| Copyright | (C) 2017 mniip |
|---|---|
| License | MIT |
| Maintainer | mniip@mniip.com |
| Stability | experimental |
| Portability | portable |
| Safe Haskell | None |
| Language | Haskell2010 |
GHC.TypeLits.Induction
Description
- induceIsZero :: KnownNat m => (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m
- inducePeano :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> 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
- 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
- induceUnary :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p (1 + 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)
- 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)
Natural number induction
induceIsZero :: KnownNat m => (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m Source #
\((\forall n. n > 0 \to P(n)) \to P(0) \to \forall m. P(m)\)
inducePeano :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m Source #
\((\forall n. P(n) \to P(n + 1)) \to P(0) \to \forall m. P(m)\)
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 Source #
\((\forall n. P(n) \to P(2n + 1)) \to \\ (\forall n. P(n) \to P(2n + 2)) \to \\ P(0) \to \\ \forall m. P(m)\)
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 Source #
\(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d + 1)) \to \\ \forall m. P(m)\)
Positive number induction
induceUnary :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p (1 + m) Source #
\((\forall n. P(n) \to P(n + 1)) \to P(1) \to \forall m > 0. P(m)\)
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) Source #
\((\forall n. P(n) \to P(2n)) \to \\ (\forall n. P(n) \to P(2n + 1)) \to \\ P(1) \to \\ \forall m > 0. P(m)\)
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) Source #
\(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d)) \to \\ (\forall d. d > 0 \to d < b \to Q(d) \to P(d)) \to \\ \forall m > 0. P (m)\)