mirror of
https://github.com/idris-lang/Idris2.git
synced 2024-12-19 01:01:59 +03:00
87 lines
2.3 KiB
Idris
87 lines
2.3 KiB
Idris
module Control.WellFounded
|
|
|
|
import Data.Nat
|
|
import Data.List
|
|
|
|
%default total
|
|
|
|
public export
|
|
data Accessible : (rel : a -> a -> Type) -> (x : a) -> Type where
|
|
Access : (rec : (y : a) -> rel y x -> Accessible rel y) ->
|
|
Accessible rel x
|
|
|
|
public export
|
|
interface WellFounded a rel where
|
|
wellFounded : (x : a) -> Accessible rel x
|
|
|
|
export
|
|
accRec : {0 rel : (arg1 : a) -> (arg2 : a) -> Type} ->
|
|
(step : (x : a) -> ((y : a) -> rel y x -> b) -> b) ->
|
|
(z : a) -> (0 acc : Accessible rel z) -> b
|
|
accRec step z (Access f) =
|
|
step z $ \yarg, lt => accRec step yarg (f yarg lt)
|
|
|
|
export
|
|
accInd : {0 rel : a -> a -> Type} -> {0 P : a -> Type} ->
|
|
(step : (x : a) -> ((y : a) -> rel y x -> P y) -> P x) ->
|
|
(z : a) -> (0 acc : Accessible rel z) -> P z
|
|
accInd step z (Access f) =
|
|
step z $ \y, lt => accInd step y (f y lt)
|
|
|
|
export
|
|
wfRec : (0 _ : WellFounded a rel) =>
|
|
(step : (x : a) -> ((y : a) -> rel y x -> b) -> b) ->
|
|
a -> b
|
|
wfRec step x = accRec step x (wellFounded {rel} x)
|
|
|
|
export
|
|
wfInd : (0 _ : WellFounded a rel) => {0 P : a -> Type} ->
|
|
(step : (x : a) -> ((y : a) -> rel y x -> P y) -> P x) ->
|
|
(myz : a) -> P myz
|
|
wfInd step myz = accInd step myz (wellFounded {rel} myz)
|
|
|
|
public export
|
|
interface Sized a where
|
|
size : a -> Nat
|
|
|
|
public export
|
|
Smaller : Sized a => a -> a -> Type
|
|
Smaller x y = size x `LT` size y
|
|
|
|
public export
|
|
SizeAccessible : Sized a => a -> Type
|
|
SizeAccessible = Accessible Smaller
|
|
|
|
export
|
|
sizeAccessible : Sized a => (x : a) -> SizeAccessible x
|
|
sizeAccessible x = Access (acc $ size x)
|
|
where
|
|
acc : (sizeX : Nat) -> (y : a) -> (size y `LT` sizeX) -> SizeAccessible y
|
|
acc (S x') y (LTESucc yLEx')
|
|
= Access (\z, zLTy => acc x' z (lteTransitive zLTy yLEx'))
|
|
|
|
export
|
|
sizeInd : Sized a => {0 P : a -> Type} ->
|
|
(step : (x : a) -> ((y : a) -> Smaller y x -> P y) -> P x) ->
|
|
(z : a) ->
|
|
P z
|
|
sizeInd step z = accInd step z (sizeAccessible z)
|
|
|
|
export
|
|
sizeRec : Sized a =>
|
|
(step : (x : a) -> ((y : a) -> Smaller y x -> b) -> b) ->
|
|
(z : a) -> b
|
|
sizeRec step z = accRec step z (sizeAccessible z)
|
|
|
|
export
|
|
implementation Sized Nat where
|
|
size = \x => x
|
|
|
|
export
|
|
implementation Sized (List a) where
|
|
size = length
|
|
|
|
export
|
|
implementation (Sized a, Sized b) => Sized (Pair a b) where
|
|
size (x,y) = size x + size y
|