GHC.TypeLits.Induction

Natural number induction

induceIsZero :: KnownNat m => (forall n. KnownNat n => p (1 + n)) -> p 0 -> p m

\((\forall n. n > 0 \to P(n)) \to P(0) \to \forall m. P(m)\)

inducePeano :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 0 -> p m

\((\forall n. P(n) \to P(n + 1)) \to P(0) \to \forall m. P(m)\)

For example:

inducePeano f z :: p 3 = f (f (f z))

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

\((\forall n. P(n) \to P(2n + 1)) \to \\ (\forall n. P(n) \to P(2n + 2)) \to \\ P(0) \to \\ \forall m. P(m)\)

For example:

induceTwosComp f1 f2 z :: p 12 = f2 (f1 (f2 z))

induceBaseComp :: forall r b q p m c s. (KnownNat b, b ~ (1 + c), KnownNat m) => s q -> r b -> (forall k. (KnownNat k, (1 + k) <= b) => q k) -> (forall k n. (KnownNat k, (1 + k) <= b, KnownNat n) => q k -> p n -> p ((1 + k) + (b * n))) -> p 0 -> p m

\(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d + 1)) \to \\ \forall m. P(m)\)

For example:

induceBaseComp (_ :: s q) (_ :: r 10) d f z :: p 123 = (d :: q 2) `f` ((d :: q 1) `f` ((d :: q 0) `f` z)

The s q parameter is necessary because presently GHC is unable to unify q under the equational constraint 1 + k <= b.

Positive number induction

induceUnary :: KnownNat m => (forall n. KnownNat n => p n -> p (1 + n)) -> p 1 -> p (1 + m)

\((\forall n. P(n) \to P(n + 1)) \to P(1) \to \forall m > 0. P(m)\)

For example:

induceUnary f o :: p 5 = f (f (f (f o)))

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)

\((\forall n. P(n) \to P(2n)) \to \\ (\forall n. P(n) \to P(2n + 1)) \to \\ P(1) \to \\ \forall m > 0. P(m)\)

For example:

inducePosBinary f0 f1 o :: p 10 = f0 (f1 (f0 o))

inducePosBase :: forall r b q p m c s. (KnownNat b, b ~ (2 + c), KnownNat m) => s q -> r b -> (forall k. (KnownNat k, (1 + k) <= b) => q k) -> (forall k n. (KnownNat k, (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)

\(\forall b. (\prod_d Q(d)) \to \\ (\forall n. \forall d. d < b \to Q(d) \to P(n) \to P(bn + d)) \to \\ (\forall d. d > 0 \to d < b \to Q(d) \to P(d)) \to \\ \forall m > 0. P (m)\)

For example:

inducePosBase (_ :: s q) (_ :: r 10) d f l :: p 123 = (d :: q 3) `f` ((d :: q 2) `f` l (d :: q 1))

The s q parameter is necessary because presently GHC is unable to unify q under the equational constraint 1 + k <= b.