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|
module Total.LogRel
import Syntax.PreorderReasoning
import Total.Reduction
import Total.Term
%prefix_record_projections off
public export
LogRel :
{ctx : SnocList Ty} ->
(P : {ctx, ty : _} -> Term ctx ty -> Type) ->
{ty : Ty} ->
(t : Term ctx ty) ->
Type
LogRel p {ty = N} t = p t
LogRel p {ty = ty ~> ty'} t =
(p t,
{ctx' : SnocList Ty} ->
(thin : ctx `Thins` ctx') ->
(u : Term ctx' ty) ->
LogRel p u ->
LogRel p (App (wkn t thin) u))
export
escape :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
{ty : Ty} ->
{0 t : Term ctx ty} ->
LogRel P t ->
P t
escape {ty = N} pt = pt
escape {ty = ty ~> ty'} (pt, app) = pt
public export
record PreserveHelper
(P : {ctx, ty : _} -> Term ctx ty -> Type)
(R : {ctx, ty : _} -> Term ctx ty -> Term ctx ty -> Type) where
constructor MkPresHelper
app :
{0 ctx : SnocList Ty} ->
{ty, ty' : Ty} ->
{0 t, u : Term ctx (ty ~> ty')} ->
{ctx' : SnocList Ty} ->
(thin : ctx `Thins` ctx') ->
(v : Term ctx' ty) ->
R t u ->
R (App (wkn t thin) v) (App (wkn u thin) v)
pres :
{0 ctx : SnocList Ty} ->
{ty : Ty} ->
{0 t, u : Term ctx ty} ->
P t ->
(0 _ : R t u) ->
P u
%name PreserveHelper help
export
backStepHelper :
{0 P : {ctx, ty : _} -> Term ctx ty -> Type} ->
(forall ctx, ty. {0 t, u : Term ctx ty} -> P t -> (0 _ : u > t) -> P u) ->
PreserveHelper P (flip (>))
backStepHelper pres =
MkPresHelper
(\thin, v, step => AppCong1 (wknStep step))
pres
export
preserve :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
{R : {ctx, ty : _} -> Term ctx ty -> Term ctx ty -> Type} ->
{ty : Ty} ->
{0 t, u : Term ctx ty} ->
PreserveHelper P R ->
LogRel P t ->
(0 _ : R t u) ->
LogRel P u
preserve help {ty = N} pt prf = help.pres pt prf
preserve help {ty = ty ~> ty'} (pt, app) prf =
(help.pres pt prf, \thin, v, rel => preserve help (app thin v rel) (help.app thin v prf))
data AllLogRel : (P : {ctx, ty : _} -> Term ctx ty -> Type) -> Terms ctx' ctx -> Type where
Base :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
{0 thin : ctx `Thins` ctx'} ->
AllLogRel P (Base thin)
(:<) :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
AllLogRel P sub ->
LogRel P t ->
AllLogRel P (sub :< t)
%name AllLogRel allRels
index :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
((i : Elem ty ctx') -> LogRel P (Var i)) ->
{sub : Terms ctx' ctx} ->
AllLogRel P sub ->
(i : Elem ty ctx) ->
LogRel P (index sub i)
index f Base i = f (index _ i)
index f (allRels :< rel) Here = rel
index f (allRels :< rel) (There i) = index f allRels i
Valid :
(P : {ctx, ty : _} -> Term ctx ty -> Type) ->
{ctx : SnocList Ty} ->
{ty : Ty} ->
(t : Term ctx ty) ->
Type
Valid p t =
{ctx' : SnocList Ty} ->
(sub : Terms ctx' ctx) ->
(allRel : AllLogRel p sub) ->
LogRel p (subst t sub)
public export
record ValidHelper (P : {ctx, ty : _} -> Term ctx ty -> Type) where
constructor MkValidHelper
var : forall ctx. {ty : Ty} -> (i : Elem ty ctx) -> LogRel P (Var i)
abs : forall ctx, ty. {ty' : Ty} -> {0 t : Term (ctx :< ty) ty'} -> LogRel P t -> P (Abs t)
zero : forall ctx. P {ctx} Zero
suc : forall ctx. {0 t : Term ctx N} -> P t -> P (Suc t)
rec :
{ctx : SnocList Ty} ->
{ty : Ty} ->
{0 t : Term ctx N} ->
{u : Term ctx ty} ->
{v : Term ctx (ty ~> ty)} ->
LogRel P t ->
LogRel P u ->
LogRel P v ->
LogRel P (Rec t u v)
step : forall ctx, ty. {0 t, u : Term ctx ty} -> P t -> (0 _ : u > t) -> P u
wkn : forall ctx, ctx', ty.
{0 t : Term ctx ty} ->
P t ->
(thin : ctx `Thins` ctx') ->
P (wkn t thin)
%name ValidHelper help
wknRel :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
ValidHelper P ->
{ty : Ty} ->
{0 t : Term ctx ty} ->
LogRel P t ->
(thin : ctx `Thins` ctx') ->
LogRel P (wkn t thin)
wknRel help {ty = N} pt thin = help.wkn pt thin
wknRel help {ty = ty ~> ty'} (pt, app) thin =
( help.wkn pt thin
, \thin', u, rel => rewrite wknHomo t thin thin' in app (thin' . thin) u rel
)
wknAllRel :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
ValidHelper P ->
{ctx : SnocList Ty} ->
{sub : Terms ctx' ctx} ->
AllLogRel P sub ->
(thin : ctx' `Thins` ctx'') ->
AllLogRel P (wknAll sub thin)
wknAllRel help Base thin = Base
wknAllRel help (allRels :< rel) thin = wknAllRel help allRels thin :< wknRel help rel thin
export
allValid :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
{ctx : SnocList Ty} ->
{ty : Ty} ->
ValidHelper P ->
(t : Term ctx ty) ->
Valid P t
allValid help (Var i) sub allRels = index help.var allRels i
allValid help (Abs t) sub allRels =
let valid = allValid help t in
(let
rec =
valid
(wknAll sub (Drop $ id @{termsLen sub}) :< Var Here)
(wknAllRel help allRels (Drop $ id @{termsLen sub}) :< help.var Here)
in
help.abs rec
, \thin, u, rel =>
let
eq :
(subst
(wkn (subst t (wknAll sub (Drop $ id @{termsLen sub}) :< Var Here)) (Keep thin))
(Base (id @{length _}) :< u) =
subst t (wknAll sub thin :< u))
eq =
rewrite lenUnique (termsLen sub) (length ctx') in
Calc $
|~ subst (wkn (subst t (wknAll sub (Drop id) :< Var Here)) (Keep thin)) (Base id :< u)
~~ subst (subst t (wknAll sub (Drop id) :< Var Here)) (restrict (Base id :< u) (Keep thin))
...(substWkn (subst t (wknAll sub (Drop id) :< Var Here)) (Keep thin) (Base id :< u))
~~ subst (subst t (wknAll sub (Drop id) :< Var Here)) (Base thin :< u)
...(cong (subst (subst t (wknAll sub (Drop id) :< Var Here)) . (:< u) . Base) $ identityLeft thin)
~~ subst t ((Base thin :< u) . wknAll sub (Drop id) :< u)
...(substHomo t (wknAll sub (Drop id) :< Var Here) (Base thin :< u))
~~ subst t (Base (thin . id) . sub :< u)
...(cong (subst t . (:< u)) $ compWknAll sub (Base thin :< u) (Drop id))
~~ subst t (Base thin . sub :< u)
...(cong (subst t . (:< u) . (. sub) . Base) $ identityRight thin)
~~ subst t (wknAll sub thin :< u)
...(cong (subst t . (:< u)) $ baseComp thin sub)
in
preserve
(backStepHelper help.step)
(valid (wknAll sub thin :< u) (wknAllRel help allRels thin :< rel))
(replace {p = (App (wkn (subst (Abs t) sub) thin) u >)}
eq
(AppBeta (length _)))
)
allValid help (App t u) sub allRels =
let (pt, app) = allValid help t sub allRels in
let rel = allValid help u sub allRels in
rewrite sym $ wknId @{termsLen sub} (subst t sub) in
app (id @{termsLen sub}) (subst u sub) rel
allValid help Zero sub allRels = help.zero
allValid help (Suc t) sub allRels =
let pt = allValid help t sub allRels in
help.suc pt
allValid help (Rec t u v) sub allRels =
let relt = allValid help t sub allRels in
let relu = allValid help u sub allRels in
let relv = allValid help v sub allRels in
help.rec relt relu relv
export
allRel :
{P : {ctx, ty : _} -> Term ctx ty -> Type} ->
{ctx : SnocList Ty} ->
{ty : Ty} ->
ValidHelper P ->
(t : Term ctx ty) ->
P t
allRel help t =
rewrite sym $ substId @{length ctx} t in escape (allValid help t (Base $ id @{length ctx}) Base)
|
