WIP: co-de bruijn #12
3 changed files with 418 additions and 0 deletions
|
|
@ -1,9 +1,11 @@
|
|||
module Quox.NatExtra
|
||||
|
||||
import public Data.Nat
|
||||
import public Data.Nat.Views
|
||||
import Data.Nat.Division
|
||||
import Data.SnocList
|
||||
import Data.Vect
|
||||
import Syntax.PreorderReasoning
|
||||
|
||||
%default total
|
||||
|
||||
|
|
@ -55,3 +57,30 @@ parameters {base : Nat} {auto 0 _ : base `GTE` 2} (chars : Vect base Char)
|
|||
export
|
||||
showHex : Nat -> String
|
||||
showHex = showAtBase $ fromList $ unpack "0123456789ABCDEF"
|
||||
|
||||
|
||||
export
|
||||
0 notEvenOdd : (a, b : Nat) -> Not (a + a = S (b + b))
|
||||
notEvenOdd 0 b prf = absurd prf
|
||||
notEvenOdd (S a) b prf =
|
||||
notEvenOdd b a $ Calc $
|
||||
|~ b + b
|
||||
~~ a + S a ..<(inj S prf)
|
||||
~~ S (a + a) ..<(plusSuccRightSucc {})
|
||||
|
||||
export
|
||||
0 doubleInj : (m, n : Nat) -> m + m = n + n -> m = n
|
||||
doubleInj 0 0 _ = Refl
|
||||
doubleInj (S m) (S n) prf =
|
||||
cong S $ doubleInj m n $
|
||||
inj S $ Calc $
|
||||
|~ S (m + m)
|
||||
~~ m + S m ...(plusSuccRightSucc {})
|
||||
~~ n + S n ...(inj S prf)
|
||||
~~ S (n + n) ..<(plusSuccRightSucc {})
|
||||
|
||||
export
|
||||
0 halfDouble : (n : Nat) -> half (n + n) = HalfEven n
|
||||
halfDouble n with (half (n + n)) | (n + n) proof nn
|
||||
_ | HalfOdd k | S (k + k) = void $ notEvenOdd n k nn
|
||||
_ | HalfEven k | k + k = rewrite doubleInj n k nn in Refl
|
||||
|
|
|
|||
388
lib/Quox/Thin.idr
Normal file
388
lib/Quox/Thin.idr
Normal file
|
|
@ -0,0 +1,388 @@
|
|||
||| thinnings, covers, partitions, etc,
|
||||
||| for co-de Bruijn representation [@egtbs]
|
||||
module Quox.Thin
|
||||
|
||||
import Quox.NatExtra
|
||||
import Data.Nat.Views
|
||||
import Data.Singleton
|
||||
import Data.DPair
|
||||
import Syntax.PreorderReasoning
|
||||
|
||||
%default total
|
||||
|
||||
|
||||
||| "order preserving embeddings", for recording a correspondence between a
|
||||
||| smaller scope and part of a larger one. the third argument is a bitmask
|
||||
||| representing this OPE, unique for a given `n`.
|
||||
public export
|
||||
data OPE : (m, n, mask : Nat) -> Type where
|
||||
[search m n]
|
||||
Stop : OPE 0 0 0
|
||||
Drop : OPE m n mask -> mask' = mask + mask -> OPE m (S n) mask'
|
||||
Keep : OPE m n mask -> mask' = (S (mask + mask)) -> OPE (S m) (S n) mask'
|
||||
%name OPE ope
|
||||
|
||||
|
||||
||| everything selected
|
||||
public export
|
||||
id : {m : Nat} -> Subset Nat (OPE m m)
|
||||
id {m = 0} = Element _ Stop
|
||||
id {m = S m} = Element _ $ Keep id.snd Refl
|
||||
|
||||
||| nothing selected
|
||||
public export
|
||||
zero : {m : Nat} -> OPE 0 m 0
|
||||
zero {m = 0} = Stop
|
||||
zero {m = S m} = Drop zero Refl
|
||||
|
||||
|
||||
infix 6 `Eqv`
|
||||
|
||||
public export
|
||||
data Eqv : OPE m1 n1 mask1 -> OPE m2 n2 mask2 -> Type where
|
||||
EqvStop : Eqv Stop Stop
|
||||
EqvDrop : {0 p : OPE m1 n1 mask1} ->
|
||||
{0 q : OPE m2 n2 mask2} ->
|
||||
Eqv p q -> Eqv (Drop p eq1) (Drop q eq2)
|
||||
EqvKeep : {0 p : OPE m1 n1 mask1} ->
|
||||
{0 q : OPE m2 n2 mask2} ->
|
||||
Eqv p q -> Eqv (Keep p eq1) (Keep q eq2)
|
||||
%name Eqv eqv
|
||||
|
||||
export
|
||||
Reflexive (OPE m n mask) Eqv where
|
||||
reflexive {x = Stop} = EqvStop
|
||||
reflexive {x = Drop {}} = EqvDrop reflexive
|
||||
reflexive {x = Keep {}} = EqvKeep reflexive
|
||||
|
||||
export
|
||||
symmetric : p `Eqv` q -> q `Eqv` p
|
||||
symmetric EqvStop = EqvStop
|
||||
symmetric (EqvDrop eqv) = EqvDrop $ symmetric eqv
|
||||
symmetric (EqvKeep eqv) = EqvKeep $ symmetric eqv
|
||||
|
||||
export
|
||||
transitive : p `Eqv` q -> q `Eqv` r -> p `Eqv` r
|
||||
transitive EqvStop EqvStop = EqvStop
|
||||
transitive (EqvDrop x) (EqvDrop y) = EqvDrop (transitive x y)
|
||||
transitive (EqvKeep x) (EqvKeep y) = EqvKeep (transitive x y)
|
||||
|
||||
export
|
||||
eqvIndices : {0 p : OPE m1 n1 mask1} -> {0 q : OPE m2 n2 mask2} ->
|
||||
p `Eqv` q -> (m1 = m2, n1 = n2, mask1 = mask2)
|
||||
eqvIndices EqvStop = (Refl, Refl, Refl)
|
||||
eqvIndices (EqvDrop eqv {eq1 = Refl, eq2 = Refl}) =
|
||||
let (Refl, Refl, Refl) = eqvIndices eqv in (Refl, Refl, Refl)
|
||||
eqvIndices (EqvKeep eqv {eq1 = Refl, eq2 = Refl}) =
|
||||
let (Refl, Refl, Refl) = eqvIndices eqv in (Refl, Refl, Refl)
|
||||
|
||||
export
|
||||
0 eqvMask : (p : OPE m1 n mask1) -> (q : OPE m2 n mask2) ->
|
||||
mask1 = mask2 -> p `Eqv` q
|
||||
eqvMask Stop Stop _ = EqvStop
|
||||
eqvMask (Drop ope1 Refl) (Drop {mask = mm2} ope2 eq2) Refl =
|
||||
EqvDrop $ eqvMask ope1 ope2 (doubleInj _ _ eq2)
|
||||
eqvMask (Drop ope1 Refl) (Keep ope2 eq2) Refl =
|
||||
void $ notEvenOdd _ _ eq2
|
||||
eqvMask (Keep ope1 eq1) (Keep ope2 eq2) Refl =
|
||||
EqvKeep $ eqvMask ope1 ope2 (doubleInj _ _ $ inj S $ trans (sym eq1) eq2)
|
||||
eqvMask (Keep ope1 eq1) (Drop ope2 eq2) Refl =
|
||||
void $ notEvenOdd _ _ $ trans (sym eq2) eq1
|
||||
|
||||
uip : (p, q : a = b) -> p = q
|
||||
uip Refl Refl = Refl
|
||||
|
||||
export
|
||||
0 eqvEq' : (p, q : OPE m n mask) -> p `Eqv` q -> p === q
|
||||
eqvEq' Stop Stop EqvStop = Refl
|
||||
eqvEq' (Drop p eq1) (Drop q eq2) (EqvDrop eqv)
|
||||
with (doubleInj _ _ $ trans (sym eq1) eq2)
|
||||
_ | Refl = cong2 Drop (eqvEq' p q eqv) (uip eq1 eq2)
|
||||
eqvEq' (Keep p eq1) (Keep q eq2) (EqvKeep eqv)
|
||||
with (doubleInj _ _ $ inj S $ trans (sym eq1) eq2)
|
||||
_ | Refl = cong2 Keep (eqvEq' p q eqv) (uip eq1 eq2)
|
||||
|
||||
|
||||
export
|
||||
0 eqvEq : (p : OPE m1 n1 mask1) -> (q : OPE m2 n2 mask2) ->
|
||||
p `Eqv` q -> p ~=~ q
|
||||
eqvEq p q eqv = let (Refl, Refl, Refl) = eqvIndices eqv in eqvEq' p q eqv
|
||||
|
||||
|
||||
public export
|
||||
data View : OPE m n mask -> Type where
|
||||
StopV : View Stop
|
||||
DropV : (mask : Nat) -> (0 ope : OPE m n mask) -> View (Drop ope Refl)
|
||||
KeepV : (mask : Nat) -> (0 ope : OPE m n mask) -> View (Keep ope Refl)
|
||||
%name Thin.View v
|
||||
|
||||
private
|
||||
0 stopEqs : OPE m 0 mask -> (m = 0, mask = 0)
|
||||
stopEqs Stop = (Refl, Refl)
|
||||
|
||||
private
|
||||
0 fromStop : (ope : OPE 0 0 0) -> ope = Stop
|
||||
fromStop Stop = Refl
|
||||
|
||||
private
|
||||
0 fromDrop : (ope : OPE m (S n) (k + k)) ->
|
||||
(inner : OPE m n k ** ope === Drop inner Refl)
|
||||
fromDrop (Drop ope eq) with (doubleInj _ _ eq)
|
||||
fromDrop (Drop ope Refl) | Refl = (ope ** Refl)
|
||||
fromDrop (Keep ope eq) = void $ notEvenOdd _ _ eq
|
||||
|
||||
private
|
||||
0 fromKeep : (ope : OPE (S m) (S n) (S (k + k))) ->
|
||||
(inner : OPE m n k ** ope === Keep inner Refl)
|
||||
fromKeep (Drop ope eq) = void $ notEvenOdd _ _ $ sym eq
|
||||
fromKeep (Keep ope eq) with (doubleInj _ _ $ inj S eq)
|
||||
fromKeep (Keep ope Refl) | Refl = (ope ** Refl)
|
||||
|
||||
private
|
||||
0 keepIsSucc : (ope : OPE m n (S (k + k))) -> IsSucc m
|
||||
keepIsSucc (Drop ope eq) = void $ notEvenOdd _ _ $ sym eq
|
||||
keepIsSucc (Keep ope _) = ItIsSucc
|
||||
|
||||
export
|
||||
view : {0 m : Nat} -> {n, mask : Nat} -> (0 ope : OPE m n mask) -> View ope
|
||||
view {n = 0} ope with 0 (fst $ stopEqs ope) | 0 (snd $ stopEqs ope)
|
||||
_ | Refl | Refl = rewrite fromStop ope in StopV
|
||||
view {n = S n} ope with (half mask)
|
||||
_ | HalfOdd mask' with 0 (keepIsSucc ope)
|
||||
_ | ItIsSucc with 0 (fromKeep ope)
|
||||
_ | (ope' ** eq) = rewrite eq in KeepV mask' ope'
|
||||
_ | HalfEven mask' with 0 (fromDrop ope)
|
||||
_ | (ope' ** eq) = rewrite eq in DropV mask' ope'
|
||||
|
||||
export
|
||||
appOpe : {0 m : Nat} -> (n : Nat) -> {mask : Nat} ->
|
||||
(0 ope : OPE m n mask) -> Singleton m
|
||||
appOpe n ope with (view ope)
|
||||
appOpe 0 Stop | StopV = Val 0
|
||||
appOpe (S n) (Drop ope' _) | DropV _ ope' = appOpe n ope'
|
||||
appOpe (S n) (Keep ope' _) | KeepV _ ope' = [|S $ appOpe n ope'|]
|
||||
|
||||
|
||||
||| inductive definition of OPE composition
|
||||
public export
|
||||
data Comp : (l : OPE n p mask1) -> (r : OPE m n mask2) ->
|
||||
(res : OPE m p mask3) -> Type where
|
||||
[search l r]
|
||||
StopZ : Comp Stop Stop Stop
|
||||
DropZ : Comp a b c -> Comp (Drop a Refl) b (Drop c Refl)
|
||||
KeepZ : Comp a b c -> Comp (Keep a Refl) (Keep b Refl) (Keep c Refl)
|
||||
KDZ : Comp a b c -> Comp (Keep a Refl) (Drop b Refl) (Drop c Refl)
|
||||
|
||||
public export
|
||||
record CompResult (ope1 : OPE n p mask1) (ope2 : OPE m n mask2) where
|
||||
constructor MkComp
|
||||
{resultMask : Nat}
|
||||
{0 result : OPE m p resultMask}
|
||||
0 comp : Comp ope1 ope2 result
|
||||
%name CompResult comp
|
||||
|
||||
||| compose two OPEs, if the middle scope size is already known at runtime
|
||||
export
|
||||
comp' : {n, p, mask1, mask2 : Nat} ->
|
||||
(0 ope1 : OPE n p mask1) -> (0 ope2 : OPE m n mask2) ->
|
||||
CompResult ope1 ope2
|
||||
comp' ope1 ope2 with (view ope1) | (view ope2)
|
||||
comp' Stop Stop | StopV | StopV =
|
||||
MkComp StopZ
|
||||
comp' (Drop ope1 Refl) ope2 | DropV _ ope1 | _ =
|
||||
MkComp $ DropZ (comp' ope1 ope2).comp
|
||||
comp' (Keep ope1 Refl) (Drop ope2 Refl) | KeepV _ ope1 | DropV _ ope2 =
|
||||
MkComp $ KDZ (comp' ope1 ope2).comp
|
||||
comp' (Keep ope1 Refl) (Keep ope2 Refl) | KeepV _ ope1 | KeepV _ ope2 =
|
||||
MkComp $ KeepZ (comp' ope1 ope2).comp
|
||||
|
||||
||| compose two OPEs, after recomputing the middle scope size using `appOpe`
|
||||
export
|
||||
comp : {p, mask1, mask2 : Nat} ->
|
||||
(0 ope1 : OPE n p mask1) -> (0 ope2 : OPE m n mask2) ->
|
||||
CompResult ope1 ope2
|
||||
comp ope1 ope2 = let Val n = appOpe p ope1 in comp' ope1 ope2
|
||||
|
||||
-- [todo] is there a quick way to compute the mask of comp?
|
||||
|
||||
|
||||
export
|
||||
app' : OPE m1 n1 mask1 -> OPE m2 n2 mask2 -> Exists (OPE (m1 + m2) (n1 + n2))
|
||||
app' Stop ope2 = Evidence _ ope2
|
||||
app' (Drop ope1 Refl) ope2 = Evidence _ $ Drop (app' ope1 ope2).snd Refl
|
||||
app' (Keep ope1 Refl) ope2 = Evidence _ $ Keep (app' ope1 ope2).snd Refl
|
||||
|
||||
export
|
||||
(++) : {n1, n2, mask1, mask2 : Nat} ->
|
||||
(0 ope1 : OPE m1 n1 mask1) -> (0 ope2 : OPE m2 n2 mask2) ->
|
||||
Subset Nat (OPE (m1 + m2) (n1 + n2))
|
||||
ope1 ++ ope2 with (view ope1)
|
||||
Stop ++ ope2 | StopV = Element _ ope2
|
||||
Drop ope1 Refl ++ ope2 | DropV mask ope1 =
|
||||
Element _ $ Drop (ope1 ++ ope2).snd Refl
|
||||
Keep ope1 Refl ++ ope2 | KeepV mask ope1 =
|
||||
Element _ $ Keep (ope1 ++ ope2).snd Refl
|
||||
|
||||
-- [todo] this mask is just (mask1 << n2) | mask2
|
||||
-- prove it and add %transform
|
||||
|
||||
|
||||
||| the tail of a non-empty OPE is its behaviour on all but the innermost slot
|
||||
public export
|
||||
data Tail : (full : OPE m1 (S n) mask1) -> (tail : OPE m2 n mask2) -> Type where
|
||||
[search full]
|
||||
DropT : Tail (Drop ope eq) ope
|
||||
KeepT : Tail (Keep ope eq) ope
|
||||
%name Tail tail
|
||||
|
||||
public export
|
||||
record TailRes (0 ope : OPE m (S n) mask) where
|
||||
constructor MkTail
|
||||
{0 scope : Nat}
|
||||
{tailMask : Nat}
|
||||
0 tail : OPE scope n tailMask
|
||||
0 isTail : Tail ope tail
|
||||
|
||||
export
|
||||
tail : {n, mask : Nat} -> (0 ope : OPE m (S n) mask) -> TailRes ope
|
||||
tail ope with (view ope)
|
||||
tail (Drop ope _) | DropV _ ope = MkTail ope DropT
|
||||
tail (Keep ope _) | KeepV _ ope = MkTail ope KeepT
|
||||
|
||||
|
||||
namespace OPEList
|
||||
||| a list of OPEs of a given outer scope size
|
||||
public export
|
||||
data OPEList : Nat -> Type where
|
||||
Nil : OPEList n
|
||||
(::) : OPE m n mask -> OPEList n -> OPEList n
|
||||
%name OPEList opes
|
||||
|
||||
namespace Tails -- 🦊
|
||||
||| `Tails opes tails` if each i'th element of `tails` is the tail of
|
||||
||| the i'th element of `opes`
|
||||
public export
|
||||
data Tails : (full : OPEList (S n)) -> (tails : OPEList n) -> Type where
|
||||
[search full]
|
||||
Nil : Tails [] []
|
||||
(::) : Tail ope tail -> Tails opes tails ->
|
||||
Tails (ope :: opes) (tail :: tails)
|
||||
|
||||
namespace Cover
|
||||
||| an OPE list is a cover if at least one of the OPEs has `Keep` as the head,
|
||||
||| and the tails are also a cover
|
||||
|||
|
||||
||| in @egtbs it is a binary relation which is fine for ×ᵣ but i don't want to
|
||||
||| write my AST in universe-of-syntaxes style. sorry
|
||||
public export data Cover : OPEList n -> Type
|
||||
|
||||
||| the "`Keep` in the head" condition of a cover
|
||||
public export data Cover1 : OPEList n -> Type
|
||||
|
||||
data Cover where
|
||||
Nil : Cover opes {n = 0}
|
||||
(::) : Cover1 opes -> Tails opes tails => Cover tails -> Cover opes
|
||||
%name Cover cov
|
||||
|
||||
data Cover1 where
|
||||
Here : Cover1 (Keep p eq :: opes)
|
||||
There : Cover1 opes -> Cover1 (ope :: opes)
|
||||
%name Cover1 cov1
|
||||
|
||||
|
||||
public export
|
||||
record Coprod (ope1 : OPE m1 n mask1) (ope2 : OPE m2 n mask2) where
|
||||
constructor MkCoprod
|
||||
{0 size : Nat}
|
||||
{sizeMask : Nat}
|
||||
{leftMask : Nat}
|
||||
{rightMask : Nat}
|
||||
{0 sub : OPE size n sizeMask}
|
||||
{0 left : OPE m1 size leftMask}
|
||||
{0 right : OPE m2 size rightMask}
|
||||
0 leftComp : Comp sub left ope1
|
||||
0 rightComp : Comp sub right ope2
|
||||
{auto 0 isCover : Cover [left, right]}
|
||||
%name Coprod cop
|
||||
|
||||
export
|
||||
coprod : {n, mask1, mask2 : Nat} ->
|
||||
(0 ope1 : OPE m1 n mask1) -> (0 ope2 : OPE m2 n mask2) ->
|
||||
Coprod ope1 ope2
|
||||
coprod ope1 ope2 with (view ope1) | (view ope2)
|
||||
coprod Stop Stop | StopV | StopV = MkCoprod StopZ StopZ
|
||||
coprod (Drop ope1 Refl) (Drop ope2 Refl) | DropV _ ope1 | DropV _ ope2 =
|
||||
let MkCoprod l r = coprod ope1 ope2 in MkCoprod (DropZ l) (DropZ r)
|
||||
coprod (Drop ope1 Refl) (Keep ope2 Refl) | DropV _ ope1 | KeepV _ ope2 =
|
||||
let MkCoprod l r = coprod ope1 ope2 in MkCoprod (KDZ l) (KeepZ r)
|
||||
coprod (Keep ope1 Refl) (Drop ope2 Refl) | KeepV _ ope1 | DropV _ ope2 =
|
||||
let MkCoprod l r = coprod ope1 ope2 in MkCoprod (KeepZ l) (KDZ r)
|
||||
coprod (Keep ope1 Refl) (Keep ope2 Refl) | KeepV _ ope1 | KeepV _ ope2 =
|
||||
let MkCoprod l r = coprod ope1 ope2 in MkCoprod (KeepZ l) (KeepZ r)
|
||||
|
||||
-- [todo] n-ary coprod
|
||||
|
||||
|
||||
public export
|
||||
record Chunks m n where
|
||||
constructor MkChunks
|
||||
{leftMask : Nat}
|
||||
{rightMask : Nat}
|
||||
0 left : OPE m (m + n) leftMask
|
||||
0 right : OPE n (m + n) rightMask
|
||||
{auto 0 isCover : Cover [left, right]}
|
||||
%name Chunks chunks
|
||||
|
||||
export
|
||||
chunks : (m, n : Nat) -> Chunks m n
|
||||
chunks 0 0 = MkChunks Stop Stop
|
||||
chunks 0 (S n) =
|
||||
let MkChunks l r = chunks 0 n in
|
||||
MkChunks (Drop l Refl) (Keep r Refl)
|
||||
chunks (S m) n =
|
||||
let MkChunks l r = chunks m n in
|
||||
MkChunks (Keep l Refl) (Drop r Refl)
|
||||
|
||||
-- [todo] the masks here are just ((2 << m) - 1) << n and (2 << n) - 1
|
||||
|
||||
|
||||
public export
|
||||
record SplitAt m n1 n2 (ope : OPE m (n1 + n2) mask) where
|
||||
constructor MkSplitAt
|
||||
{leftMask, rightMask : Nat}
|
||||
{0 leftScope, rightScope : Nat}
|
||||
0 left : OPE leftScope n1 leftMask
|
||||
0 right : OPE rightScope n2 rightMask
|
||||
0 scopePrf : m = leftScope + rightScope
|
||||
0 opePrf : ope `Eqv` (left `app'` right).snd
|
||||
%name SplitAt split
|
||||
|
||||
export
|
||||
splitAt : (n1 : Nat) -> {n2, mask : Nat} -> (0 ope : OPE m (n1 + n2) mask) ->
|
||||
SplitAt m n1 n2 ope
|
||||
splitAt 0 ope = MkSplitAt zero ope Refl reflexive
|
||||
splitAt (S n1) ope with (view ope)
|
||||
splitAt (S n1) (Drop ope Refl) | DropV _ ope with (splitAt n1 ope)
|
||||
_ | MkSplitAt left right scopePrf opePrf =
|
||||
MkSplitAt (Drop left Refl) right scopePrf (EqvDrop opePrf)
|
||||
splitAt (S n1) (Keep ope Refl) | KeepV _ ope with (splitAt n1 ope)
|
||||
_ | MkSplitAt left right scopePrf opePrf =
|
||||
MkSplitAt (Keep left Refl) right (cong S scopePrf) (EqvKeep opePrf)
|
||||
|
||||
|
||||
public export
|
||||
record Thinned f n where
|
||||
constructor Th
|
||||
{0 scope : Nat}
|
||||
{scopeMask : Nat}
|
||||
0 ope : OPE scope n scopeMask
|
||||
term : f scope
|
||||
%name Thinned s, t, u
|
||||
|
||||
export
|
||||
pure : {n : Nat} -> f n -> Thinned f n
|
||||
pure term = Th id.snd term
|
||||
|
||||
export
|
||||
join : {n : Nat} -> Thinned (Thinned f) n -> Thinned f n
|
||||
join (Th ope1 (Th ope2 term)) = Th (comp ope1 ope2).result term
|
||||
|
|
@ -17,6 +17,7 @@ modules =
|
|||
Quox.No,
|
||||
Quox.Loc,
|
||||
Quox.OPE,
|
||||
Quox.Thin,
|
||||
Quox.Pretty,
|
||||
Quox.Syntax,
|
||||
Quox.Syntax.Dim,
|
||||
|
|
|
|||
Loading…
Reference in a new issue