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-rw-r--r--src/Total/LogRel.idr258
1 files changed, 135 insertions, 123 deletions
diff --git a/src/Total/LogRel.idr b/src/Total/LogRel.idr
index b088e6f..0b73706 100644
--- a/src/Total/LogRel.idr
+++ b/src/Total/LogRel.idr
@@ -11,24 +11,24 @@ import Total.Term.CoDebruijn
public export
LogRel :
{ctx : SnocList Ty} ->
- (P : {ctx, ty : _} -> FullTerm ty ctx -> Type) ->
+ (P : {ctx, ty : _} -> CoTerm ty ctx -> Type) ->
{ty : Ty} ->
- (t : FullTerm ty ctx) ->
+ (t : CoTerm ty ctx) ->
Type
LogRel p {ty = N} t = p t
LogRel p {ty = ty ~> ty'} t =
(p t,
{ctx' : SnocList Ty} ->
(thin : ctx `Thins` ctx') ->
- (u : FullTerm ty ctx') ->
+ (u : CoTerm ty ctx') ->
LogRel p u ->
LogRel p (app (wkn t thin) u))
export
escape :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
{ty : Ty} ->
- {0 t : FullTerm ty ctx} ->
+ {0 t : CoTerm ty ctx} ->
LogRel P t ->
P t
escape {ty = N} pt = pt
@@ -36,22 +36,22 @@ escape {ty = ty ~> ty'} (pt, app) = pt
public export
record PreserveHelper
- (P : {ctx, ty : _} -> FullTerm ty ctx -> Type)
- (R : {ctx, ty : _} -> FullTerm ty ctx -> FullTerm ty ctx -> Type) where
+ (P : {ctx, ty : _} -> CoTerm ty ctx -> Type)
+ (R : {ctx, ty : _} -> CoTerm ty ctx -> CoTerm ty ctx -> Type) where
constructor MkPresHelper
0 app :
{ctx : SnocList Ty} ->
{ty, ty' : Ty} ->
- (t, u : FullTerm (ty ~> ty') ctx) ->
+ (t, u : CoTerm (ty ~> ty') ctx) ->
{ctx' : SnocList Ty} ->
(thin : ctx `Thins` ctx') ->
- (v : FullTerm ty ctx') ->
+ (v : CoTerm ty ctx') ->
R t u ->
R (app (wkn t thin) v) (app (wkn u thin) v)
pres :
{0 ctx : SnocList Ty} ->
{ty : Ty} ->
- {0 t, u : FullTerm ty ctx} ->
+ {0 t, u : CoTerm ty ctx} ->
P t ->
(0 _ : R t u) ->
P u
@@ -60,25 +60,26 @@ record PreserveHelper
export
backStepHelper :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- (forall ctx, ty. {0 t, u : FullTerm ty ctx} -> P t -> (0 _ : toTerm u > toTerm t) -> P u) ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+ (forall ctx, ty. {0 t, u : CoTerm ty ctx} -> P t -> (0 _ : toTerm u > toTerm t) -> P u) ->
PreserveHelper P (flip (>) `on` CoDebruijn.toTerm)
backStepHelper pres =
MkPresHelper
- (\t, u, thin, v, step =>
- rewrite toTermApp (wkn u thin) v in
+ (\t, u, thin, v, step => ?appStep
+ {- rewrite toTermApp (wkn u thin) v in
rewrite toTermApp (wkn t thin) v in
rewrite sym $ wknToTerm t thin in
rewrite sym $ wknToTerm u thin in
- AppCong1 (wknStep step))
+ AppCong1 (wknStep step)-}
+ )
pres
export
preserve :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- {0 R : {ctx, ty : _} -> FullTerm ty ctx -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+ {0 R : {ctx, ty : _} -> CoTerm ty ctx -> CoTerm ty ctx -> Type} ->
{ty : Ty} ->
- {0 t, u : FullTerm ty ctx} ->
+ {0 t, u : CoTerm ty ctx} ->
PreserveHelper P R ->
LogRel P t ->
(0 _ : R t u) ->
@@ -87,12 +88,12 @@ 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 : _} -> FullTerm ty ctx -> Type) -> CoTerms ctx ctx' -> Type where
- Lin :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- AllLogRel P [<]
+data AllLogRel : (P : {ctx, ty : _} -> CoTerm ty ctx -> Type) -> Subst CoTerm ctx ctx' -> Type where
+ Base :
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+ AllLogRel P (Base thin)
(:<) :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
AllLogRel P sub ->
LogRel P t ->
AllLogRel P (sub :< t)
@@ -100,57 +101,55 @@ data AllLogRel : (P : {ctx, ty : _} -> FullTerm ty ctx -> Type) -> CoTerms ctx c
%name AllLogRel allRels
index :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
((i : Elem ty ctx') -> LogRel P (var i)) ->
- {sub : CoTerms ctx ctx'} ->
+ {sub : Subst CoTerm 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
restrict :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- {0 sub : CoTerms ctx' ctx''} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+ {0 sub : Subst CoTerm ctx' ctx''} ->
AllLogRel P sub ->
(thin : ctx `Thins` ctx') ->
AllLogRel P (restrict sub thin)
-restrict [<] Empty = [<]
-restrict (allRels :< rel) (Drop thin) = restrict allRels thin
-restrict (allRels :< rel) (Keep thin) = restrict allRels thin :< rel
Valid :
- (P : {ctx, ty : _} -> FullTerm ty ctx -> Type) ->
+ (P : {ctx, ty : _} -> CoTerm ty ctx -> Type) ->
{ctx : SnocList Ty} ->
{ty : Ty} ->
- (t : FullTerm ty ctx) ->
+ (t : CoTerm ty ctx) ->
Type
Valid p t =
{ctx' : SnocList Ty} ->
- (sub : CoTerms ctx ctx') ->
+ (sub : Subst CoTerm ctx ctx') ->
(allRel : AllLogRel p sub) ->
LogRel p (subst t sub)
public export
-record ValidHelper (P : {ctx, ty : _} -> FullTerm ty ctx -> Type) where
+record ValidHelper (P : {ctx, ty : _} -> CoTerm ty ctx -> Type) where
constructor MkValidHelper
- var : forall ctx. Len ctx => {ty : Ty} -> (i : Elem ty ctx) -> LogRel P (var i)
- abs : forall ctx, ty. {ty' : Ty} -> {0 t : FullTerm ty' (ctx :< ty)} -> LogRel P t -> P (abs t)
- zero : forall ctx. Len ctx => P {ctx} zero
- suc : forall ctx. {0 t : FullTerm N ctx} -> P t -> P (suc t)
+ var : forall ctx. {ty : Ty} -> (i : Elem ty ctx) -> LogRel P (var i)
+ abs : forall ctx, ty. {ty' : Ty} -> {0 t : CoTerm ty' (ctx :< ty)} -> LogRel P t -> P (abs t)
+ zero : forall ctx. P {ctx} zero
+ suc : forall ctx. {0 t : CoTerm N ctx} -> P t -> P (suc t)
rec :
{ctx : SnocList Ty} ->
{ty : Ty} ->
- {0 t : FullTerm N ctx} ->
- {u : FullTerm ty ctx} ->
- {v : FullTerm (ty ~> ty) ctx} ->
+ {0 t : CoTerm N ctx} ->
+ {u : CoTerm ty ctx} ->
+ {v : CoTerm (ty ~> ty) ctx} ->
LogRel P t ->
LogRel P u ->
LogRel P v ->
LogRel P (rec t u v)
- step : forall ctx, ty. {0 t, u : FullTerm ty ctx} -> P t -> (0 _ : toTerm u > toTerm t) -> P u
+ step : forall ctx, ty. {0 t, u : CoTerm ty ctx} -> P t -> (0 _ : toTerm u > toTerm t) -> P u
wkn : forall ctx, ctx', ty.
- {0 t : FullTerm ty ctx} ->
+ {0 t : CoTerm ty ctx} ->
P t ->
(thin : ctx `Thins` ctx') ->
P (wkn t thin)
@@ -158,101 +157,109 @@ record ValidHelper (P : {ctx, ty : _} -> FullTerm ty ctx -> Type) where
%name ValidHelper help
wknRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
{ty : Ty} ->
- {0 t : FullTerm ty ctx} ->
+ {0 t : CoTerm ty ctx} ->
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
+ , \thin', u, rel => ?appURel
+ -- rewrite wknHomo t thin' thin in
+ -- app (thin' . thin) u rel
)
wknAllRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
{ctx : SnocList Ty} ->
- {sub : CoTerms ctx ctx'} ->
+ {sub : Subst CoTerm ctx ctx'} ->
AllLogRel P sub ->
(thin : ctx' `Thins` ctx'') ->
- AllLogRel P (wknAll sub thin)
-wknAllRel help [<] thin = [<]
+ AllLogRel P (wkn sub thin)
+wknAllRel help Base thin = Base
wknAllRel help (allRels :< rel) thin = wknAllRel help allRels thin :< wknRel help rel thin
shiftRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
{ctx, ctx' : SnocList Ty} ->
- {sub : CoTerms ctx ctx'} ->
+ {sub : Subst CoTerm ctx ctx'} ->
AllLogRel P sub ->
- AllLogRel P (shift {ty} sub)
-shiftRel help [<] = [<]
+ AllLogRel P (shift sub)
+shiftRel help Base = Base
shiftRel help (allRels :< rel) =
shiftRel help allRels :<
- replace {p = LogRel P} (sym $ dropIsWkn _) (wknRel help rel (Drop id))
+ ?shift1
+ -- replace {p = LogRel P} (sym $ dropIsWkn _) (wknRel help rel (Drop id))
liftRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
{ctx, ctx' : SnocList Ty} ->
{ty : Ty} ->
- {sub : CoTerms ctx ctx'} ->
+ {sub : Subst CoTerm ctx ctx'} ->
AllLogRel P sub ->
- AllLogRel P (lift {ty} sub)
+ AllLogRel P (lift {z = ty} sub)
liftRel help allRels = shiftRel help allRels :< help.var Here
-absValid :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- ValidHelper P ->
- {ctx : SnocList Ty} ->
- {ty, ty' : Ty} ->
- (t : CoTerm ty' (ctx ++ used)) ->
- (thin : used `Thins` [<ty]) ->
- (valid : {ctx' : SnocList Ty} ->
- (sub : CoTerms (ctx ++ used) ctx') ->
- AllLogRel P sub ->
- LogRel P (subst' t sub)) ->
- {ctx' : SnocList Ty} ->
- (sub : CoTerms ctx ctx') ->
- AllLogRel P sub ->
- LogRel P (subst' (Abs (MakeBound t thin)) sub)
-absValid help t (Drop Empty) valid sub allRels =
- ( help.abs (valid (shift sub) (shiftRel help allRels))
- , \thin, u, rel =>
- preserve
- (backStepHelper help.step)
- (valid (wknAll sub thin) (wknAllRel help allRels thin))
- ?betaStep
- )
-absValid help t (Keep Empty) valid sub allRels =
- ( help.abs (valid (lift sub) (liftRel help allRels))
- , \thin, u, rel =>
- preserve (backStepHelper help.step)
- (valid (wknAll sub thin :< u) (wknAllRel help allRels thin :< rel))
- ?betaStep'
- )
+-- absValid :
+-- {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+-- ValidHelper P ->
+-- {ctx : SnocList Ty} ->
+-- {ty, ty' : Ty} ->
+-- (t : FullTerm ty' (ctx ++ used)) ->
+-- (thin : used `Thins` [<ty]) ->
+-- (valid : {ctx' : SnocList Ty} ->
+-- (sub : Subst CoTerm (ctx ++ used) ctx') ->
+-- AllLogRel P sub ->
+-- LogRel P (subst' t sub)) ->
+-- {ctx' : SnocList Ty} ->
+-- (sub : Subst CoTerm ctx ctx') ->
+-- AllLogRel P sub ->
+-- LogRel P (subst' (Abs (MakeBound t thin)) sub)
+-- absValid help t (Drop Empty) valid sub allRels =
+-- ( help.abs (valid (shift sub) (shiftRel help allRels))
+-- , \thin, u, rel =>
+-- preserve
+-- (backStepHelper help.step)
+-- (valid (wkn sub thin) (wknRel help allRels thin))
+-- ?betaStep
+-- )
+-- absValid help t (Keep Empty) valid sub allRels =
+-- ( help.abs (valid (lift sub) (liftRel help allRels))
+-- , \thin, u, rel =>
+-- preserve (backStepHelper help.step)
+-- (valid (wkn sub thin :< u) (wknRel help allRels thin :< rel))
+-- ?betaStep'
+-- )
+
+%hint
+retractSingleton : Singleton ctx' -> ctx `Thins` ctx' -> Singleton ctx
+retractSingleton v Id = v
+retractSingleton v Empty = Val [<]
+retractSingleton (Val (ctx' :< ty)) (Drop thin) = retractSingleton (Val ctx') thin
+retractSingleton (Val (ctx' :< ty)) (Keep thin) = pure (:< ty) <*> retractSingleton (Val ctx') thin
export
allValid :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
(s : Singleton ctx) =>
{ty : Ty} ->
- (t : FullTerm ty ctx) ->
+ (t : CoTerm ty ctx) ->
Valid P t
allValid' :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
ValidHelper P ->
(s : Singleton ctx) =>
{ty : Ty} ->
- (t : CoTerm ty ctx) ->
+ (t : FullTerm ty ctx) ->
{ctx' : SnocList Ty} ->
- (sub : CoTerms ctx ctx') ->
+ (sub : Subst CoTerm ctx ctx') ->
AllLogRel P sub ->
LogRel P (subst' t sub)
@@ -260,14 +267,17 @@ allValid help (t `Over` thin) sub allRels =
allValid' help t (restrict sub thin) (restrict allRels thin)
allValid' help Var sub allRels = index help.var allRels Here
-allValid' help {s = Val ctx} (Abs {ty, ty'} (MakeBound t thin)) sub allRels =
- let s' = [| Val ctx ++ retractSingleton (Val [<ty]) thin |] in
- absValid help t thin (allValid' help t) sub allRels
+allValid' help {s = Val ctx} (Const {ty, ty'} t) sub allRels =
+ ?constValid
+allValid' help {s = Val ctx} (Abs {ty, ty'} t) sub allRels =
+ -- let s' = [| Val ctx ++ retractSingleton (Val [<ty]) thin |] in
+ ?absValid
allValid' help (App (MakePair t u _)) sub allRels =
let (pt, app) = allValid help t sub allRels in
let rel = allValid help u sub allRels in
- rewrite sym $ wknId (subst t sub) in
- app id (subst u sub) rel
+ ?appValid
+ -- rewrite sym $ wknId (subst t sub) in
+ -- app id (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
@@ -284,7 +294,7 @@ allValid' help (Rec (MakePair t (MakePair u v _ `Over` thin) _)) sub allRels =
-- -- (let
-- -- rec =
-- -- valid
--- -- (wknAll sub (Drop id) :< Var Here)
+-- -- (wkn sub (Drop id) :< Var Here)
-- -- (wknAllRel help allRels (Drop id) :< help.var Here)
-- -- in
-- -- help.abs rec
@@ -292,28 +302,28 @@ allValid' help (Rec (MakePair t (MakePair u v _ `Over` thin) _)) sub allRels =
-- -- let
-- -- eq :
-- -- (subst
--- -- (wkn (subst t (wknAll sub (Drop Thinning.id) :< Var Here)) (Keep thin))
+-- -- (wkn (subst t (wkn sub (Drop Thinning.id) :< Var Here)) (Keep thin))
-- -- (Base Thinning.id :< u) =
-- -- subst t (wknAll sub thin :< u))
-- -- eq =
-- -- 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 (wkn (subst t (wkn sub (Drop id) :< Var Here)) (Keep thin)) (Base id :< u)
+-- -- ~~ subst (subst t (wkn sub (Drop id) :< Var Here)) (restrict (Base id :< u) (Keep thin))
+-- -- ...(substWkn (subst t (wkn sub (Drop id) :< Var Here)) (Keep thin) (Base id :< u))
+-- -- ~~ subst (subst t (wkn sub (Drop id) :< Var Here)) (Base thin :< u)
+-- -- ...(cong (subst (subst t (wkn sub (Drop id) :< Var Here)) . (:< u) . Base) $ identityLeft thin)
+-- -- ~~ subst t ((Base thin :< u) . wkn sub (Drop id) :< u)
+-- -- ...(substHomo t (wkn 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)
+-- -- ~~ subst t (wkn 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))
+-- -- (valid (wkn sub thin :< u) (wknRel help allRels thin :< rel))
-- -- (replace {p = (App (wkn (subst (Abs t) sub) thin) u >)}
-- -- eq
-- -- (AppBeta (length _)))
@@ -333,23 +343,25 @@ allValid' help (Rec (MakePair t (MakePair u v _ `Over` thin) _)) sub allRels =
-- -- let relv = allValid help v sub allRels in
-- -- help.rec relt relu relv
-idRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
- {ctx : SnocList Ty} ->
- ValidHelper P ->
- AllLogRel P {ctx} (Base Thinning.id)
-idRel help {ctx = [<]} = [<]
-idRel help {ctx = ctx :< ty} = liftRel help (idRel help)
+-- idRel :
+-- {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
+-- {ctx : SnocList Ty} ->
+-- ValidHelper P ->
+-- AllLogRel P {ctx} (Base Thinning.id)
+-- idRel help {ctx = [<]} = [<]
+-- idRel help {ctx = ctx :< ty} = liftRel help (idRel help)
export
allRel :
- {0 P : {ctx, ty : _} -> FullTerm ty ctx -> Type} ->
+ {0 P : {ctx, ty : _} -> CoTerm ty ctx -> Type} ->
{ctx : SnocList Ty} ->
{ty : Ty} ->
ValidHelper P ->
- (t : FullTerm ty ctx) ->
+ (t : CoTerm ty ctx) ->
P t
allRel help t =
- rewrite sym $ substId t in
- escape {P} $
- allValid help @{Val ctx} t (Base id) (idRel help)
+ let rel = allValid help t (Base Id) Base in
+ ?allRel_rhs
+ -- rewrite sym $ substId t in
+ -- escape {P} $
+ -- allValid help @{Val ctx} t Base (idRel help)
generated by cgit v1.2.3 (git 2.49.0) at 2025-04-30 15:59:00 +0000