88 lines
4.1 KiB
Haskell
88 lines
4.1 KiB
Haskell
-- |
|
|
-- Module : GHC.TypeLits.Induction
|
|
-- Description : Induction for GHC TypeLits
|
|
-- Copyright : (C) 2017-2018 mniip
|
|
-- License : MIT
|
|
-- Maintainer : mniip@mniip.com
|
|
-- Stability : stable
|
|
-- Portability : portable
|
|
{-# LANGUAGE TypeOperators, KindSignatures, DataKinds, PolyKinds, GADTs, RankNTypes, StandaloneDeriving, InstanceSigs, ScopedTypeVariables #-}
|
|
{-# LANGUAGE Safe #-}
|
|
{-# OPTIONS_GHC -fno-warn-tabs #-}
|
|
|
|
module GHC.TypeLits.Induction
|
|
(
|
|
-- * Natural number induction
|
|
induceIsZero,
|
|
inducePeano,
|
|
induceTwosComp,
|
|
induceBaseComp,
|
|
|
|
-- * Positive number induction
|
|
induceUnary,
|
|
inducePosBinary,
|
|
inducePosBase
|
|
)
|
|
where
|
|
|
|
import Data.Proxy
|
|
import GHC.TypeLits
|
|
import GHC.TypeLits.Singletons
|
|
|
|
-- | \((\forall n. n > 0 \to P(n)) \to P(0) \to \forall m. P(m)\)
|
|
induceIsZero :: KnownNat m => (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m
|
|
induceIsZero = go natSingleton
|
|
where
|
|
go :: NatIsZero m -> (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m
|
|
go IsZero y z = z
|
|
go IsNonZero y z = y
|
|
|
|
-- | \((\forall n. P(n) \to P(n + 1)) \to P(0) \to \forall m. P(m)\)
|
|
inducePeano :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m
|
|
inducePeano = go natSingleton
|
|
where
|
|
go :: NatPeano m -> (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m
|
|
go PeanoZero f z = z
|
|
go (PeanoSucc n) f z = f (go n f z)
|
|
|
|
-- | \((\forall n. P(n) \to P(2n + 1)) \to \\ (\forall n. P(n) \to P(2n + 2)) \to \\ P(0) \to \\ \forall m. P(m)\)
|
|
induceTwosComp :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> (forall n. KnownNat n => p n -> p (2 + 2 * n)) -> p 0 -> p m
|
|
induceTwosComp = go natSingleton
|
|
where
|
|
go :: NatTwosComp m -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> (forall n. KnownNat n => p n -> p (2 + 2 * n)) -> p 0 -> p m
|
|
go TwosCompZero f g z = z
|
|
go (TwosCompx2p1 n) f g z = f (go n f g z)
|
|
go (TwosCompx2p2 n) f g z = g (go n f g z)
|
|
|
|
-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d + 1)) \to \\ \forall m. P(m)\)
|
|
induceBaseComp :: forall r b q p m. (KnownNat b, KnownNat m) => r b -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
|
|
induceBaseComp _ = go natSingleton
|
|
where
|
|
go :: forall m. NatBaseComp Proxy b m -> (forall m. (KnownNat m, 1 + m <= b) => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (1 + k + b * n)) -> p 0 -> p m
|
|
go BaseCompZero q f z = z
|
|
go (BaseCompxBp1p k n) q f z = f q (go n q f z)
|
|
|
|
-- | \((\forall n. P(n) \to P(n + 1)) \to P(1) \to \forall m > 0. P(m)\)
|
|
induceUnary :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p (1 + m)
|
|
induceUnary = go posSingleton
|
|
where
|
|
go :: Unary m -> (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p m
|
|
go UnaryOne f o = o
|
|
go (UnarySucc n) f o = f (go n f o)
|
|
|
|
-- | \((\forall n. P(n) \to P(2n)) \to \\ (\forall n. P(n) \to P(2n + 1)) \to \\ P(1) \to \\ \forall m > 0. P(m)\)
|
|
inducePosBinary :: KnownNat m => (forall n. KnownNat n => p n -> p (2 * n)) -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> p 1 -> p (1 + m)
|
|
inducePosBinary = go posSingleton
|
|
where
|
|
go :: PosBinary m -> (forall n. KnownNat n => p n -> p (2 * n)) -> (forall n. KnownNat n => p n -> p (1 + 2 * n)) -> p 1 -> p m
|
|
go BinOne f g z = z
|
|
go (Bin0 n) f g z = f (go n f g z)
|
|
go (Bin1 n) f g z = g (go n f g z)
|
|
|
|
-- | \(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d)) \to \\ (\forall d. d > 0 \to d < b \to Q(d) \to P(d)) \to \\ \forall m > 0. P (m)\)
|
|
inducePosBase :: forall r b q p m. (KnownNat b, KnownNat m) => r b -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p (1 + m)
|
|
inducePosBase _ = go posSingleton
|
|
where
|
|
go :: forall m. PosBase Proxy b m -> (forall m. KnownNat m => q m) -> (forall k n. (KnownNat b, 1 + k <= b, KnownNat n) => q k -> p n -> p (k + b * n)) -> (forall k n. (KnownNat k, 1 + k <= b, k ~ (1 + n)) => q k -> p k) -> p m
|
|
go (BaseLead n) q f g = g q
|
|
go (BaseDigit k n) q f g = f q (go n q f g)
|