g $ sub1 u)
PiEta :
TypeWf env f ->
TermWf env t (Pi n f g) ->
TermWf env u (Pi n f g) ->
TermConv (env :< f)
(App (wkn t $ drop (id _) n) (Var Var.here))
(App (wkn u $ drop (id _) n) (Var Var.here))
g ->
---
TermConv env t u (Pi n f g)
AppConv :
TermConv env t u (Pi n f g) ->
TermConv env a b f ->
---
TermConv env (App t a) (App u b) (subst g $ sub1 a)
%name EnvWf envWf
%name TypeWf wf
%name TypeConv conv
%name TermWf wf
%name TermConv conv
-- Environment Equality --------------------------------------------------------
envWfRespEnvEq : EnvWf env1 -> EnvEq env1 env2 -> EnvWf env2
typeWfRespEnvEq : TypeWf env1 a -> EnvEq env1 env2 -> TypeWf env2 a
typeConvRespEnvEq : TypeConv env1 a b -> EnvEq env1 env2 -> TypeConv env2 a b
export
termWfRespEnvEq : TermWf env1 t a -> EnvEq env1 env2 -> TermWf env2 t a
termConvRespEnvEq : TermConv env1 t u b -> EnvEq env1 env2 -> TermConv env2 t u b
envWfRespEnvEq envWf Base = envWf
envWfRespEnvEq (envWf :< wf) (prf :< eq) =
envWfRespEnvEq envWf prf :<
rewrite sym eq in typeWfRespEnvEq wf prf
typeWfRespEnvEq (SetType envWf) prf = SetType (envWfRespEnvEq envWf prf)
typeWfRespEnvEq (PiType wf wf1) prf =
PiType
(typeWfRespEnvEq wf prf)
(typeWfRespEnvEq wf1 $ prf :< Refl)
typeWfRespEnvEq (LiftType wf) prf = LiftType (termWfRespEnvEq wf prf)
typeConvRespEnvEq (ReflType wf) prf = ReflType (typeWfRespEnvEq wf prf)
typeConvRespEnvEq (SymType conv) prf = SymType (typeConvRespEnvEq conv prf)
typeConvRespEnvEq (TransType conv conv1) prf =
TransType
(typeConvRespEnvEq conv prf)
(typeConvRespEnvEq conv1 prf)
typeConvRespEnvEq (PiTypeConv wf conv conv1) prf =
PiTypeConv
(typeWfRespEnvEq wf prf)
(typeConvRespEnvEq conv prf)
(typeConvRespEnvEq conv1 $ prf :< Refl)
typeConvRespEnvEq (LiftConv conv) prf = LiftConv (termConvRespEnvEq conv prf)
termWfRespEnvEq (VarTerm {i} envWf) prf =
rewrite indexEqIsEq prf i in
VarTerm (envWfRespEnvEq envWf prf)
termWfRespEnvEq (PiTerm wf wf1) prf =
PiTerm
(termWfRespEnvEq wf prf)
(termWfRespEnvEq wf1 $ prf :< Refl)
termWfRespEnvEq (AbsTerm wf wf1) prf =
AbsTerm
(typeWfRespEnvEq wf prf)
(termWfRespEnvEq wf1 $ prf :< Refl)
termWfRespEnvEq (AppTerm wf wf1) prf = AppTerm (termWfRespEnvEq wf prf) (termWfRespEnvEq wf1 prf)
termWfRespEnvEq (ConvTerm wf conv) prf =
ConvTerm
(termWfRespEnvEq wf prf)
(typeConvRespEnvEq conv prf)
termConvRespEnvEq (ReflTerm wf) prf = ReflTerm (termWfRespEnvEq wf prf)
termConvRespEnvEq (SymTerm conv) prf = SymTerm (termConvRespEnvEq conv prf)
termConvRespEnvEq (TransTerm conv conv1) prf =
TransTerm
(termConvRespEnvEq conv prf)
(termConvRespEnvEq conv1 prf)
termConvRespEnvEq (ConvTermConv conv conv1) prf =
ConvTermConv (termConvRespEnvEq conv prf) (typeConvRespEnvEq conv1 prf)
termConvRespEnvEq (PiTermConv wf conv conv1) prf =
PiTermConv
(typeWfRespEnvEq wf prf)
(termConvRespEnvEq conv prf)
(termConvRespEnvEq conv1 $ prf :< Refl)
termConvRespEnvEq (PiBeta wf wf1 wf2) prf =
PiBeta
(typeWfRespEnvEq wf prf)
(termWfRespEnvEq wf1 $ prf :< Refl)
(termWfRespEnvEq wf2 prf)
termConvRespEnvEq (PiEta wf wf1 wf2 conv) prf =
PiEta
(typeWfRespEnvEq wf prf)
(termWfRespEnvEq wf1 prf)
(termWfRespEnvEq wf2 prf)
(termConvRespEnvEq conv $ prf :< Refl)
termConvRespEnvEq (AppConv conv conv1) prf =
AppConv
(termConvRespEnvEq conv prf)
(termConvRespEnvEq conv1 prf)
-- Well Formed Environment -----------------------------------------------------
typeWfImpliesEnvWf : TypeWf env a -> EnvWf env
typeConvImpliesEnvWf : TypeConv env a b -> EnvWf env
termWfImpliesEnvWf : TermWf env t a -> EnvWf env
termConvImpliesEnvWf : TermConv env t u a -> EnvWf env
typeWfImpliesEnvWf (SetType envWf) = envWf
typeWfImpliesEnvWf (PiType wf wf1) = typeWfImpliesEnvWf wf
typeWfImpliesEnvWf (LiftType wf) = termWfImpliesEnvWf wf
typeConvImpliesEnvWf (ReflType wf) = typeWfImpliesEnvWf wf
typeConvImpliesEnvWf (SymType conv) = typeConvImpliesEnvWf conv
typeConvImpliesEnvWf (TransType conv conv1) = typeConvImpliesEnvWf conv
typeConvImpliesEnvWf (PiTypeConv wf conv conv1) = typeConvImpliesEnvWf conv
typeConvImpliesEnvWf (LiftConv conv) = termConvImpliesEnvWf conv
termWfImpliesEnvWf (VarTerm envWf) = envWf
termWfImpliesEnvWf (PiTerm wf wf1) = termWfImpliesEnvWf wf
termWfImpliesEnvWf (AbsTerm wf wf1) = typeWfImpliesEnvWf wf
termWfImpliesEnvWf (AppTerm wf wf1) = termWfImpliesEnvWf wf
termWfImpliesEnvWf (ConvTerm wf conv) = termWfImpliesEnvWf wf
termConvImpliesEnvWf (ReflTerm wf) = termWfImpliesEnvWf wf
termConvImpliesEnvWf (SymTerm conv) = termConvImpliesEnvWf conv
termConvImpliesEnvWf (TransTerm conv conv1) = termConvImpliesEnvWf conv
termConvImpliesEnvWf (ConvTermConv conv conv1) = termConvImpliesEnvWf conv
termConvImpliesEnvWf (PiTermConv wf conv conv1) = termConvImpliesEnvWf conv
termConvImpliesEnvWf (PiBeta wf wf1 wf2) = typeWfImpliesEnvWf wf
termConvImpliesEnvWf (PiEta wf wf1 wf2 conv) = typeWfImpliesEnvWf wf
termConvImpliesEnvWf (AppConv conv conv1) = termConvImpliesEnvWf conv
-- Weakening Preservation ------------------------------------------------------
export
wknPresTypeWf :
{0 env1 : Env sx} ->
TypeWf env1 a ->
EnvWf env2 ->
IsExtension thin env2 env1 ->
TypeWf env2 (wkn a thin)
export
wknPresTypeConv :
{0 env1 : Env sx} ->
TypeConv env1 a b ->
EnvWf env2 ->
IsExtension thin env2 env1 ->
TypeConv env2 (wkn a thin) (wkn b thin)
export
wknPresTermWf :
{0 env1 : Env sx} ->
TermWf env1 t a ->
EnvWf env2 ->
IsExtension thin env2 env1 ->
TermWf env2 (wkn t thin) (wkn a thin)
export
wknPresTermConv :
{0 env1 : Env sx} ->
TermConv env1 t u a ->
EnvWf env2 ->
IsExtension thin env2 env1 ->
TermConv env2 (wkn t thin) (wkn u thin) (wkn a thin)
wknPresTypeWf (SetType envWf1) envWf ext = SetType envWf
wknPresTypeWf (PiType {f, n} wf wf1) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
PiType wf' $
typeWfRespEnvEq
(wknPresTypeWf wf1 (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin))
wknPresTypeWf (LiftType wf) envWf ext = LiftType (wknPresTermWf wf envWf ext)
wknPresTypeConv (ReflType wf) envWf ext = ReflType (wknPresTypeWf wf envWf ext)
wknPresTypeConv (SymType conv) envWf ext = SymType (wknPresTypeConv conv envWf ext)
wknPresTypeConv (TransType conv conv1) envWf ext =
TransType
(wknPresTypeConv conv envWf ext)
(wknPresTypeConv conv1 envWf ext)
wknPresTypeConv (PiTypeConv {f, n} wf conv conv1) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
PiTypeConv wf' (wknPresTypeConv conv envWf ext) $
typeConvRespEnvEq
(wknPresTypeConv conv1 (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin))
wknPresTypeConv (LiftConv conv) envWf ext = LiftConv (wknPresTermConv conv envWf ext)
wknPresTermWf (VarTerm {i} envWf1) envWf ext =
rewrite wknIndex ext i in
VarTerm envWf
wknPresTermWf (PiTerm {f, n} wf wf1) envWf ext =
let wf' = wknPresTermWf wf envWf ext in
PiTerm wf' $
termWfRespEnvEq
(wknPresTermWf wf1 (envWf :< LiftType wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin))
wknPresTermWf (AbsTerm {f, n} wf wf1) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
AbsTerm wf' $
termWfRespEnvEq
(wknPresTermWf wf1 (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin))
wknPresTermWf (AppTerm {g, u, n} wf wf1) envWf ext =
rewrite wknSub1 g u thin in
AppTerm (wknPresTermWf wf envWf ext) (wknPresTermWf wf1 envWf ext)
wknPresTermWf (ConvTerm wf conv) envWf ext =
ConvTerm (wknPresTermWf wf envWf ext) (wknPresTypeConv conv envWf ext)
wknPresTermConv (ReflTerm wf) envWf ext = ReflTerm (wknPresTermWf wf envWf ext)
wknPresTermConv (SymTerm conv) envWf ext = SymTerm (wknPresTermConv conv envWf ext)
wknPresTermConv (TransTerm conv conv1) envWf ext =
TransTerm
(wknPresTermConv conv envWf ext)
(wknPresTermConv conv1 envWf ext)
wknPresTermConv (ConvTermConv conv conv1) envWf ext =
ConvTermConv
(wknPresTermConv conv envWf ext)
(wknPresTypeConv conv1 envWf ext)
wknPresTermConv (PiTermConv {f, n} wf conv conv1) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
PiTermConv wf' (wknPresTermConv conv envWf ext) $
termConvRespEnvEq
(wknPresTermConv conv1 (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin))
wknPresTermConv (PiBeta {f, g, t, u, n} wf wf1 wf2) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
rewrite wknSub1 g u thin in
rewrite wknSub1 t u thin in
PiBeta wf'
(termWfRespEnvEq
(wknPresTermWf wf1 (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin)))
(wknPresTermWf wf2 envWf ext)
wknPresTermConv (PiEta {f, n, t, u} wf wf1 wf2 conv) envWf ext =
let wf' = wknPresTypeWf wf envWf ext in
PiEta wf'
(wknPresTermWf wf1 envWf ext)
(wknPresTermWf wf2 envWf ext)
(termConvRespEnvEq
(rewrite sym $ wknEta t thin n in
rewrite sym $ wknEta u thin n in
wknPresTermConv conv (envWf :< wf') $
replace {p = \thin' => IsExtension (keep thin n) (Add env2 (f `Over` thin')) (env1 :< f)}
(compRightIdentity thin)
(KeepWf ext))
((:<) {t = f `Over` thin} Base (sym $ wknId $ expand $ f `Over` thin)))
wknPresTermConv (AppConv {g, n, a} conv conv1) envWf ext =
rewrite wknSub1 g a thin in
AppConv
(wknPresTermConv conv envWf ext)
(wknPresTermConv conv1 envWf ext)
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