quox/lib/Quox/Syntax/Term/Subst.idr

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module Quox.Syntax.Term.Subst
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import Quox.No
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import Quox.Syntax.Term.Base
import Quox.Syntax.Term.Tighten
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import Data.SnocVect
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%default total
namespace CanDSubst
public export
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interface CanDSubst (0 tm : TermLike) where
(//) : tm dfrom n -> Lazy (DSubst dfrom dto) -> tm dto n
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||| does the minimal reasonable work:
||| - deletes the closure around an atomic constant like `TYPE`
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level dim-closure
||| - otherwise, wraps in a new closure
export
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CanDSubst Term where
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s // Shift SZ = s
TYPE l // _ = TYPE l
DCloT s ph // th = DCloT s $ ph . th
s // th = DCloT s th
private
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subDArgs : Elim dfrom n -> DSubst dfrom dto -> Elim dto n
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subDArgs (f :% d) th = subDArgs f th :% (d // th)
subDArgs e th = DCloE e th
||| does the minimal reasonable work:
||| - deletes the closure around a term variable
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level dim-closure
||| - immediately looks up bound variables in a
||| top-level sequence of dimension applications
||| - otherwise, wraps in a new closure
export
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CanDSubst Elim where
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e // Shift SZ = e
F x // _ = F x
B i // _ = B i
f :% d // th = subDArgs (f :% d) th
DCloE e ph // th = DCloE e $ ph . th
e // th = DCloE e th
namespace DSubst.ScopeTermN
export %inline
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(//) : ScopeTermN s dfrom n -> Lazy (DSubst dfrom dto) ->
ScopeTermN s dto n
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S ns (Y body) // th = S ns $ Y $ body // th
S ns (N body) // th = S ns $ N $ body // th
namespace DSubst.DScopeTermN
export %inline
(//) : {s : Nat} ->
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DScopeTermN s dfrom n -> Lazy (DSubst dfrom dto) ->
DScopeTermN s dto n
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S ns (Y body) // th = S ns $ Y $ body // pushN s th
S ns (N body) // th = S ns $ N $ body // th
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export %inline FromVar (Elim d) where fromVar = B
export %inline FromVar (Term d) where fromVar = E . fromVar
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||| does the minimal reasonable work:
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||| - deletes the closure around a *free* name
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||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level closure
||| - immediately looks up a bound variable
||| - otherwise, wraps in a new closure
export
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CanSubstSelf (Elim d) where
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F x // _ = F x
B i // th = th !! i
CloE e ph // th = assert_total CloE e $ ph . th
e // th = case force th of
Shift SZ => e
th => CloE e th
namespace CanTSubst
public export
interface CanTSubst (0 tm : TermLike) where
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(//) : tm d from -> Lazy (TSubst d from to) -> tm d to
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||| does the minimal reasonable work:
||| - deletes the closure around an atomic constant like `TYPE`
||| - deletes an identity substitution
||| - composes (lazily) with an existing top-level closure
||| - goes inside `E` in case it is a simple variable or something
||| - otherwise, wraps in a new closure
export
CanTSubst Term where
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TYPE l // _ = TYPE l
E e // th = E $ e // th
CloT s ph // th = CloT s $ ph . th
s // th = case force th of
Shift SZ => s
th => CloT s th
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namespace ScopeTermN
export %inline
(//) : {s : Nat} ->
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ScopeTermN s d from -> Lazy (TSubst d from to) ->
ScopeTermN s d to
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S ns (Y body) // th = S ns $ Y $ body // pushN s th
S ns (N body) // th = S ns $ N $ body // th
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namespace DScopeTermN
export %inline
(//) : {s : Nat} ->
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DScopeTermN s d from -> Lazy (TSubst d from to) ->
DScopeTermN s d to
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S ns (Y body) // th = S ns $ Y $ body // map (// shift s) th
S ns (N body) // th = S ns $ N $ body // th
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export %inline CanShift (Term d) where s // by = s // Shift by
export %inline CanShift (Elim d) where e // by = e // Shift by
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export %inline
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{s : Nat} -> CanShift (ScopeTermN s d) where
b // by = b // Shift by
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export %inline
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comp : DSubst dfrom dto -> TSubst dfrom from mid -> TSubst dto mid to ->
TSubst dto from to
comp th ps ph = map (// th) ps . ph
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public export %inline
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dweakT : (by : Nat) -> Term d n -> Term (by + d) n
dweakT by t = t // shift by
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public export %inline
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dweakE : (by : Nat) -> Elim d n -> Elim (by + d) n
dweakE by t = t // shift by
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public export %inline
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weakT : (by : Nat) -> Term d n -> Term d (by + n)
weakT by t = t // shift by
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public export %inline
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weakE : (by : Nat) -> Elim d n -> Elim d (by + n)
weakE by t = t // shift by
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parameters {s : Nat}
namespace ScopeTermBody
export %inline
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(.term) : ScopedBody s (Term d) n -> Term d (s + n)
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(Y b).term = b
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(N b).term = weakT s b
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namespace ScopeTermN
export %inline
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(.term) : ScopeTermN s d n -> Term d (s + n)
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t.term = t.body.term
namespace DScopeTermBody
export %inline
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(.term) : ScopedBody s (\d => Term d n) d -> Term (s + d) n
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(Y b).term = b
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(N b).term = dweakT s b
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namespace DScopeTermN
export %inline
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(.term) : DScopeTermN s d n -> Term (s + d) n
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t.term = t.body.term
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export %inline
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subN : ScopeTermN s d n -> SnocVect s (Elim d n) -> Term d n
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subN (S _ (Y body)) es = body // fromSnocVect es
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subN (S _ (N body)) _ = body
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export %inline
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sub1 : ScopeTerm d n -> Elim d n -> Term d n
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sub1 t e = subN t [< e]
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export %inline
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dsubN : DScopeTermN s d n -> SnocVect s (Dim d) -> Term d n
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dsubN (S _ (Y body)) ps = body // fromSnocVect ps
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dsubN (S _ (N body)) _ = body
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export %inline
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dsub1 : DScopeTerm d n -> Dim d -> Term d n
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dsub1 t p = dsubN t [< p]
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public export %inline
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(.zero) : DScopeTerm d n -> Term d n
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body.zero = dsub1 body $ K Zero
public export %inline
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(.one) : DScopeTerm d n -> Term d n
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body.one = dsub1 body $ K One
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public export
0 CloTest : TermLike -> Type
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CloTest tm = forall d, n. tm d n -> Bool
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interface PushSubsts (0 tm : TermLike) (0 isClo : CloTest tm) | tm where
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pushSubstsWith : DSubst dfrom dto -> TSubst dto from to ->
tm dfrom from -> Subset (tm dto to) (No . isClo)
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public export
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0 NotClo : {isClo : CloTest tm} -> PushSubsts tm isClo => Pred (tm d n)
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NotClo = No . isClo
public export
0 NonClo : (tm : TermLike) -> {isClo : CloTest tm} ->
PushSubsts tm isClo => TermLike
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NonClo tm d n = Subset (tm d n) NotClo
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public export %inline
nclo : {isClo : CloTest tm} -> (0 _ : PushSubsts tm isClo) =>
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(t : tm d n) -> (0 nc : NotClo t) => NonClo tm d n
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nclo t = Element t nc
parameters {0 isClo : CloTest tm} {auto _ : PushSubsts tm isClo}
||| if the input term has any top-level closures, push them under one layer of
||| syntax
export %inline
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pushSubsts : tm d n -> NonClo tm d n
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pushSubsts s = pushSubstsWith id id s
export %inline
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pushSubstsWith' : DSubst dfrom dto -> TSubst dto from to ->
tm dfrom from -> tm dto to
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pushSubstsWith' th ph x = fst $ pushSubstsWith th ph x
mutual
public export
isCloT : CloTest Term
isCloT (CloT {}) = True
isCloT (DCloT {}) = True
isCloT (E e) = isCloE e
isCloT _ = False
public export
isCloE : CloTest Elim
isCloE (CloE {}) = True
isCloE (DCloE {}) = True
isCloE _ = False
mutual
export
PushSubsts Term Subst.isCloT where
pushSubstsWith th ph (TYPE l) =
nclo $ TYPE l
pushSubstsWith th ph (Pi qty a body) =
nclo $ Pi qty (a // th // ph) (body // th // ph)
pushSubstsWith th ph (Lam body) =
nclo $ Lam (body // th // ph)
pushSubstsWith th ph (Sig a b) =
nclo $ Sig (a // th // ph) (b // th // ph)
pushSubstsWith th ph (Pair s t) =
nclo $ Pair (s // th // ph) (t // th // ph)
pushSubstsWith th ph (Enum tags) =
nclo $ Enum tags
pushSubstsWith th ph (Tag tag) =
nclo $ Tag tag
pushSubstsWith th ph (Eq ty l r) =
nclo $ Eq (ty // th // ph) (l // th // ph) (r // th // ph)
pushSubstsWith th ph (DLam body) =
nclo $ DLam (body // th // ph)
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pushSubstsWith _ _ Nat = nclo Nat
pushSubstsWith _ _ Zero = nclo Zero
pushSubstsWith th ph (Succ n) = nclo $ Succ $ n // th // ph
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pushSubstsWith th ph (BOX pi ty) = nclo $ BOX pi $ ty // th // ph
pushSubstsWith th ph (Box val) = nclo $ Box $ val // th // ph
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pushSubstsWith th ph (E e) =
let Element e nc = pushSubstsWith th ph e in nclo $ E e
pushSubstsWith th ph (CloT s ps) =
pushSubstsWith th (comp th ps ph) s
pushSubstsWith th ph (DCloT s ps) =
pushSubstsWith (ps . th) ph s
export
PushSubsts Elim Subst.isCloE where
pushSubstsWith th ph (F x) =
nclo $ F x
pushSubstsWith th ph (B i) =
let res = ph !! i in
case nchoose $ isCloE res of
Left yes => assert_total pushSubsts res
Right no => Element res no
pushSubstsWith th ph (f :@ s) =
nclo $ (f // th // ph) :@ (s // th // ph)
pushSubstsWith th ph (CasePair pi p r b) =
nclo $ CasePair pi (p // th // ph) (r // th // ph) (b // th // ph)
pushSubstsWith th ph (CaseEnum pi t r arms) =
nclo $ CaseEnum pi (t // th // ph) (r // th // ph)
(map (\b => b // th // ph) arms)
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pushSubstsWith th ph (CaseNat pi pi' n r z s) =
nclo $ CaseNat pi pi' (n // th // ph) (r // th // ph)
(z // th // ph) (s // th // ph)
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pushSubstsWith th ph (CaseBox pi x r b) =
nclo $ CaseBox pi (x // th // ph) (r // th // ph) (b // th // ph)
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pushSubstsWith th ph (f :% d) =
nclo $ (f // th // ph) :% (d // th)
pushSubstsWith th ph (s :# a) =
nclo $ (s // th // ph) :# (a // th // ph)
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pushSubstsWith th ph (Coe ty p q val) =
nclo $ Coe (ty // th // ph) (p // th) (q // th) (val // th // ph)
pushSubstsWith th ph (Comp ty p q val r zero one) =
nclo $ Comp (ty // th // ph) (p // th) (q // th)
(val // th // ph) (r // th)
(zero // th // ph) (one // th // ph)
pushSubstsWith th ph (TypeCase ty ret arms def) =
nclo $ TypeCase (ty // th // ph) (ret // th // ph)
(map (\t => t // th // ph) arms) (def // th // ph)
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pushSubstsWith th ph (CloE e ps) =
pushSubstsWith th (comp th ps ph) e
pushSubstsWith th ph (DCloE e ps) =
pushSubstsWith (ps . th) ph e
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private %inline
CompHY : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
(r : Dim d) -> (zero, one : DScopeTerm d n) -> Elim d n
CompHY {ty, p, q, val, r, zero, one} =
let ty' = SY ty.names $ ty.term // (B VZ ::: shift 2) in
Comp {
ty = dsub1 ty q, p, q,
val = E $ Coe ty p q val, r,
zero = SY zero.names $ E $ Coe ty' (B VZ) (weakD 1 q) zero.term,
one = SY one.names $ E $ Coe ty' (B VZ) (weakD 1 q) one.term
}
public export %inline
CompH' : (ty : DScopeTerm d n) ->
(p, q : Dim d) -> (val : Term d n) -> (r : Dim d) ->
(zero : DScopeTerm d n) ->
(one : DScopeTerm d n) ->
Elim d n
CompH' {ty, p, q, val, r, zero, one} =
case dsqueeze ty of
S _ (N ty) => Comp {ty, p, q, val, r, zero, one}
S _ (Y _) => CompHY {ty, p, q, val, r, zero, one}
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||| heterogeneous composition, using Comp and Coe (and subst)
|||
||| comp [i ⇒ A] @p @q s { (r=0) j ⇒ t₀; (r=1) j ⇒ t₁ }
||| ≔
||| comp [Aq/i] @p @q (coe [i ⇒ A] @p @q s) {
||| (r=0) j ⇒ coe [i ⇒ A] @j @q t₀;
||| (r=1) j ⇒ coe [i ⇒ A] @j @q t₁
||| }
public export %inline
CompH : (i : BaseName) -> (ty : Term (S d) n) ->
(p, q : Dim d) -> (val : Term d n) -> (r : Dim d) ->
(j0 : BaseName) -> (zero : Term (S d) n) ->
(j1 : BaseName) -> (one : Term (S d) n) ->
Elim d n
CompH {i, ty, p, q, val, r, j0, zero, j1, one} =
CompH' {ty = SY [< i] ty, p, q, val, r,
zero = SY [< j0] zero, one = SY [< j0] one}