bit mask OPE stuff

This commit is contained in:
rhiannon morris 2022-11-15 15:44:49 +01:00
parent 42acbfc4ac
commit 5e11433c9f
6 changed files with 638 additions and 290 deletions

View file

@ -84,58 +84,85 @@ modNatViaIntegerNZ m n _ = assert_total modNatViaInteger m n
public export
data Parity = Even | Odd
public export
data ViewLsb : Nat -> Parity -> Type where
Lsb0 : (n : Nat) -> ViewLsb (2 * n) Even
Lsb1 : (n : Nat) -> ViewLsb (S (2 * n)) Odd
data ViewLsb : Nat -> Type where
Lsb0 : (n : Nat) -> (0 eq : n' = 2 * n) -> ViewLsb n'
Lsb1 : (n : Nat) -> (0 eq : n' = S (2 * n)) -> ViewLsb n'
%name ViewLsb p, q
export
fromLsb0 : ViewLsb n Even -> Subset Nat (\n' => n = 2 * n')
fromLsb0 (Lsb0 n') = Element n' Refl
public export data IsLsb0 : ViewLsb n -> Type where ItIsLsb0 : IsLsb0 (Lsb0 n eq)
public export data IsLsb1 : ViewLsb n -> Type where ItIsLsb1 : IsLsb1 (Lsb1 n eq)
export
fromLsb1 : ViewLsb n Odd -> Subset Nat (\n' => n = S (2 * n'))
fromLsb1 (Lsb1 n') = Element n' Refl
private
viewLsb' : (m, d : Nat) -> (0 _ : m `LT` 2) -> Exists $ ViewLsb (m + 2 * d)
viewLsb' 0 d p = Evidence Even (Lsb0 d)
viewLsb' 1 d p = Evidence Odd (Lsb1 d)
viewLsb' (S (S _)) _ (LTESucc p) = void $ absurd p
export
viewLsb : (n : Nat) -> Exists $ ViewLsb n
viewLsb n =
let 0 nz = the (NonZero 2) %search in
rewrite DivisionTheorem n 2 nz nz in
rewrite multCommutative (divNatNZ n 2 nz) 2 in
viewLsb' (modNatNZ n 2 nz) (divNatNZ n 2 nz) (boundModNatNZ n 2 nz)
export
0 lsbMutex' : n = (2 * a) -> n = S (2 * b) -> Void
lsbMutex' ev od {n = 0} impossible
lsbMutex' ev od {n = 1} {a = S a} {b = 0} =
0 lsbMutex : n = (2 * a) -> n = S (2 * b) -> Void
lsbMutex ev od {n = 0} impossible
lsbMutex ev od {n = 1} {a = S a} {b = 0} =
let ev = injective ev in
let s = sym $ plusSuccRightSucc a (a + 0) in
absurd $ trans ev s
lsbMutex' ev od {n = S (S n)} {a = S a} {b = S b} =
lsbMutex ev od {n = S (S n)} {a = S a} {b = S b} =
let ev = injective $
trans (injective ev) (sym $ plusSuccRightSucc a (a + 0)) in
let od = trans (injective $ injective od)
(sym $ plusSuccRightSucc b (b + 0)) in
lsbMutex' ev od
lsbMutex ev od
public export %hint
doubleIsLsb0 : (p : ViewLsb (2 * n)) -> IsLsb0 p
doubleIsLsb0 (Lsb0 k eq) = ItIsLsb0
doubleIsLsb0 (Lsb1 k eq) = void $ absurd $ lsbMutex Refl eq
public export %hint
sDoubleIsLsb1 : (p : ViewLsb (S (2 * n))) -> IsLsb1 p
sDoubleIsLsb1 (Lsb1 k eq) = ItIsLsb1
sDoubleIsLsb1 (Lsb0 k eq) = void $ absurd $ lsbMutex Refl (sym eq)
export
0 lsbMutex : ViewLsb n Even -> ViewLsb n Odd -> Void
lsbMutex p q = lsbMutex' (fromLsb0 p).snd (fromLsb1 q).snd
fromLsb0 : (p : ViewLsb n) -> (0 _ : IsLsb0 p) =>
Subset Nat (\n' => n = 2 * n')
fromLsb0 (Lsb0 n' eq) @{ItIsLsb0} = Element n' eq
export
doubleInj : {m, n : Nat} -> 2 * m = 2 * n -> m = n
fromLsb1 : (p : ViewLsb n) -> (0 _ : IsLsb1 p) =>
Subset Nat (\n' => n = S (2 * n'))
fromLsb1 (Lsb1 n' eq) @{ItIsLsb1} = Element n' eq
private
viewLsb' : (m, d : Nat) -> (0 _ : m `LT` 2) -> ViewLsb (m + 2 * d)
viewLsb' 0 d p = Lsb0 d Refl
viewLsb' 1 d p = Lsb1 d Refl
viewLsb' (S (S _)) _ (LTESucc p) = void $ absurd p
export
viewLsb : (n : Nat) -> ViewLsb n
viewLsb n =
let 0 nz : NonZero 2 = %search in
rewrite DivisionTheorem n 2 nz nz in
rewrite multCommutative (divNatNZ n 2 nz) 2 in
viewLsb' (modNatNZ n 2 nz) (divNatNZ n 2 nz) (boundModNatNZ n 2 nz)
export
0 doubleInj : {m, n : Nat} -> 2 * m = 2 * n -> m = n
doubleInj eq =
multRightCancel m n 2 %search $
trans (multCommutative m 2) $ trans eq (multCommutative 2 n)
export
0 sDoubleInj : {m, n : Nat} -> S (2 * m) = S (2 * n) -> m = n
sDoubleInj eq = doubleInj $ injective eq
export
0 lsbOdd : (n : Nat) -> (eq ** viewLsb (S (2 * n)) = Lsb1 n eq)
lsbOdd n with (viewLsb (S (2 * n)))
_ | Lsb0 _ eq = void $ absurd $ lsbMutex Refl (sym eq)
_ | Lsb1 n' eq with (sDoubleInj eq)
lsbOdd n | (Lsb1 n eq) | Refl = (eq ** Refl)
export
0 lsbEven : (n : Nat) -> (eq ** viewLsb (2 * n) = Lsb0 n eq)
lsbEven n with (viewLsb (2 * n))
_ | Lsb1 _ eq = void $ absurd $ lsbMutex Refl eq
_ | Lsb0 n' eq with (doubleInj eq)
lsbEven n | Lsb0 n eq | Refl = (eq ** Refl)

View file

@ -5,67 +5,124 @@ import Quox.OPE.Length
import Quox.OPE.Sub
import Data.DPair
import Control.Function
import Data.Nat
%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
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)
public export
Subscope : Scope a -> Type
record Comp {a : Type} {xs, ys, zs : Scope a}
(p : ys `Sub` zs) (q : xs `Sub` ys) (pq : xs `Sub` zs) where
constructor MkComp
0 comp : Comp' p.lte q.lte pq.lte
export
compPrf' : (p : ys `Sub'` zs) -> (q : xs `Sub'` ys) -> Comp' p q (p . q)
compPrf' End End = CEE
compPrf' (Keep p) (Keep q) = CKK $ compPrf' p q
compPrf' (Keep p) (Drop q) = CKD $ compPrf' p q
compPrf' (Drop p) q = CD0 $ compPrf' 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 %inline
compPrf : (0 sy : Length ys) => (0 sz : Length zs) =>
(p : ys `Sub` zs) -> (q : xs `Sub` ys) ->
Comp p q ((p . q) @{sy} @{sz})
compPrf p q = MkComp $
replace {p = Comp' p.lte q.lte} (sym $ compLte p q) $
compPrf' p.lte q.lte
export
compZero' : (sx : Length xs) => (sy : Length ys) =>
(p : xs `Sub'` ys) -> Comp' p (zero' @{sx}) (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 %inline
compZero : (sx : Length xs) => (sy : Length ys) =>
(p : xs `Sub` ys) -> Comp p (zero @{sx}) (zero @{sy})
compZero p = MkComp $
rewrite zeroLte {sx} in
rewrite zeroLte {sx = sy} in
compZero' {}
export
compIdLeft' : (sy : Length ys) =>
(p : xs `Sub'` ys) -> Comp' (refl' @{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 %inline
compIdLeft : (sy : Length ys) =>
(p : xs `Sub` ys) -> Comp (refl @{sy}) p p
compIdLeft {sy} p = MkComp $
rewrite reflLte {sx = sy} in compIdLeft' {}
export
compIdRight' : (sx : Length xs) =>
(p : xs `Sub'` ys) -> Comp' p (refl' @{sx}) p
compIdRight' {sx = Z} End = CEE
compIdRight' {sx = S _} (Keep p) = CKK (compIdRight' p)
compIdRight' (Drop p) = CD0 (compIdRight' p)
export %inline
compIdRight : (sx : Length xs) =>
(p : xs `Sub` ys) -> Comp p (refl @{sx}) p
compIdRight {sx} p = MkComp $ rewrite reflLte {sx} in compIdRight' {}
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
export %inline
0 compAssoc : (sx : Length xs) => (sy : Length ys) => (sz : Length zs) =>
(p : ys `Sub` zs) -> (q : xs `Sub` ys) -> (r : ws `Sub` xs) ->
comp @{sy} @{sz} p (comp @{sx} @{sy} q r) =
comp @{sx} @{sz} (comp @{sy} @{sz} p q) r
compAssoc p q r = lteEq $
trans (transLte {}) $
trans (cong (p.lte .) (transLte {})) $
sym $
trans (transLte {}) $
trans (cong (. r.lte) (transLte {})) $
sym $ compAssoc' {}
public export
0 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
public export
record SubMap {a : Type} {xs, ys, zs : Scope a}
(p : xs `Sub` zs) (q : ys `Sub` zs) where
@ -73,10 +130,23 @@ record SubMap {a : Type} {xs, ys, zs : Scope a}
thin : xs `Sub` ys
0 comp : Comp q thin p
parameters (p : xs `Sub` ys)
export
0 submap' : SubMap p q -> SubMap' p.lte q.lte
submap' (SM thin comp) = SM' {thin = thin.lte, comp = comp.comp}
parameters (p : xs `Sub'` ys)
export
subId' : SubMap' p p
subId' = SM' refl' (compIdRight' p)
export
subZero' : SubMap' Sub.zero' p
subZero' = SM' zero' (compZero' p)
parameters {auto sx : Length xs} (p : xs `Sub` ys)
export
subId : SubMap p p
subId = SM id (compIdRight p)
subId = SM refl (compIdRight p)
export
subZero : SubMap Sub.zero p

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@ -8,17 +8,83 @@ import Quox.OPE.Sub
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)
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)
parameters (p : xs `Sub'` zs) (q : ys `Sub'` zs)
public export
0 Cover' : Type
Cover' = Cover'_ True p q
public export
0 Partition' : Type
Partition' = Cover'_ False p q
public export
Cover : xs `Sub` zs -> ys `Sub` zs -> Type
Cover = Cover_ True
record Cover_ {a : Type} {xs, ys, zs : Scope a} (overlap : Bool)
(p : xs `Sub` zs) (q : ys `Sub` zs) where
constructor MkCover
0 cover : Cover'_ overlap p.lte q.lte
public export
Partition : xs `Sub` zs -> ys `Sub` zs -> Type
Partition = Cover_ False
parameters (p : xs `Sub` zs) (q : ys `Sub` zs)
public export
0 Cover : Type
Cover = Cover_ True p q
public export
0 Partition : Type
Partition = Cover_ False p q
private %inline
covCast' : {0 p, p' : xs `Sub'` zs} -> {0 q, q' : ys `Sub'` zs} ->
(0 pe : p = p') -> (0 qe : q = q') ->
Cover'_ overlap p' q' -> Cover'_ overlap p q
covCast' pe qe c = replace {p = id} (sym $ cong2 (Cover'_ overlap) pe qe) c
parameters {0 overlap : Bool} {0 p : xs `Sub` zs} {0 q : ys `Sub` zs}
export %inline
left : Cover_ overlap p q -> Cover_ overlap (keep p) (drop q)
left (MkCover cover) = MkCover $ covCast' (keepLte p) (dropLte q) $ CL cover
export %inline
right : Cover_ overlap p q -> Cover_ overlap (drop p) (keep q)
right (MkCover cover) = MkCover $ covCast' (dropLte p) (keepLte q) $ CR cover
export %inline
both : Cover_ overlap p q -> Cover (keep p) (keep q)
both (MkCover cover) = MkCover $ covCast' (keepLte p) (keepLte q) $ C2 cover
export
allLeft' : (sx : Length xs) => Partition' (refl' @{sx}) (zero' @{sx})
allLeft' {sx = Z} = CE
allLeft' {sx = S s} = CL allLeft'
export
allRight' : (sx : Length xs) => Partition' (zero' @{sx}) (refl' @{sx})
allRight' {sx = Z} = CE
allRight' {sx = S s} = CR allRight'
export
both' : (sx : Length xs) => Cover' (refl' @{sx}) (refl' @{sx})
both' {sx = Z} = CE
both' {sx = S s} = C2 both'
export %inline
allLeft : (sx : Length xs) => Partition (refl @{sx}) (zero @{sx})
allLeft = MkCover $ rewrite reflLte {sx} in rewrite zeroLte {sx} in allLeft'
export %inline
allRight : (sx : Length xs) => Partition (zero @{sx}) (refl @{sx})
allRight = MkCover $ rewrite reflLte {sx} in rewrite zeroLte {sx} in allRight'
export %inline
allBoth : (sx : Length xs) => Cover (refl @{sx}) (refl @{sx})
allBoth = MkCover $ rewrite reflLte {sx} in both'

View file

@ -27,3 +27,13 @@ public export %hint
toLength : (xs : Scope a) -> Length xs
toLength [<] = Z
toLength (sx :< x) = S (toLength sx)
public export %hint
lengthApp : Length xs -> Length ys -> Length (xs ++ ys)
lengthApp sx Z = sx
lengthApp sx (S sy) = S (lengthApp sx sy)
public export
0 lengthIrrel : (sx1, sx2 : Length xs) -> sx1 = sx2
lengthIrrel Z Z = Refl
lengthIrrel (S sx1) (S sx2) = cong S $ lengthIrrel sx1 sx2

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@ -17,11 +17,21 @@ record Split {a : Type} (xs, ys, zs : Scope a) (p : xs `Sub` ys ++ zs) where
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
split : Length ys =>
(zs : Scope a) -> (p : xs `Sub` ys ++ zs) -> Split xs ys zs p
split [<] p = MkSplit p zero Refl (endRight p)
split (zs :< z) p @{ys} with (p.view @{S (lengthApp ys %search)})
split (zs :< z) (SubM (S (2 * n)) (Keep p) v0) | (KEEP v Refl) =
case split zs (sub v) of
MkSplit l r Refl t =>
MkSplit l (keep r) Refl $
rewrite viewIrrel v0 (KEEP v Refl) in
trans (cong keep {a = sub v} t) $
sym $ keepAppRight l r
split (zs :< z) (SubM (2 * n) (Drop p) v0) | (DROP v Refl) =
case split zs (sub v) of
MkSplit l r Refl t =>
MkSplit l (drop r) Refl $
rewrite viewIrrel v0 (DROP v Refl) in
trans (cong drop {a = sub v} t) $
sym $ dropAppRight l r

View file

@ -9,6 +9,8 @@ import Data.SnocList.Quantifiers
%default total
infix 0 `Sub'`, `Sub`
public export
data Sub' : Scope a -> Scope a -> Type where
End : [<] `Sub'` [<]
@ -24,7 +26,6 @@ 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
@ -67,9 +68,9 @@ lteNilRight End = Refl
public export
id' : Length xs => xs `Sub'` xs
id' @{Z} = End
id' @{S s} = Keep id'
refl' : Length xs => xs `Sub'` xs
refl' @{Z} = End
refl' @{S s} = Keep refl'
public export
zero' : Length xs => [<] `Sub'` xs
@ -82,12 +83,21 @@ single' @{S _} Here = Keep zero'
single' @{S _} (There p) = Drop $ single' p
namespace Sub
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
trans' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` zs
trans' End End = End
trans' (Keep p) (Keep q) = Keep $ trans' p q
trans' (Keep p) (Drop q) = Drop $ trans' p q
trans' (Drop p) q = Drop $ trans' p q
comp' : ys `Sub'` zs -> xs `Sub'` ys -> xs `Sub'` zs
comp' = (.)
public export %inline
trans' : xs `Sub'` ys -> ys `Sub'` zs -> xs `Sub'` zs
trans' = flip (.)
public export
app' : xs1 `Sub'` ys1 -> xs2 `Sub'` ys2 -> (xs1 ++ xs2) `Sub'` (ys1 ++ ys2)
@ -116,11 +126,13 @@ subAny' (Drop q) x = There (subAny' q x)
public export
data SubView : (lte : xs `Sub'` ys) -> (mask : Nat) -> Type where
[search lte]
END : SubView End 0
END : (0 eq : n = 0) -> SubView End n
KEEP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
(0 v : SubView p n) -> SubView (Keep {z} p) (S (2 * n))
(0 v : SubView p n) -> (0 eq : n' = S (2 * n)) ->
SubView (Keep {z} p) n'
DROP : {n : Nat} -> {0 p : xs `Sub'` ys} ->
(0 v : SubView p n) -> SubView (Drop {z} p) (2 * n)
(0 v : SubView p n) -> (0 eq : n' = 2 * n) ->
SubView (Drop {z} p) n'
%name SubView v
public export
@ -129,162 +141,214 @@ record Sub {a : Type} (xs, ys : Scope a) where
mask : Nat
0 lte : xs `Sub'` ys
0 view0 : SubView lte mask
%name Sub m
%name Sub sub
public export %inline
sub : {mask : Nat} -> {0 lte : xs `Sub'` ys} -> (0 view0 : SubView lte mask) ->
xs `Sub` ys
sub v = SubM _ _ v
private
0 uip : (p, q : a = b) -> p = q
uip Refl Refl = Refl
export
0 viewIrrel : Length ys =>
{m : Nat} -> {lte : xs `Sub'` ys} ->
(v1, v2 : SubView lte m) -> v1 === v2
viewIrrel (END eq1) (END eq2) = rewrite uip eq1 eq2 in Refl
viewIrrel (KEEP v1 eq1) (KEEP v2 eq2) with (sDoubleInj $ trans (sym eq1) eq2)
_ | Refl = rewrite viewIrrel v1 v2 in rewrite uip eq1 eq2 in Refl
viewIrrel (DROP v1 eq1) (DROP v2 eq2) with (doubleInj $ trans (sym eq1) eq2)
_ | Refl = rewrite viewIrrel v1 v2 in rewrite uip eq1 eq2 in Refl
export
0 viewLteEq : {p, q : xs `Sub'` ys} -> SubView p m -> SubView q m -> p = q
viewLteEq (END _) (END _) = Refl
viewLteEq (KEEP {n = m} pv pe) (KEEP {n} qv qe) =
let eq = sDoubleInj {m, n} $ trans (sym pe) qe in
rewrite viewLteEq pv (rewrite eq in qv) in Refl
viewLteEq (KEEP pv pe) (DROP qv qe) = absurd $ lsbMutex qe pe
viewLteEq (DROP pv pe) (KEEP qv qe) = absurd $ lsbMutex pe qe
viewLteEq (DROP {n = m} pv pe) (DROP {n} qv qe) =
let eq = doubleInj {m, n} $ trans (sym pe) qe in
rewrite viewLteEq pv (rewrite eq in qv) in Refl
private
0 lteEq' : (p : xs `Sub'` ys) ->
(v1 : SubView p m1) -> (v2 : SubView p m2) ->
SubM m1 p v1 = SubM m2 p v2
lteEq' End (END Refl) (END Refl) = Refl
lteEq' {m1 = S (2 * n)} {m2 = S (2 * n)} (Keep p) (KEEP v1 Refl) (KEEP v2 Refl) =
cong (\v => SubM (S (2 * n)) (Keep p) (KEEP v Refl)) $
case lteEq' p v1 v2 of Refl => Refl
lteEq' {m1 = 2 * n} {m2 = 2 * n} (Drop p) (DROP v1 Refl) (DROP v2 Refl) =
cong (\v => SubM (2 * n) (Drop p) (DROP v Refl)) $
case lteEq' p v1 v2 of Refl => Refl
lteEq' {m1 = _} {m2 = S _} _ _ (END _) impossible
lteEq' {m1 = Z} {m2 = _} _ (KEEP _ _) _ impossible
lteEq' {m1 = _} {m2 = Z} _ _ (KEEP _ _) impossible
export
0 lteEq : {p, q : xs `Sub` ys} -> p.lte = q.lte -> p = q
lteEq {p = SubM pm pl pv} {q = SubM qm ql qv} eq =
rewrite eq in lteEq' ql (rewrite sym eq in pv) qv
public export %inline
end : [<] `Sub` [<]
end = SubM 0 End (END Refl)
public export %inline
keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
keep (SubM mask lte view0) = SubM (S (2 * mask)) (Keep lte) (KEEP view0 Refl)
public export %inline
drop : xs `Sub` ys -> xs `Sub` ys :< z
drop (SubM mask lte view0) = SubM (2 * mask) (Drop lte) (DROP view0 Refl)
export
0 keepLte : (p : xs `Sub` ys) -> (keep p).lte = Keep p.lte
keepLte (SubM mask lte view0) = Refl
export
0 dropLte : (p : xs `Sub` ys) -> (drop p).lte = Drop p.lte
dropLte (SubM mask lte view0) = Refl
public export
data Eqv : {xs, ys : Scope a} -> (p, q : xs `Sub` ys) -> Type where
EQV : {0 p, q : xs `Sub` ys} -> p.mask = q.mask -> p `Eqv` q
%name Eqv eqv
export
Reflexive (xs `Sub` ys) Eqv where
reflexive = EQV Refl
export
Symmetric (xs `Sub` ys) Eqv where
symmetric (EQV eq) = EQV $ sym eq
export
Transitive (xs `Sub` ys) Eqv where
transitive (EQV eq1) (EQV eq2) = EQV $ trans eq1 eq2
export
0 eqvToEq : {p, q : xs `Sub` ys} -> p `Eqv` q -> p = q
eqvToEq {p = SubM _ _ pv} {q = SubM _ _ qv} (EQV Refl) with (viewLteEq pv qv)
_ | Refl = rewrite viewIrrel pv qv in Refl
export
0 eqToEqv : {p, q : xs `Sub` ys} -> p = q -> p `Eqv` q
eqToEqv Refl = reflexive
export
getMask : SubView lte mask -> Subset Nat (Equal mask)
private
0 ltemNilLeftZero' : SubView {xs = [<]} lte mask -> mask = 0
ltemNilLeftZero' END = Refl
ltemNilLeftZero' (DROP v) = cong (2 *) $ ltemNilLeftZero' v
export
ltemNilLeftZero : (0 _ : SubView {xs = [<]} lte mask) -> mask = 0
ltemNilLeftZero v = irrelevantEq $ ltemNilLeftZero' v
getMask (END Refl) = Element 0 Refl
getMask (KEEP v eq) = Element _ eq
getMask (DROP v eq) = Element _ eq
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)
splitExEq : {0 f, g : a -> b} ->
(0 p : (x : a ** f x = g x)) -> Exists (\x => f x = g x)
splitExEq p = Evidence p.fst (irr p.snd)
where irr : forall a, b. (0 eq : a = b) -> a = b
irr Refl = Refl
private
0 lteNil2End : (p : [<] `Sub'` [<]) -> p = End
lteNil2End End = Refl
private
0 ltemEnd' : SubView p n -> p = End -> n = 0
ltemEnd' END Refl = Refl
0 ltemEnd : SubView p n -> p = End -> n = 0
ltemEnd (END eq) Refl = eq
private
0 ltemEven' : {p : xs `Sub'` (ys :< y)} ->
n = 2 * n' -> SubView p n -> (q ** p = Drop q)
ltemEven' eq (KEEP q) = absurd $ lsbMutex' eq Refl
ltemEven' eq (DROP q) = (_ ** Refl)
0 ltemEvenDrop : {p : xs `Sub'` (ys :< y)} ->
n = 2 * n' -> SubView p n -> (q ** p = Drop q)
ltemEvenDrop eq (KEEP _ eq') = absurd $ lsbMutex eq eq'
ltemEvenDrop eq (DROP {}) = (_ ** Refl)
private
ltemEven : {0 p : xs `Sub'` (ys :< y)} ->
(0 _ : SubView 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' ->
SubView (Drop lte) n -> SubView lte n'
fromDROP' eq (DROP {n} p) =
0 fromDROP : {lte : xs `Sub'` ys :< z} -> n = 2 * n' -> lte = Drop lte' ->
SubView lte n -> SubView lte' n'
fromDROP eq Refl (DROP {n} p Refl) =
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} ->
0 ltemOddKeep : {p : (xs :< x) `Sub'` (ys :< x)} -> {n' : Nat} ->
n = S (2 * n') -> SubView p n -> (q ** p = Keep q)
ltemOdd' eq (KEEP q) = (_ ** Refl)
ltemOdd' eq (DROP q) = absurd $ lsbMutex' Refl eq
ltemOdd' eq END impossible
ltemOddKeep eq (KEEP {}) = (_ ** Refl)
ltemOddKeep eq (DROP _ eq') = absurd $ lsbMutex eq' eq
ltemOddKeep eq (END _) impossible
private
ltemOdd : (0 _ : SubView p (S (2 * n))) -> Exists (\q => p = Keep q)
ltemOdd q =
let 0 res = ltemOdd' Refl q in
Evidence res.fst (irrelevantEq res.snd)
0 ltemOddLeft : {0 n, n' : Nat} -> {lte : xs `Sub'` ys :< z} ->
n = S (2 * n') -> SubView lte n ->
(xs' ** xs = xs' :< z)
ltemOddLeft eq {lte = Keep p} _ = (_ ** Refl)
ltemOddLeft eq {lte = Drop p} (DROP n' eq') = void $ lsbMutex eq' eq
private
0 ltemOddHead' : {p : (xs :< x) `Sub'` (ys :< y)} -> {n' : Nat} ->
n = S (2 * n') -> SubView 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 _ : SubView p (S (2 * n))) -> x = y
ltemOddHead q = irrelevantEq $ ltemOddHead' Refl q
private
0 fromKEEP' : {lte : xs `Sub'` ys} -> n = S (2 * n') ->
SubView (Keep lte) n -> SubView lte n'
fromKEEP' eq (KEEP {n} p) =
let eq = doubleInj (injective eq) {m = n, n = n'} in
0 fromKEEP : {lte : xs :< z `Sub'` ys :< z} ->
n = S (2 * n') -> lte = Keep lte' ->
SubView lte n -> SubView lte' n'
fromKEEP eq Refl (KEEP {n} p Refl) =
let eq = sDoubleInj eq {m = n, n = n'} in
rewrite sym eq in p
export
view : Length xs => Length ys =>
(m : Sub xs ys) -> SubView 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 : Sub xs ys) -> SubView m.lte m.mask
view : Length ys => (m : Sub xs ys) -> SubView m.lte m.mask
view @{Z} (SubM mask lte view0) with 0 (lteNilRight lte)
_ | Refl =
rewrite lteNil2End lte in
rewrite ltemEnd view0 (lteNil2End lte) in
END Refl
view @{S sy} (SubM mask lte view0) {ys = ys :< z} with (viewLsb mask)
_ | Lsb0 n eqn with (splitExEq $ ltemEvenDrop eqn view0)
_ | Evidence lte' eql =
rewrite eqn in rewrite eql in
DROP (fromDROP eqn eql view0) Refl
_ | Lsb1 n eqn with (splitExEq $ ltemOddLeft eqn view0)
_ | Evidence xs' Refl =
let Evidence q eqq = splitExEq $ ltemOddKeep eqn view0 in
rewrite eqn in rewrite eqq in
KEEP (fromKEEP eqn eqq view0) Refl
public export %inline
(.view) : Length ys => (m : Sub xs ys) -> SubView m.lte m.mask
(.view) = view
public export %inline
review : Length ys => {m : Nat} -> {0 lte : xs `Sub'` ys} ->
(0 _ : SubView lte m) -> SubView lte m
review v = view $ sub v
export
ltemLen : Length xs => Length ys =>
xs `Sub` ys -> length xs `LTE` length ys
ltemLen @{sx} @{sy} sub@(SubM m l _) with (sub.view)
ltemLen @{sx} @{sy} sub@(SubM 0 End _) | END = LTEZero
ltemLen @{S sx} @{S sy} sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP q) =
LTESucc $ ltemLen $ SubM n p q
ltemLen @{sx} @{S sy} sub@(SubM (2 * n) (Drop p) _) | (DROP q) =
lteSuccRight $ ltemLen $ SubM n p q
ltemLen : (sy : Length ys) => xs `Sub` ys -> length xs `LTE` sy.nat
ltemLen (SubM mask lte view0) {sy} with (review view0)
ltemLen (SubM _ _ _) | END eq = LTEZero
ltemLen (SubM _ _ _) {sy = S _} | KEEP v eq = LTESucc $ ltemLen $ sub v
ltemLen (SubM _ _ _) {sy = S _} | DROP v eq = lteSuccRight $ ltemLen $ sub v
export
ltemNilRight : xs `Sub` [<] -> xs = [<]
ltemNilRight m = irrelevantEq $ lteNilRight m.lte
public export %inline
end : [<] `Sub` [<]
end = SubM 0 End END
public export %inline
keep : xs `Sub` ys -> xs :< z `Sub` ys :< z
keep (SubM mask lte view0) = SubM (S (2 * mask)) (Keep lte) (KEEP view0)
public export %inline
drop : xs `Sub` ys -> xs `Sub` ys :< z
drop (SubM mask lte view0) = SubM (2 * mask) (Drop lte) (DROP view0)
export %inline
dropLast : Length xs => Length ys =>
(xs :< x) `Sub` ys -> xs `Sub` ys
dropLast @{sx} @{sy} sub@(SubM mask lte _) with (sub.view)
dropLast sub@(SubM (S (2 * n)) (Keep p) _) | (KEEP v) =
SubM (2 * n) (Drop p) (DROP v)
dropLast @{_} @{S sy} sub@(SubM (2 * n) (Drop p) _) | DROP v =
drop $ dropLast $ SubM n p v
dropLast : Length ys => (xs :< x) `Sub` ys -> xs `Sub` ys
dropLast (SubM _ _ view0) @{sy} with (review view0)
dropLast (SubM _ _ _) | KEEP v _ = drop $ sub v
dropLast (SubM _ _ _) @{S _} | DROP v _ = drop $ dropLast $ sub v
export
@ -293,11 +357,9 @@ Uninhabited (xs :< x `Sub` [<]) where
export
Length xs => Uninhabited (xs :< x `Sub` xs) where
uninhabited @{sx} sub@(SubM mask lte view0) with (sub.view)
uninhabited @{S sx} sub@(SubM (S (2 * n)) (Keep p) _) | KEEP v =
uninhabited $ SubM n p v
uninhabited @{S sx} sub@(SubM (2 * n) (Drop p) _) | DROP v =
uninhabited $ dropLast $ SubM n p v
uninhabited (SubM _ _ view0) @{sx} with (review view0)
uninhabited (SubM _ _ _) @{S _} | KEEP v _ = uninhabited $ sub v
uninhabited (SubM _ _ _) @{S _} | DROP v _ = uninhabited $ dropLast $ sub v
export
@ -305,103 +367,206 @@ refl : Length xs => xs `Sub` xs
refl @{Z} = end
refl @{S s} = keep refl
export Reflexive (Scope a) Sub where reflexive = refl
export
0 reflLte : {sx : Length xs} -> (refl @{sx}).lte = refl' @{sx}
reflLte {sx = Z} = Refl
reflLte {sx = S s} = trans (keepLte _) $ cong Keep reflLte
public export %inline
Reflexive (Scope a) Sub where reflexive = refl
mutual
private
antisym_ : Length xs => Length ys =>
{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` xs} ->
SubView p m1 -> SubView q m2 -> xs = ys
antisym_ END END = Refl
antisym_ (KEEP v1) (KEEP v2 {z}) @{S sx} @{S sy} =
antisym_ (END _) (END _) = Refl
antisym_ (KEEP v1 _) (KEEP v2 _ {z}) @{S sx} @{S sy} =
cong (:< z) $ antisym (SubM _ _ v1) (SubM _ _ v2)
antisym_ (KEEP v1) (DROP v2) {p = Keep p} {q = Drop q} =
antisym_ (KEEP {}) (DROP {}) {p = Keep p} {q = Drop q} =
void $ succNotLTEpred $ lteLen q `transitive` lteLen p
antisym_ (DROP v1) (KEEP v2) {p = Drop p} {q = Keep q} =
antisym_ (DROP {}) (KEEP {}) {p = Drop p} {q = Keep q} =
void $ succNotLTEpred $ lteLen p `transitive` lteLen q
antisym_ (DROP v1) (DROP v2) {p = Drop p} {q = Drop q} =
antisym_ (DROP {}) (DROP {}) {p = Drop p} {q = Drop q} =
void $ succNotLTEpred $ lteLen p `transitive` lteSuccLeft (lteLen q)
export
antisym : Length xs => Length ys => xs `Sub` ys -> ys `Sub` xs -> xs = ys
antisym p q = antisym_ p.view q.view
antisym v1 v2 = antisym_ v1.view v2.view
export
public export %inline
Antisymmetric (Scope a) Sub where
antisymmetric p q = antisym p q
mutual
private
trans_ : Length xs => Length ys => Length zs =>
trans_ : Length ys => Length zs =>
{0 p : xs `Sub'` ys} -> {0 q : ys `Sub'` zs} ->
SubView p m1 -> SubView q m2 -> xs `Sub` zs
trans_ END END = end
trans_ (KEEP v1) (KEEP v2) @{S sx} @{S sy} @{S sz} =
keep $ SubM _ _ v1 `trans` SubM _ _ v2
trans_ (DROP v1) (KEEP v2) @{sx} @{S sy} @{S sz} =
drop $ SubM _ _ v1 `trans` SubM _ _ v2
trans_ v1 (DROP v2) @{sx} @{sy} @{S sz} =
trans_ (END _) (END _) = end
trans_ (KEEP v1 _) (KEEP v2 _) @{S sy} @{S sz} = keep $ sub v1 `trans` sub v2
trans_ (DROP v1 _) (KEEP v2 _) @{S sy} @{S sz} = drop $ sub v1 `trans` sub v2
trans_ v1 (DROP v2 _) @{sy} @{S sz} =
let Element m1' eq = getMask v1 in
drop $ SubM m1' _ (rewrite sym eq in v1) `trans` SubM _ _ v2
drop $ SubM m1' _ (rewrite sym eq in v1) `trans` sub v2
export
trans : Length xs => Length ys => Length zs =>
xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
trans : Length ys => Length zs => xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
trans p q = trans_ p.view q.view
export
(.) : Length xs => Length ys => Length zs =>
xs `Sub` ys -> ys `Sub` zs -> xs `Sub` zs
(.) = trans
public export %inline
(.) : Length ys => Length zs => ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
(.) = flip trans
export
public export %inline
comp : Length ys => Length zs => ys `Sub` zs -> xs `Sub` ys -> xs `Sub` zs
comp = (.)
public export %inline
Transitive (Scope a) Sub where
transitive p q = trans p q
transitive p q = q . p
export
public export
zero : Length xs => [<] `Sub` xs
zero @{Z} = end
zero @{S s} = drop zero
export
0 zeroLte : {sx : Length xs} -> (zero @{sx}).lte = zero' @{sx}
zeroLte {sx = Z} = Refl
zeroLte {sx = S s} = trans (dropLte zero) $ cong Drop zeroLte
public export
single : Length xs => x `Elem` xs -> [< x] `Sub` xs
single @{S sx} Here = keep zero
single @{S sx} (There p) = drop $ single p
export
0 singleLte : {sx : Length xs} -> (p : x `Elem` xs) ->
(single p @{sx}).lte = single' p @{sx}
singleLte {sx = S s} Here = trans (keepLte zero) $ cong Keep zeroLte
singleLte {sx = S s} (There p) = trans (dropLte _) $ cong Drop $ singleLte p
mutual
private
app0 : Length ys2 =>
(l : xs1 `Sub` ys1) ->
{rm : Nat} -> {0 rl : xs2 `Sub'` ys2} -> (rv : SubView rl rm) ->
(xs1 ++ xs2) `Sub` (ys1 ++ ys2)
app0 l (END eq) = l
app0 l (KEEP v Refl) @{S sy} = keep $ l ++ sub v
app0 l (DROP v Refl) @{S sy} = drop $ l ++ sub v
export
app : Length ys2 =>
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
app l r = app0 l r.view
public export %inline
(++) : Length ys2 =>
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
(++) = app
export
(++) : Length xs2 => Length ys2 =>
xs1 `Sub` ys1 -> xs2 `Sub` ys2 -> (xs1 ++ xs2) `Sub` (ys1 ++ ys2)
(++) sub1 sub2@(SubM {}) @{sx2} @{sy2} with (sub2.view)
(++) sub1 sub2@(SubM {}) | END = sub1
(++) sub1 sub2@(SubM {}) @{S sx2} @{S sy2} | KEEP v = keep $ sub1 ++ SubM _ _ v
(++) sub1 sub2@(SubM {}) @{sx2} @{S sy2} | DROP v = drop $ sub1 ++ SubM _ _ v
0 appLte : {sy : Length ys2} ->
(l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
(app l r @{sy}).lte = app' l.lte r.lte
appLte l r@(SubM 0 End (END Refl)) =
cong (\v => (app0 l v).lte) (viewIrrel (view end) (END Refl))
appLte l r@(SubM (S (2 * n)) (Keep p) (KEEP v Refl)) {sy = S sy} =
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (KEEP v Refl))) $
trans (keepLte _) $
cong Keep $ appLte l (sub v)
appLte l r@(SubM 0 (Drop p) (DROP {n = 0} v Refl)) {sy = S sy} =
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (DROP v Refl))) $
trans (dropLte _) $
cong Drop $ appLte l (sub v)
appLte l r@(SubM (2 * S n) (Drop p) (DROP {n = S n} v Refl)) {sy = S sy} =
trans (cong (\v => (app0 l v @{S sy}).lte) (viewIrrel _ (DROP v Refl))) $
trans (dropLte _) $
cong Drop $ appLte l (sub v)
appLte {xs2 = _ :< _, ys2 = [<]} _ (SubM _ _ _) impossible
export
0 keepAppRight : {sy : Length ys2} ->
(l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
(l ++ keep r) @{S sy} = keep ((l ++ r) @{sy})
keepAppRight l (SubM mask _ _) = rewrite (lsbOdd mask).snd in Refl
export
0 dropAppRight : (l : xs1 `Sub` ys1) -> (r : xs2 `Sub` ys2) ->
(l ++ drop r) @{S sy} = drop ((l ++ r) @{sy})
dropAppRight l r = lteEq $
trans (appLte l (drop r)) $
trans (cong (app' l.lte) (dropLte r)) $
trans (cong Drop (sym (appLte l r))) $
sym $ dropLte (l ++ r)
export
0 endRight : {xs, ys : Scope a} -> (sub : xs `Sub` ys) -> sub = (sub ++ Sub.end)
endRight sub = Refl
private
0 transDrop_ : {p : xs `Sub'` ys} -> {q : ys `Sub'` zs} ->
(pv : SubView p m1) -> (qv : SubView q m2) ->
trans_ pv (DROP qv qe) @{sy} @{S sz} =
drop (trans_ pv qv @{sy} @{sz})
transDrop_ {sy = Z} {sz = sz} (END Refl) qv =
cong (\v => drop $ trans_ {m2, q} (END Refl) v) $ viewIrrel {}
transDrop_ {sy = S sy} {sz = sz} (KEEP pv Refl) qv {p = Keep p} =
cong2 (\x, y => drop (trans_ @{S sy} x y)) (viewIrrel {}) (viewIrrel {})
transDrop_ {sy = S sy} {sz = S sz} (DROP pv Refl) qv =
cong2 (\x, y => drop (trans_ @{S sy} @{S sz} x y))
(viewIrrel {}) (viewIrrel {})
transDrop_ {ys = _ :< _, zs = [<]} _ _ impossible
private
0 transLte_ : {p : xs `Sub'` ys} -> {q : ys `Sub'` zs} ->
(pv : SubView p m1) -> (qv : SubView q m2) ->
(trans_ pv qv @{sy} @{sz}).lte = trans' p q
transLte_ (END _) (END _) = Refl
transLte_ (KEEP pv pe) (KEEP qv qe) {sy = S _} {sz = S _} =
trans (keepLte _) $ cong Keep $ transLte_ {}
transLte_ (DROP pv pe) (KEEP qv qe) {sy = S _} {sz = S _} =
trans (dropLte _) $ cong Drop $ transLte_ {}
transLte_ pv (DROP qv qe) {sz = S _} =
trans (cong lte $ transDrop_ pv qv) $
trans (dropLte _) $ cong Drop $ transLte_ pv qv
transLte_ {ys = _ :< _, zs = [<]} _ _ impossible
export
0 transLte : {sy : Length ys} -> {sz : Length zs} ->
(p : xs `Sub` ys) -> (q : ys `Sub` zs) ->
(trans p q @{sy} @{sz}).lte = trans' p.lte q.lte
transLte p q = transLte_ p.view q.view
public export
0 compLte : {sy : Length ys} -> {sz : Length zs} ->
(q : ys `Sub` zs) -> (p : xs `Sub` ys) ->
((q . p) @{sy} @{sz}).lte = q.lte . p.lte
compLte q p = transLte p q
||| if `p` holds for all elements of the main list,
||| it holds for all elements of the sublist
export
subAll : Length xs => Length ys =>
xs `Sub` ys -> All prop ys -> All prop xs
subAll sub@(SubM {}) ps @{sx} @{sy} with (sub.view)
subAll sub@(SubM {}) [<] | END = [<]
subAll sub@(SubM {}) (ps :< p) @{S sx} @{S sy} | KEEP v =
subAll (SubM _ _ v) ps :< p
subAll sub@(SubM {}) (ps :< p) @{sx} @{S sy} | DROP v =
subAll (SubM _ _ v) ps
subAll : Length ys => xs `Sub` ys -> All prop ys -> All prop xs
subAll (SubM _ _ v) ps @{sy} with (review v)
subAll (SubM _ _ _) [<] | END _ = [<]
subAll (SubM _ _ _) (ps :< p) @{S sy} | KEEP v _ = subAll (sub v) ps :< p
subAll (SubM _ _ _) (ps :< p) @{S sy} | DROP v _ = subAll (sub v) ps
||| if `p` holds for one element of the sublist,
||| it holds for one element of the main list
export
subAny : Length xs => Length ys =>
xs `Sub` ys -> Any prop xs -> Any prop ys
subAny sub@(SubM {}) p @{sx} @{sy} with (sub.view)
subAny sub@(SubM {}) p | END impossible
subAny sub@(SubM {}) (Here p) | KEEP v = Here p
subAny sub@(SubM {}) (There p) @{S sx} @{S sy} | KEEP v =
There $ subAny (SubM _ _ v) p
subAny sub@(SubM {}) p @{sx} @{S sy} | DROP v =
There $ subAny (SubM _ _ v) p
subAny : Length ys => xs `Sub` ys -> Any prop xs -> Any prop ys
subAny (SubM _ _ v) p @{sy} with (review v)
subAny (SubM _ _ _) (Here p) | KEEP v _ = Here p
subAny (SubM _ _ _) (There p) @{S sy} | KEEP v _ = There $ subAny (sub v) p
subAny (SubM _ _ _) p @{S sy} | DROP v _ = There $ subAny (sub v) p