split OPE stuff into modules

This commit is contained in:
rhiannon morris 2022-11-06 12:39:33 +01:00
parent 4b64399891
commit 5c3f2510fe
9 changed files with 446 additions and 366 deletions

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@ -2,368 +2,9 @@
||| a smaller scope and part of a larger one.
module Quox.OPE
import Quox.NatExtra
import public Data.DPair
import public Data.SnocList
import public Data.SnocList.Elem
%default total
LTE_n = Nat.LTE
%hide Nat.LTE
public export
Scope : Type -> Type
Scope = SnocList
public export
data LTE : Scope a -> Scope a -> Type where
End : [<] `LTE` [<]
Keep : xs `LTE` ys -> xs :< z `LTE` ys :< z
Drop : xs `LTE` ys -> xs `LTE` ys :< z
%name LTE p, q
-- [todo] bitmask representation???
export
dropLast : (xs :< x) `LTE` ys -> xs `LTE` ys
dropLast (Keep p) = Drop p
dropLast (Drop p) = Drop $ dropLast p
export
Uninhabited (xs :< x `LTE` [<]) where uninhabited _ impossible
export
Uninhabited (xs :< x `LTE` xs) where
uninhabited (Keep p) = uninhabited p
uninhabited (Drop p) = uninhabited $ dropLast p
export
0 lteLen : xs `LTE` ys -> length xs `LTE_n` length ys
lteLen End = LTEZero
lteLen (Keep p) = LTESucc $ lteLen p
lteLen (Drop p) = lteSuccRight $ lteLen p
export
0 lteNilRight : xs `LTE` [<] -> xs = [<]
lteNilRight End = Refl
export
0 lteNilLeftDrop : (p : [<] `LTE` (xs :< x)) -> Exists (\q => p = Drop q)
lteNilLeftDrop (Drop q) = Evidence q Refl
export
0 lteNil2End : (p : [<] `LTE` [<]) -> p = End
lteNil2End End = Refl
public export
data Length : Scope a -> Type where
Z : Length [<]
S : (s : Length xs) -> Length (xs :< x)
%name Length s
%builtin Natural Length
namespace Length
public export
(.nat) : Length xs -> Nat
(Z).nat = Z
(S s).nat = S s.nat
%transform "Length.nat" Length.(.nat) xs = believe_me xs
export
0 lengthOk : (s : Length xs) -> s.nat = length xs
lengthOk Z = Refl
lengthOk (S s) = cong S $ lengthOk s
export %hint
lengthLeft : xs `LTE` ys -> Length xs
lengthLeft End = Z
lengthLeft (Keep p) = S (lengthLeft p)
lengthLeft (Drop p) = lengthLeft p
export %hint
lengthRight : xs `LTE` ys -> Length ys
lengthRight End = Z
lengthRight (Keep p) = S (lengthRight p)
lengthRight (Drop p) = S (lengthRight p)
export
id : Length xs => xs `LTE` xs
id @{Z} = End
id @{S s} = Keep id
export
zero : Length xs => [<] `LTE` xs
zero @{Z} = End
zero @{S s} = Drop zero
export
single : Length xs => x `Elem` xs -> [< x] `LTE` xs
single @{S _} Here = Keep zero
single @{S _} (There p) = Drop $ single p
export
(.) : ys `LTE` zs -> xs `LTE` ys -> xs `LTE` zs
End . End = End
Keep p . Keep q = Keep (p . q)
Keep p . Drop q = Drop (p . q)
Drop p . q = Drop (p . q)
export
(++) : xs1 `LTE` ys1 -> xs2 `LTE` ys2 -> (xs1 ++ xs2) `LTE` (ys1 ++ ys2)
p ++ End = p
p ++ Keep q = Keep (p ++ q)
p ++ Drop q = Drop (p ++ q)
public export
data LTEMaskView : (lte : xs `LTE` ys) -> (mask : Nat) -> Type where
[search lte]
END : LTEMaskView End 0
KEEP : (0 _ : LTEMaskView p n) -> LTEMaskView (Keep p) (S (2 * n))
DROP : (0 _ : LTEMaskView p n) -> LTEMaskView (Drop p) (2 * n)
%name LTEMaskView p, q
record LTEMask {a : Type} (xs, ys : Scope a) where
constructor LTEM
mask : Nat
0 lte : xs `LTE` ys
0 view0 : LTEMaskView lte mask
%name LTEMask m
namespace View
private
0 lteMaskEnd' : LTEMaskView p n -> p = End -> n = 0
lteMaskEnd' END Refl = Refl
private
0 lteMaskDrop' : LTEMaskView p n -> p = Drop q -> (n' ** n = 2 * n')
lteMaskDrop' (DROP p {n = n'}) Refl = (n' ** Refl)
private
0 lteMaskEven' : {p : xs `LTE` (ys :< y)} ->
n = 2 * n' -> LTEMaskView p n -> (q ** p = Drop q)
lteMaskEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
lteMaskEven' eq (DROP q) = (_ ** Refl)
private
lteMaskEven : {0 p : xs `LTE` (ys :< y)} ->
(0 _ : LTEMaskView p (2 * n)) -> Exists (\q => p = Drop q)
lteMaskEven q =
let 0 res = lteMaskEven' Refl q in
Evidence res.fst (irrelevantEq res.snd)
private
0 fromDROP' : {lte : xs `LTE` ys} -> n = 2 * n' ->
LTEMaskView (Drop lte) n -> LTEMaskView lte n'
fromDROP' eq (DROP {n} p) =
let eq = doubleInj eq {m = n, n = n'} in
rewrite sym eq in p
private
0 fromDROP : LTEMaskView (Drop lte) (2 * n) -> LTEMaskView lte n
fromDROP = fromDROP' Refl
private
0 lteMaskOdd' : {p : (xs :< x) `LTE` (ys :< x)} -> {n' : Nat} ->
n = S (2 * n') -> LTEMaskView p n -> (q ** p = Keep q)
lteMaskOdd' eq (KEEP q) = (_ ** Refl)
lteMaskOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
lteMaskOdd' _ END impossible
private
lteMaskOdd : (0 _ : LTEMaskView p (S (2 * n))) -> Exists (\q => p = Keep q)
lteMaskOdd q =
let 0 res = lteMaskOdd' Refl q in
Evidence res.fst (irrelevantEq res.snd)
private
0 lteMaskOddHead' : {p : (xs :< x) `LTE` (ys :< y)} -> {n' : Nat} ->
n = S (2 * n') -> LTEMaskView p n -> x = y
lteMaskOddHead' eq (KEEP q) = Refl
lteMaskOddHead' eq (DROP q) = absurd $ lsbMutex' Refl eq
lteMaskOddHead' eq END impossible
private
lteMaskOddHead : {0 p : (xs :< x) `LTE` (ys :< y)} ->
(0 _ : LTEMaskView p (S (2 * n))) -> x = y
lteMaskOddHead q = irrelevantEq $ lteMaskOddHead' Refl q
private
0 fromKEEP' : {lte : xs `LTE` ys} -> n = S (2 * n') ->
LTEMaskView (Keep lte) n -> LTEMaskView lte n'
fromKEEP' eq (KEEP {n} p) =
let eq = doubleInj (injective eq) {m = n, n = n'} in
rewrite sym eq in p
private
0 fromKEEP : LTEMaskView (Keep lte) (S (2 * n)) -> LTEMaskView lte n
fromKEEP = fromKEEP' Refl
export
view : (sx : Length xs) => (sy : Length ys) =>
(m : LTEMask xs ys) -> LTEMaskView m.lte m.mask
view @{Z} @{Z} (LTEM {lte, view0, _}) =
rewrite lteNil2End lte in
rewrite lteMaskEnd' view0 (lteNil2End lte) in
END
view @{S _} @{Z} (LTEM {lte, _}) = void $ absurd lte
view @{Z} @{S sy} (LTEM mask lte view0) =
rewrite (lteNilLeftDrop lte).snd in
rewrite (lteMaskDrop' view0 (lteNilLeftDrop lte).snd).snd in
DROP $ let DROP p = view0 in p
view @{S sx} @{S sy} (LTEM mask lte view0) with (viewLsb mask)
view @{S sx} @{S sy} (LTEM (2 * n) lte view0)
| Evidence Even (Lsb0 n) with (lteMaskEven view0)
view @{S sx} @{S sy} (LTEM (2 * m) (Drop lte) view0)
| Evidence Even (Lsb0 m) | Evidence lte Refl =
DROP $ fromDROP view0
view @{S sx} @{S sy} (LTEM (S (2 * n)) lte view0)
| Evidence Odd (Lsb1 n) with (lteMaskOddHead view0)
view @{S sx} @{S sy} (LTEM (S (2 * n)) lte view0)
| Evidence Odd (Lsb1 n) | Refl with (lteMaskOdd view0)
view @{S sx} @{S sy} (LTEM (S (2 * n)) (Keep lte) view0)
| Evidence Odd (Lsb1 n) | Refl | Evidence lte Refl =
KEEP $ fromKEEP view0
public export
record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `LTE` ys ++ zs) where
constructor MkSplit
{0 leftSub, rightSub : Scope a}
leftThin : leftSub `LTE` ys
rightThin : rightSub `LTE` zs
0 eqScope : xs = leftSub ++ rightSub
0 eqThin : p ~=~ leftThin ++ rightThin
export
split : (zs : Scope a) -> (p : xs `LTE` ys ++ zs) -> Split xs ys zs p
split [<] p = MkSplit p zero Refl Refl
split (zs :< z) (Keep p) with (split zs p)
split (zs :< z) (Keep (l ++ r)) | MkSplit l r Refl Refl =
MkSplit l (Keep r) Refl Refl
split (zs :< z) (Drop p) {xs} with (split zs p)
split (zs :< z) (Drop (l ++ r)) {xs = _} | MkSplit l r Refl Refl =
MkSplit l (Drop r) Refl Refl
public export
data Comp : ys `LTE` zs -> xs `LTE` ys -> xs `LTE` zs -> Type where
CEE : Comp End End End
CKK : Comp p q pq -> Comp (Keep p) (Keep q) (Keep pq)
CKD : Comp p q pq -> Comp (Keep p) (Drop q) (Drop pq)
CD0 : Comp p q pq -> Comp (Drop p) q (Drop pq)
export
comp : (p : ys `LTE` zs) -> (q : xs `LTE` ys) -> Comp p q (p . q)
comp End End = CEE
comp (Keep p) (Keep q) = CKK (comp p q)
comp (Keep p) (Drop q) = CKD (comp p q)
comp (Drop p) q = CD0 (comp p q)
export
0 compOk : Comp p q r -> r = (p . q)
compOk CEE = Refl
compOk (CKK z) = cong Keep $ compOk z
compOk (CKD z) = cong Drop $ compOk z
compOk (CD0 z) = cong Drop $ compOk z
export
compZero : (sx : Length xs) => (sy : Length ys) =>
(p : xs `LTE` ys) -> Comp p (OPE.zero @{sx}) (OPE.zero @{sy})
compZero {sx = Z, sy = Z} End = CEE
compZero {sx = S _, sy = S _} (Keep p) = CKD (compZero p)
compZero {sy = S _} (Drop p) = CD0 (compZero p)
export
compIdLeft : (sy : Length ys) =>
(p : xs `LTE` ys) -> Comp (OPE.id @{sy}) p p
compIdLeft {sy = Z} End = CEE
compIdLeft {sy = S _} (Keep p) = CKK (compIdLeft p)
compIdLeft {sy = S _} (Drop p) = CKD (compIdLeft p)
export
compIdRight : (sx : Length xs) =>
(p : xs `LTE` ys) -> Comp p (OPE.id @{sx}) p
compIdRight {sx = Z} End = CEE
compIdRight {sx = S _} (Keep p) = CKK (compIdRight p)
compIdRight (Drop p) = CD0 (compIdRight p)
export
0 compAssoc : (p : ys `LTE` zs) -> (q : xs `LTE` ys) -> (r : ws `LTE` xs) ->
p . (q . r) = (p . q) . r
compAssoc End End End = Refl
compAssoc (Keep p) (Keep q) (Keep r) = cong Keep $ compAssoc p q r
compAssoc (Keep p) (Keep q) (Drop r) = cong Drop $ compAssoc p q r
compAssoc (Keep p) (Drop q) r = cong Drop $ compAssoc p q r
compAssoc (Drop p) q r = cong Drop $ compAssoc p q r
compAssoc End (Drop _) _ impossible
public export
Scoped : Type -> Type
Scoped a = Scope a -> Type
public export
Subscope : Scope a -> Type
Subscope ys = Exists (`LTE` ys)
public export
record SubMap {a : Type} {xs, ys, zs : Scope a}
(p : xs `LTE` zs) (q : ys `LTE` zs) where
constructor SM
thin : xs `LTE` ys
0 comp : Comp q thin p
parameters (p : xs `LTE` ys)
export
subId : SubMap p p
subId = SM id (compIdRight p)
export
subZero : SubMap OPE.zero p
subZero = SM zero (compZero p)
public export
data All : (a -> Type) -> Scoped a where
Lin : All p [<]
(:<) : All p xs -> p x -> All p (xs :< x)
%name OPE.All ps, qs
export
mapAll : (forall x. p x -> q x) -> All p xs -> All q xs
mapAll f [<] = [<]
mapAll f (x :< y) = mapAll f x :< f y
export
subAll : xs `LTE` ys -> All p ys -> All p xs
subAll End [<] = [<]
subAll (Keep q) (ps :< x) = subAll q ps :< x
subAll (Drop q) (ps :< x) = subAll q ps
public export
data Cover_ : (overlap : Bool) -> xs `LTE` zs -> ys `LTE` zs -> Type where
CE : Cover_ ov End End
CL : Cover_ ov p q -> Cover_ ov (Keep p) (Drop q)
CR : Cover_ ov p q -> Cover_ ov (Drop p) (Keep q)
C2 : Cover_ ov p q -> Cover_ True (Keep p) (Keep q)
public export
Cover : xs `LTE` zs -> ys `LTE` zs -> Type
Cover = Cover_ True
public export
Partition : xs `LTE` zs -> ys `LTE` zs -> Type
Partition = Cover_ False
import public Quox.OPE.Basics
import public Quox.OPE.Length
import public Quox.OPE.Sub
import public Quox.OPE.Split
import public Quox.OPE.Comp
import public Quox.OPE.Cover

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module Quox.OPE.Basics
%default total
public export
Scope : Type -> Type
Scope = SnocList
public export
Scoped : Type -> Type
Scoped a = Scope a -> Type
public export
data All : (a -> Type) -> Scoped a where
Lin : All p [<]
(:<) : All p xs -> p x -> All p (xs :< x)
%name OPE.Basics.All ps, qs
public export
mapAll : (forall x. p x -> q x) -> All p xs -> All q xs
mapAll f [<] = [<]
mapAll f (x :< y) = mapAll f x :< f y

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module Quox.OPE.Comp
import Quox.OPE.Basics
import Quox.OPE.Length
import Quox.OPE.Sub
import Data.DPair
%default total
public export
data Comp : ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs -> Type where
CEE : Comp End End End
CKK : Comp p q pq -> Comp (Keep p) (Keep q) (Keep pq)
CKD : Comp p q pq -> Comp (Keep p) (Drop q) (Drop pq)
CD0 : Comp p q pq -> Comp (Drop p) q (Drop pq)
export
comp : (p : ys `Sub` zs) -> (q : xs `Sub` ys) -> Comp p q (p . q)
comp End End = CEE
comp (Keep p) (Keep q) = CKK (comp p q)
comp (Keep p) (Drop q) = CKD (comp p q)
comp (Drop p) q = CD0 (comp p q)
export
0 compOk : Comp p q r -> r = (p . q)
compOk CEE = Refl
compOk (CKK z) = cong Keep $ compOk z
compOk (CKD z) = cong Drop $ compOk z
compOk (CD0 z) = cong Drop $ compOk z
export
compZero : (sx : Length xs) => (sy : Length ys) =>
(p : xs `Sub` ys) -> Comp p (Sub.zero @{sx}) (Sub.zero @{sy})
compZero {sx = Z, sy = Z} End = CEE
compZero {sx = S _, sy = S _} (Keep p) = CKD (compZero p)
compZero {sy = S _} (Drop p) = CD0 (compZero p)
export
compIdLeft : (sy : Length ys) =>
(p : xs `Sub` ys) -> Comp (Sub.id @{sy}) p p
compIdLeft {sy = Z} End = CEE
compIdLeft {sy = S _} (Keep p) = CKK (compIdLeft p)
compIdLeft {sy = S _} (Drop p) = CKD (compIdLeft p)
export
compIdRight : (sx : Length xs) =>
(p : xs `Sub` ys) -> Comp p (Sub.id @{sx}) p
compIdRight {sx = Z} End = CEE
compIdRight {sx = S _} (Keep p) = CKK (compIdRight p)
compIdRight (Drop p) = CD0 (compIdRight p)
export
0 compAssoc : (p : ys `Sub` zs) -> (q : xs `Sub` ys) -> (r : ws `Sub` xs) ->
p . (q . r) = (p . q) . r
compAssoc End End End = Refl
compAssoc (Keep p) (Keep q) (Keep r) = cong Keep $ compAssoc p q r
compAssoc (Keep p) (Keep q) (Drop r) = cong Drop $ compAssoc p q r
compAssoc (Keep p) (Drop q) r = cong Drop $ compAssoc p q r
compAssoc (Drop p) q r = cong Drop $ compAssoc p q r
compAssoc End (Drop _) _ impossible
public export
Subscope : Scope a -> Type
Subscope ys = Exists (`Sub` ys)
public export
record SubMap {a : Type} {xs, ys, zs : Scope a}
(p : xs `Sub` zs) (q : ys `Sub` zs) where
constructor SM
thin : xs `Sub` ys
0 comp : Comp q thin p
parameters (p : xs `Sub` ys)
export
subId : SubMap p p
subId = SM id (compIdRight p)
export
subZero : SubMap Sub.zero p
subZero = SM zero (compZero p)

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module Quox.OPE.Cover
import Quox.OPE.Basics
import Quox.OPE.Length
import Quox.OPE.Sub
%default total
public export
data Cover_ : (overlap : Bool) -> xs `Sub` zs -> ys `Sub` zs -> Type where
CE : Cover_ ov End End
CL : Cover_ ov p q -> Cover_ ov (Keep p) (Drop q)
CR : Cover_ ov p q -> Cover_ ov (Drop p) (Keep q)
C2 : Cover_ ov p q -> Cover_ True (Keep p) (Keep q)
public export
Cover : xs `Sub` zs -> ys `Sub` zs -> Type
Cover = Cover_ True
public export
Partition : xs `Sub` zs -> ys `Sub` zs -> Type
Partition = Cover_ False

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module Quox.OPE.Length
import Quox.OPE.Basics
%default total
public export
data Length : Scope a -> Type where
Z : Length [<]
S : (s : Length xs) -> Length (xs :< x)
%name Length s
%builtin Natural Length
public export
(.nat) : Length xs -> Nat
(Z).nat = Z
(S s).nat = S s.nat
%transform "Length.nat" Length.(.nat) xs = believe_me xs
export
0 ok : (s : Length xs) -> s.nat = length xs
ok Z = Refl
ok (S s) = cong S $ ok s

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module Quox.OPE.Split
import Quox.OPE.Basics
import Quox.OPE.Length
import Quox.OPE.Sub
%default total
public export
record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `Sub` ys ++ zs) where
constructor MkSplit
{0 leftSub, rightSub : Scope a}
leftThin : leftSub `Sub` ys
rightThin : rightSub `Sub` zs
0 eqScope : xs = leftSub ++ rightSub
0 eqThin : p ~=~ leftThin ++ rightThin
export
split : (zs : Scope a) -> (p : xs `Sub` ys ++ zs) -> Split xs ys zs p
split [<] p = MkSplit p zero Refl Refl
split (zs :< z) (Keep p) with (split zs p)
split (zs :< z) (Keep (l ++ r)) | MkSplit l r Refl Refl =
MkSplit l (Keep r) Refl Refl
split (zs :< z) (Drop p) {xs} with (split zs p)
split (zs :< z) (Drop (l ++ r)) {xs = _} | MkSplit l r Refl Refl =
MkSplit l (Drop r) Refl Refl

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module Quox.OPE.Sub
import Quox.OPE.Basics
import Quox.OPE.Length
import Quox.NatExtra
import Data.DPair
import Data.SnocList.Elem
%default total
public export
data Sub : Scope a -> Scope a -> Type where
End : [<] `Sub` [<]
Keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
Drop : xs `Sub` ys -> xs `Sub` ys :< z
%name Sub p, q
export
keepInjective : Keep p = Keep q -> p = q
keepInjective Refl = Refl
export
dropInjective : Drop p = Drop q -> p = q
dropInjective Refl = Refl
-- these need to be `public export` so that
-- `id`, `zero`, and maybe others can reduce
public export %hint
lengthLeft : xs `Sub` ys -> Length xs
lengthLeft End = Z
lengthLeft (Keep p) = S (lengthLeft p)
lengthLeft (Drop p) = lengthLeft p
public export %hint
lengthRight : xs `Sub` ys -> Length ys
lengthRight End = Z
lengthRight (Keep p) = S (lengthRight p)
lengthRight (Drop p) = S (lengthRight p)
export
dropLast : (xs :< x) `Sub` ys -> xs `Sub` ys
dropLast (Keep p) = Drop p
dropLast (Drop p) = Drop $ dropLast p
export
Uninhabited (xs :< x `Sub` [<]) where uninhabited _ impossible
export
Uninhabited (xs :< x `Sub` xs) where
uninhabited (Keep p) = uninhabited p
uninhabited (Drop p) = uninhabited $ dropLast p
export
0 lteLen : xs `Sub` ys -> length xs `LTE` length ys
lteLen End = LTEZero
lteLen (Keep p) = LTESucc $ lteLen p
lteLen (Drop p) = lteSuccRight $ lteLen p
export
0 lteNilRight : xs `Sub` [<] -> xs = [<]
lteNilRight End = Refl
public export
id : Length xs => xs `Sub` xs
id @{Z} = End
id @{S s} = Keep id
public export
zero : Length xs => [<] `Sub` xs
zero @{Z} = End
zero @{S s} = Drop zero
public export
single : Length xs => x `Elem` xs -> [< x] `Sub` xs
single @{S _} Here = Keep zero
single @{S _} (There p) = Drop $ single p
public export
(.) : ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
End . End = End
Keep p . Keep q = Keep (p . q)
Keep p . Drop q = Drop (p . q)
Drop p . q = Drop (p . q)
public export
(++) : xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
p ++ End = p
p ++ Keep q = Keep (p ++ q)
p ++ Drop q = Drop (p ++ q)
export
0 appZeroRight : (p : xs `Sub` ys) -> p ++ zero @{len} {xs = [<]} = p
appZeroRight {len = Z} p = Refl
public export
subAll : xs `Sub` ys -> All p ys -> All p xs
subAll End [<] = [<]
subAll (Keep q) (ps :< x) = subAll q ps :< x
subAll (Drop q) (ps :< x) = subAll q ps
public export
data SubMaskView : (lte : xs `Sub` ys) -> (mask : Nat) -> Type where
[search lte]
END : SubMaskView End 0
KEEP : {n : Nat} -> {0 p : xs `Sub` ys} ->
(0 v : SubMaskView p n) -> SubMaskView (Keep {z} p) (S (2 * n))
DROP : {n : Nat} -> {0 p : xs `Sub` ys} ->
(0 v : SubMaskView p n) -> SubMaskView (Drop {z} p) (2 * n)
%name SubMaskView v
public export
record SubMask {a : Type} (xs, ys : Scope a) where
constructor SubM
mask : Nat
0 lte : xs `Sub` ys
0 view0 : SubMaskView lte mask
%name SubMask m
private
0 ltemNilLeftZero' : SubMaskView {xs = [<]} lte mask -> mask = 0
ltemNilLeftZero' END = Refl
ltemNilLeftZero' (DROP v) = cong (2 *) $ ltemNilLeftZero' v
export
ltemNilLeftZero : (0 _ : SubMaskView {xs = [<]} lte mask) -> mask = 0
ltemNilLeftZero v = irrelevantEq $ ltemNilLeftZero' v
private
0 lteNilLeftDrop0 : (p : [<] `Sub` (xs :< x)) -> (q ** p = Drop q)
lteNilLeftDrop0 (Drop q) = (q ** Refl)
private
lteNilLeftDrop : (0 p : [<] `Sub` (xs :< x)) -> Exists (\q => p = Drop q)
lteNilLeftDrop q =
let 0 res = lteNilLeftDrop0 q in
Evidence res.fst (irrelevantEq res.snd)
private
0 lteNil2End : (p : [<] `Sub` [<]) -> p = End
lteNil2End End = Refl
private
0 ltemEnd' : SubMaskView p n -> p = End -> n = 0
ltemEnd' END Refl = Refl
private
0 ltemEven' : {p : xs `Sub` (ys :< y)} ->
n = 2 * n' -> SubMaskView p n -> (q ** p = Drop q)
ltemEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
ltemEven' eq (DROP q) = (_ ** Refl)
private
ltemEven : {0 p : xs `Sub` (ys :< y)} ->
(0 _ : SubMaskView p (2 * n)) -> Exists (\q => p = Drop q)
ltemEven q =
let 0 res = ltemEven' Refl q in
Evidence res.fst (irrelevantEq res.snd)
private
0 fromDROP' : {lte : xs `Sub` ys} -> n = 2 * n' ->
SubMaskView (Drop lte) n -> SubMaskView lte n'
fromDROP' eq (DROP {n} p) =
let eq = doubleInj eq {m = n, n = n'} in
rewrite sym eq in p
private
0 ltemOdd' : {p : (xs :< x) `Sub` (ys :< x)} -> {n' : Nat} ->
n = S (2 * n') -> SubMaskView p n -> (q ** p = Keep q)
ltemOdd' eq (KEEP q) = (_ ** Refl)
ltemOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
ltemOdd' eq END impossible
private
ltemOdd : (0 _ : SubMaskView p (S (2 * n))) -> Exists (\q => p = Keep q)
ltemOdd q =
let 0 res = ltemOdd' Refl q in
Evidence res.fst (irrelevantEq res.snd)
private
0 ltemOddHead' : {p : (xs :< x) `Sub` (ys :< y)} -> {n' : Nat} ->
n = S (2 * n') -> SubMaskView p n -> x = y
ltemOddHead' eq (KEEP q) = Refl
ltemOddHead' eq (DROP q) = absurd $ lsbMutex' Refl eq
ltemOddHead' eq END impossible
private
ltemOddHead : {0 p : (xs :< x) `Sub` (ys :< y)} ->
(0 _ : SubMaskView p (S (2 * n))) -> x = y
ltemOddHead q = irrelevantEq $ ltemOddHead' Refl q
private
0 fromKEEP' : {lte : xs `Sub` ys} -> n = S (2 * n') ->
SubMaskView (Keep lte) n -> SubMaskView lte n'
fromKEEP' eq (KEEP {n} p) =
let eq = doubleInj (injective eq) {m = n, n = n'} in
rewrite sym eq in p
export
view : Length xs => Length ys =>
(m : SubMask xs ys) -> SubMaskView m.lte m.mask
view @{Z} @{Z} (SubM {lte, view0, _}) =
rewrite lteNil2End lte in
rewrite ltemEnd' view0 (lteNil2End lte) in
END
view @{S _} @{Z} (SubM {lte, _}) = void $ absurd lte
view @{Z} @{S sy} (SubM mask lte view0) with (ltemNilLeftZero view0)
view @{Z} @{S sy} (SubM 0 lte view0)
| Refl with (lteNilLeftDrop lte)
view @{Z} @{S sy} (SubM 0 (Drop lte) view0)
| Refl | Evidence lte Refl =
DROP {n = 0} $ let DROP {n = 0} p = view0 in p
view @{S sx} @{S sy} (SubM mask lte view0) with (viewLsb mask)
view @{S sx} @{S sy} (SubM (2 * n) lte view0)
| Evidence Even (Lsb0 n) with (ltemEven view0)
view @{S sx} @{S sy} (SubM (2 * m) (Drop lte) view0)
| Evidence Even (Lsb0 m) | Evidence lte Refl =
DROP $ fromDROP' Refl view0
view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
| Evidence Odd (Lsb1 n) with (ltemOddHead view0)
view @{S sx} @{S sy} (SubM (S (2 * n)) lte view0)
| Evidence Odd (Lsb1 n) | Refl with (ltemOdd view0)
view @{S sx} @{S sy} (SubM (S (2 * n)) (Keep lte) view0)
| Evidence Odd (Lsb1 n) | Refl | Evidence lte Refl =
KEEP $ fromKEEP' Refl view0
export
(.view) : Length xs => Length ys =>
(m : SubMask xs ys) -> SubMaskView m.lte m.mask
(.view) = view
export
ltemLen : Length xs => Length ys =>
xs `SubMask` ys -> length xs `LTE` length ys
ltemLen @{sx} @{sy} lte@(SubM m l _) with (lte.view)
ltemLen @{sx} @{sy} lte@(SubM 0 End _) | END = LTEZero
ltemLen @{S sx} @{S sy} lte@(SubM (S (2 * n)) (Keep p) _) | (KEEP q) =
LTESucc $ ltemLen $ SubM n p q
ltemLen @{sx} @{S sy} lte@(SubM (2 * n) (Drop p) _) | (DROP q) =
lteSuccRight $ ltemLen $ SubM n p q
export
ltemNilRight : xs `SubMask` [<] -> xs = [<]
ltemNilRight m = irrelevantEq $ lteNilRight m.lte

View file

@ -2,7 +2,6 @@ module Quox.Syntax.Var
import Quox.Name
import Quox.Pretty
import Quox.OPE
import Data.Nat
import Data.List

View file

@ -11,6 +11,12 @@ modules =
Quox.NatExtra,
Quox.Unicode,
Quox.OPE,
Quox.OPE.Basics,
Quox.OPE.Length,
Quox.OPE.Sub,
Quox.OPE.Split,
Quox.OPE.Comp,
Quox.OPE.Cover,
Quox.Pretty,
Quox.Syntax,
Quox.Syntax.Dim,