Data.Constraint.Nat · constraints-0.14.4

familytype family Min (m :: Nat) (n :: Nat) :: Nat where

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Min m n = If (n <=? m) n m

familytype family Max (m :: Nat) (n :: Nat) :: Nat where

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Max m n = If (n <=? m) m n

familytype family Lcm (m :: Nat) (n :: Nat) :: Nat where

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Lcm m m = m

familytype family Gcd (m :: Nat) (n :: Nat) :: Nat where

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Gcd m m = m

typetype Divides (n :: Nat) (m :: Nat) = n ~ Gcd n m

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familytype family Div (a :: Natural) (b :: Natural) :: Natural

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Division (round down) of natural numbers. Div x 0 is undefined (i.e., it cannot be reduced).

familytype family Mod (a :: Natural) (b :: Natural) :: Natural

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Modulus of natural numbers. Mod x 0 is undefined (i.e., it cannot be reduced).

familytype family Log2 (a :: Natural) :: Natural

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Log base 2 (round down) of natural numbers. Log 0 is undefined (i.e., it cannot be reduced).

valueplusNat :: (KnownNat n, KnownNat m) :- KnownNat (n + m)

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valueminusNat :: (KnownNat n, KnownNat m, m <= n) :- KnownNat (n - m)

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valuetimesNat :: (KnownNat n, KnownNat m) :- KnownNat (n * m)

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valuepowNat :: (KnownNat n, KnownNat m) :- KnownNat (n ^ m)

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valueminNat :: (KnownNat n, KnownNat m) :- KnownNat (Min n m)

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valuemaxNat :: (KnownNat n, KnownNat m) :- KnownNat (Max n m)

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valuegcdNat :: (KnownNat n, KnownNat m) :- KnownNat (Gcd n m)

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valuelcmNat :: (KnownNat n, KnownNat m) :- KnownNat (Lcm n m)

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valuedivNat :: (KnownNat n, KnownNat m, 1 <= m) :- KnownNat (Div n m)

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valuemodNat :: (KnownNat n, KnownNat m, 1 <= m) :- KnownNat (Mod n m)

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valuelog2Nat :: (KnownNat n, 1 <= n) :- KnownNat (Log2 n)

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valueplusZero :: Dict ((n + 0) ~ n)

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valueminusZero :: Dict ((n - 0) ~ n)

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valuetimesZero :: Dict ((n * 0) ~ 0)

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valuetimesOne :: Dict ((n * 1) ~ n)

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valuepowZero :: Dict ((n ^ 0) ~ 1)

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valuepowOne :: Dict ((n ^ 1) ~ n)

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valuemaxZero :: Dict (Max n 0 ~ n)

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valueminZero :: Dict (Min n 0 ~ 0)

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valuegcdZero :: Dict (Gcd 0 a ~ a)

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valuegcdOne :: Dict (Gcd 1 a ~ 1)

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valuelcmZero :: Dict (Lcm 0 a ~ 0)

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valuelcmOne :: Dict (Lcm 1 a ~ a)

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valueplusAssociates :: Dict (((m + n) + o) ~ (m + (n + o)))

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valuetimesAssociates :: Dict (((m * n) * o) ~ (m * (n * o)))

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valueminAssociates :: Dict (Min (Min m n) o ~ Min m (Min n o))

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valuemaxAssociates :: Dict (Max (Max m n) o ~ Max m (Max n o))

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valuegcdAssociates :: Dict (Gcd (Gcd a b) c ~ Gcd a (Gcd b c))

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valuelcmAssociates :: Dict (Lcm (Lcm a b) c ~ Lcm a (Lcm b c))

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valueplusCommutes :: Dict ((m + n) ~ (n + m))

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valuetimesCommutes :: Dict ((m * n) ~ (n * m))

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valueminCommutes :: Dict (Min m n ~ Min n m)

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valuemaxCommutes :: Dict (Max m n ~ Max n m)

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valuegcdCommutes :: Dict (Gcd a b ~ Gcd b a)

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valuelcmCommutes :: Dict (Lcm a b ~ Lcm b a)

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valueplusDistributesOverTimes :: Dict ((n * (m + o)) ~ ((n * m) + (n * o)))

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valuetimesDistributesOverPow :: Dict ((n ^ (m + o)) ~ ((n ^ m) * (n ^ o)))

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valuetimesDistributesOverGcd :: Dict ((n * Gcd m o) ~ Gcd (n * m) (n * o))

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valuetimesDistributesOverLcm :: Dict ((n * Lcm m o) ~ Lcm (n * m) (n * o))

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valueminDistributesOverPlus :: Dict ((n + Min m o) ~ Min (n + m) (n + o))

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valueminDistributesOverTimes :: Dict ((n * Min m o) ~ Min (n * m) (n * o))

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valueminDistributesOverPow1 :: Dict ((Min n m ^ o) ~ Min (n ^ o) (m ^ o))

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valueminDistributesOverPow2 :: Dict ((n ^ Min m o) ~ Min (n ^ m) (n ^ o))

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valueminDistributesOverMax :: Dict (Max n (Min m o) ~ Min (Max n m) (Max n o))

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valuemaxDistributesOverPlus :: Dict ((n + Max m o) ~ Max (n + m) (n + o))

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valuemaxDistributesOverTimes :: Dict ((n * Max m o) ~ Max (n * m) (n * o))

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valuemaxDistributesOverPow1 :: Dict ((Max n m ^ o) ~ Max (n ^ o) (m ^ o))

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valuemaxDistributesOverPow2 :: Dict ((n ^ Max m o) ~ Max (n ^ m) (n ^ o))

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valuemaxDistributesOverMin :: Dict (Min n (Max m o) ~ Max (Min n m) (Min n o))

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valuegcdDistributesOverLcm :: Dict (Gcd (Lcm a b) c ~ Lcm (Gcd a c) (Gcd b c))

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valuelcmDistributesOverGcd :: Dict (Lcm (Gcd a b) c ~ Gcd (Lcm a c) (Lcm b c))

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valueminIsIdempotent :: Dict (Min n n ~ n)

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valuemaxIsIdempotent :: Dict (Max n n ~ n)

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valuelcmIsIdempotent :: Dict (Lcm n n ~ n)

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valuegcdIsIdempotent :: Dict (Gcd n n ~ n)

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valueplusIsCancellative :: ((n + m) ~ (n + o)) :- (m ~ o)

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valuetimesIsCancellative :: (1 <= n, (n * m) ~ (n * o)) :- (m ~ o)

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valuedividesPlus :: (Divides a b, Divides a c) :- Divides a (b + c)

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valuedividesTimes :: Divides a b :- Divides a (b * c)

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valuedividesMin :: (Divides a b, Divides a c) :- Divides a (Min b c)

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valuedividesMax :: (Divides a b, Divides a c) :- Divides a (Max b c)

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valuedividesPow :: (1 <= n, Divides a b) :- Divides a (b ^ n)

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valuedividesGcd :: (Divides a b, Divides a c) :- Divides a (Gcd b c)

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valuedividesLcm :: (Divides a c, Divides b c) :- Divides (Lcm a b) c

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valueplusMonotone1 :: (a <= b) :- ((a + c) <= (b + c))

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valueplusMonotone2 :: (b <= c) :- ((a + b) <= (a + c))

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valuetimesMonotone1 :: (a <= b) :- ((a * c) <= (b * c))

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valuetimesMonotone2 :: (b <= c) :- ((a * b) <= (a * c))

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valuepowMonotone1 :: (a <= b) :- ((a ^ c) <= (b ^ c))

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valuepowMonotone2 :: (b <= c) :- ((a ^ b) <= (a ^ c))

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valueminMonotone1 :: (a <= b) :- (Min a c <= Min b c)

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valueminMonotone2 :: (b <= c) :- (Min a b <= Min a c)

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valuemaxMonotone1 :: (a <= b) :- (Max a c <= Max b c)

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valuemaxMonotone2 :: (b <= c) :- (Max a b <= Max a c)

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valuedivMonotone1 :: (a <= b) :- (Div a c <= Div b c)

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valuedivMonotone2 :: (b <= c) :- (Div a c <= Div a b)

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valueeuclideanNat :: (1 <= c) :- (a ~ ((c * Div a c) + Mod a c))

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valueplusMod :: (1 <= c) :- (Mod (a + b) c ~ Mod (Mod a c + Mod b c) c)

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valuetimesMod :: (1 <= c) :- (Mod (a * b) c ~ Mod (Mod a c * Mod b c) c)

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valuemodBound :: (1 <= n) :- (Mod m n <= n)

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valuelog2Pow :: Dict (Log2 (2 ^ n) ~ n)

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valuedividesDef :: Divides a b :- (Mod b a ~ 0)

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valuetimesDiv :: Dict ((a * Div b a) <= b)

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valueeqLe :: (a ~ b) :- (a <= b)

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valueleEq :: (a <= b, b <= a) :- (a ~ b)

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valueleId :: Dict (a <= a)

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valueleTrans :: (b <= c, a <= b) :- (a <= c)

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valueleZero :: (a <= 0) :- (a ~ 0)

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valuezeroLe :: Dict (0 <= a)

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valueplusMinusInverse1 :: Dict (((m + n) - n) ~ m)

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valueplusMinusInverse2 :: (m <= n) :- (((m + n) - m) ~ n)

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valueplusMinusInverse3 :: (n <= m) :- (((m - n) + n) ~ m)

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