type-int-0.4: Type Level 2s- and 16s- Complement IntegersContentsIndex
Data.Type.Binary.Internals
Portabilitynon-portable (FD and MPTC)
Stabilityexperimental
MaintainerEdward Kmett <ekmett@gmail.com>
Description

Simple type-level binary numbers, positive and negative with infinite precision. This forms a nice commutative ring with multiplicative identity like we would expect from a representation for Z.

The numbers are represented as a Boolean Ring over a countable set of variables, in which for every element in the set there exists an n in N and a b in {T,F} such that for all n'>=n in N, x_i = b.

For uniqueness we always choose the least such n when representing numbers this allows us to run most computations backwards. When we can't, and such a fundep would be implied, we obtain it by combining semi-operations that together yield the appropriate class fundep list.

Reuses T and F from the Type.Boolean as the infinite tail of the 2s complement binary number.

TODO: TDivMod, TGCD

Synopsis
data O a
data I a
class TSucc n m | n -> m, m -> n
tSucc :: TSucc n m => n -> m
tPred :: TSucc n m => m -> n
class TCBinary c a | a -> c
class TCBinary Closure a => TBinary a where
fromTBinary :: Integral b => a -> b
fromTBinary :: (TBinary a, Integral b) => a -> b
class TNeg a b | a -> b, b -> a
tNeg :: TNeg a b => a -> b
class TIsNegative n b | n -> b
tIsNegative :: TIsNegative n b => n -> b
class TIsPositive n b | n -> b
tIsPositive :: TIsPositive n b => n -> b
class TIsZero n b | n -> b
tIsZero :: TIsZero n b => n -> b
class TEven a b | a -> b
tEven :: TEven a b => a -> b
class TOdd a b | a -> b
tOdd :: TOdd a b => a -> b
class TAdd a b c | a b -> c, a c -> b, b c -> a
tAdd :: TAdd a b c => a -> b -> c
tSub :: TAdd a b c => c -> a -> b
class TMul a b c | a b -> c
tMul :: TMul a b c => a -> b -> c
class TPow a b c | a b -> c
tPow :: TPow a b c => a -> b -> c
class TShift a b c' | a b -> c'
tShift :: TShift a b c => a -> b -> c
class TGetBit a b c | a b -> c
tGetBit :: TGetBit a b c => a -> b -> c
class TSetBit a b c | a b -> c
tSetBit :: TSetBit a b c => a -> b -> c
class TChangeBit a b c d | a b c -> d
tChangeBit :: TChangeBit a b c d => a -> b -> c -> d
class TUnSetBit a b c | a b -> c
tUnSetBit :: TUnSetBit a b c => a -> b -> c
class TComplementBit a b c | a b -> c
tComplementBit :: TComplementBit a b c => a -> b -> c
class TCountBits a b | a -> b
tCountBits :: TCountBits a b => a -> b
class TAbs a b | a -> b
tAbs :: TAbs a b => a -> b
class TNF a b | a -> b
tNF :: TNF a b => a -> b
t2n :: TNF (O a) b => a -> b
t2np1 :: TNF (I a) b => a -> b
class TShift' a b c | a b -> c
class TNF' a b c | a -> b c
class TAddC' a b c d | a b c -> d
class TAdd' a b c | a b -> c
tAdd' :: TAdd' a b c => a -> b -> c
class TSub' a b c | a b -> c
tSub' :: TSub' a b c => a -> b -> c
class TCountBits' a b t | a t -> b
class TBool d => LSB a d a' | a -> d a', d a' -> a
tLSB :: LSB a d a' => a -> d -> a'
tBSL :: LSB a d a' => a' -> d -> a
class LSB (I a) T a => XI a
class LSB (O a) F a => XO a
Documentation
data O a
show/hide Instances
TAddC' T T F (O T)
(TCBinary c a, XO a) => TCBinary c (O a)
TAnd F (O b) F
TImplies F (O b) T
TMul (O a) b c => TMul a (O b) c
TOr T (O b) T
(TPow a k c, TMul c c d) => TPow a (O k) d
(TPow' a k c, TMul c c d) => TPow' a (O k) d
(TShift' a b c, TShift' c b d) => TShift' a (O b) d
TSucc a b => TAddC' F (I a) T (O b)
TAddC' F (O a) F (O a)
TAddC' F (O a) T (I a)
TAddC' T (I a) F (O a)
TSucc b a => TAddC' T (O a) F (I b)
TAddC' T (O a) T (O a)
TAnd T (O b) (O b)
TImplies T (O b) (O b)
TOr F (O b) (O b)
(TShift' a b c, TShift' c b d) => TShift' a (I b) (O d)
TXOr' F (O b) (O b)
TNot b c => TXOr' T (I b) (O c)
TNot b c => TXOr' T (O b) (I c)
TBinary (O a) => Show (O a)
(TBinary a, XO a) => TBinary (O a)
TSucc (O T) T
(Trichotomy a b, XO a) => Trichotomy (I (O a)) b
(Trichotomy a b, XI a) => Trichotomy (O (I a)) b
(Trichotomy a b, XO a) => Trichotomy (O (O a)) b
Trichotomy (O T) Negative
LSB (O T) F T
TAnd (O a) F F
TCountBits' a n F => TCountBits' (O a) n F
(TCountBits' a n F, TSucc m n) => TCountBits' (O a) n T
TEq (O m) F F
TEq (O m) T F
TImplies (O a) F T
TNF' (O F) F F
(TNF' (O a) c b, TIf b (O c) F d) => TNF' (O (O a)) d b
TOr (O a) T T
TShift' (O a) T a
LSB (O a) F a => X (O a) F a
TSucc a b => TAddC' (I a) F T (O b)
TAddC' (I a) T F (O a)
TAddC' (O a) F F (O a)
TAddC' (O a) F T (I a)
TSucc b a => TAddC' (O a) T F (I b)
TAddC' (O a) T T (O a)
LSB (I (O n)) T (O n)
LSB (O (I n)) F (I n)
LSB (O (O n)) F (O n)
TAnd (O a) T (O a)
TImplies (O a) T (O a)
TOr (O a) F (I a)
TShift' (O a) F (O a)
TNot b c => TXOr' (I b) T (O c)
TXOr' (O b) F (O b)
TNot b c => TXOr' (O b) T (I c)
THex2Binary' a b => THex2Binary' (D0 a) (O (O (O (O b))))
THex2Binary' a b => THex2Binary' (D1 a) (I (O (O (O b))))
THex2Binary' a b => THex2Binary' (D2 a) (O (I (O (O b))))
THex2Binary' a b => THex2Binary' (D3 a) (I (I (O (O b))))
THex2Binary' a b => THex2Binary' (D4 a) (O (O (I (O b))))
THex2Binary' a b => THex2Binary' (D5 a) (I (O (I (O b))))
THex2Binary' a b => THex2Binary' (D6 a) (O (I (I (O b))))
THex2Binary' a b => THex2Binary' (D7 a) (I (I (I (O b))))
THex2Binary' a b => THex2Binary' (D8 a) (O (O (O (I b))))
THex2Binary' a b => THex2Binary' (D9 a) (I (O (O (I b))))
THex2Binary' a b => THex2Binary' (DA a) (O (I (O (I b))))
THex2Binary' a b => THex2Binary' (DB a) (I (I (O (I b))))
THex2Binary' a b => THex2Binary' (DC a) (O (O (I (I b))))
THex2Binary' a b => THex2Binary' (DD a) (I (O (I (I b))))
THex2Binary' a b => THex2Binary' (DE a) (O (I (I (I b))))
TNot a b => TNot (I a) (O b)
TNot a b => TNot (O a) (I b)
(TSucc n m, XI n, XO m) => TSucc (I n) (O m)
TSucc (O (I n)) (I (I n))
TSucc (O (O n)) (I (O n))
(TAnd a b c, TNF (O c) c') => TAnd (I a) (O b) c'
(TAnd a b c, TNF (O c) c') => TAnd (O a) (I b) c'
(TAnd a b c, TNF (O c) c') => TAnd (O a) (O b) c'
TEq (I m) (O n) F
TEq (O m) (I n) F
TEq m n b => TEq (O m) (O n) b
(TImplies a b c, TNF (I c) c') => TImplies (I a) (O b) c'
(TImplies a b c, TNF (I c) c') => TImplies (O a) (I b) c'
(TImplies a b c, TNF (O c) c') => TImplies (O a) (O b) c'
TNF' (O a) c b => TNF' (I (O a)) (I c) T
TNF' (I a) c b => TNF' (O (I a)) (O c) T
TNF' (O T) (O T) T
(TOr a b c, TNF (I c) c') => TOr (I a) (O b) c'
(TOr a b c, TNF (I c) c') => TOr (O a) (I b) c'
(TOr a b c, TNF (O c) c') => TOr (O a) (O b) c'
(TXOr' a b c, TNF (I c) c') => TXOr' (I a) (O b) c'
(TXOr' a b c, TNF (I c) c') => TXOr' (O a) (I b) c'
(TXOr' a b c, TNF (O c) c') => TXOr' (O a) (O b) c'
TAddC' a b T c => TAddC' (I a) (I b) F (O c)
TAddC' a b F c => TAddC' (I a) (O b) F (I c)
TAddC' a b T c => TAddC' (I a) (O b) T (O c)
TAddC' a b F c => TAddC' (O a) (I b) F (I c)
TAddC' a b T c => TAddC' (O a) (I b) T (O c)
TAddC' a b F c => TAddC' (O a) (O b) F (O c)
TAddC' a b F c => TAddC' (O a) (O b) T (I c)
data I a
show/hide Instances
TAddC' F F T (I F)
TPow a F (I F)
(TCBinary c a, XI a) => TCBinary c (I a)
TSucc F (I F)
TAnd F (I b) F
TImplies F (I b) T
(TMul (O a) b c, TAdd' a c d) => TMul a (I b) d
TOr T (I b) T
(TPow a k c, TMul c c d, TMul a d e) => TPow a (I k) e
(TPow' a k c, TMul c c d, TMul a d e) => TPow' a (I k) e
TAddC' F (I a) F (I a)
TSucc a b => TAddC' F (I a) T (O b)
TAddC' F (O a) T (I a)
TAddC' T (I a) F (O a)
TAddC' T (I a) T (I a)
TSucc b a => TAddC' T (O a) F (I b)
TAnd T (I b) (I b)
TImplies T (I b) (I b)
TOr F (I b) (I b)
(TShift' a b c, TShift' c b d) => TShift' a (I b) (O d)
TXOr' F (I b) (I b)
TNot b c => TXOr' T (I b) (O c)
TNot b c => TXOr' T (O b) (I c)
TBinary (I a) => Show (I a)
(TBinary a, XI a) => TBinary (I a)
Trichotomy (I F) Positive
(Trichotomy a b, XI a) => Trichotomy (I (I a)) b
(Trichotomy a b, XO a) => Trichotomy (I (O a)) b
(Trichotomy a b, XI a) => Trichotomy (O (I a)) b
LSB (I F) T F
TAnd (I a) F F
(TCountBits' a n F, TSucc n m) => TCountBits' (I a) m F
TCountBits' a m F => TCountBits' (I a) m T
TEq (I m) F F
TEq (I m) T F
TImplies (I a) F T
(TNF' (I a) c b, TIf b (I c) T d) => TNF' (I (I a)) d b
TNF' (I T) T F
TOr (I a) T T
TShift' (I a) T a
LSB (I a) T a => X (I a) T a
TAddC' (I a) F F (I a)
TSucc a b => TAddC' (I a) F T (O b)
TAddC' (I a) T F (O a)
TAddC' (I a) T T (I a)
TAddC' (O a) F T (I a)
TSucc b a => TAddC' (O a) T F (I b)
LSB (I (I n)) T (I n)
LSB (I (O n)) T (O n)
LSB (O (I n)) F (I n)
TAnd (I a) T (I a)
TImplies (I a) T (I a)
TOr (I a) F (I a)
TOr (O a) F (I a)
TShift' (I a) F (I a)
TXOr' (I b) F (I b)
TNot b c => TXOr' (I b) T (O c)
TNot b c => TXOr' (O b) T (I c)
THex2Binary' a b => THex2Binary' (D1 a) (I (O (O (O b))))
THex2Binary' a b => THex2Binary' (D2 a) (O (I (O (O b))))
THex2Binary' a b => THex2Binary' (D3 a) (I (I (O (O b))))
THex2Binary' a b => THex2Binary' (D4 a) (O (O (I (O b))))
THex2Binary' a b => THex2Binary' (D5 a) (I (O (I (O b))))
THex2Binary' a b => THex2Binary' (D6 a) (O (I (I (O b))))
THex2Binary' a b => THex2Binary' (D7 a) (I (I (I (O b))))
THex2Binary' a b => THex2Binary' (D8 a) (O (O (O (I b))))
THex2Binary' a b => THex2Binary' (D9 a) (I (O (O (I b))))
THex2Binary' a b => THex2Binary' (DA a) (O (I (O (I b))))
THex2Binary' a b => THex2Binary' (DB a) (I (I (O (I b))))
THex2Binary' a b => THex2Binary' (DC a) (O (O (I (I b))))
THex2Binary' a b => THex2Binary' (DD a) (I (O (I (I b))))
THex2Binary' a b => THex2Binary' (DE a) (O (I (I (I b))))
THex2Binary' a b => THex2Binary' (DF a) (I (I (I (I b))))
TNot a b => TNot (I a) (O b)
TNot a b => TNot (O a) (I b)
(TSucc n m, XI n, XO m) => TSucc (I n) (O m)
TSucc (O (I n)) (I (I n))
TSucc (O (O n)) (I (O n))
(TAnd a b c, TNF (I c) c') => TAnd (I a) (I b) c'
(TAnd a b c, TNF (O c) c') => TAnd (I a) (O b) c'
(TAnd a b c, TNF (O c) c') => TAnd (O a) (I b) c'
TEq m n b => TEq (I m) (I n) b
TEq (I m) (O n) F
TEq (O m) (I n) F
(TImplies a b c, TNF (I c) c') => TImplies (I a) (I b) c'
(TImplies a b c, TNF (I c) c') => TImplies (I a) (O b) c'
(TImplies a b c, TNF (I c) c') => TImplies (O a) (I b) c'
TNF' (I F) (I F) T
TNF' (O a) c b => TNF' (I (O a)) (I c) T
TNF' (I a) c b => TNF' (O (I a)) (O c) T
(TOr a b c, TNF (I c) c') => TOr (I a) (I b) c'
(TOr a b c, TNF (I c) c') => TOr (I a) (O b) c'
(TOr a b c, TNF (I c) c') => TOr (O a) (I b) c'
(TXOr' a b c, TNF (O c) c') => TXOr' (I a) (I b) c'
(TXOr' a b c, TNF (I c) c') => TXOr' (I a) (O b) c'
(TXOr' a b c, TNF (I c) c') => TXOr' (O a) (I b) c'
TAddC' a b T c => TAddC' (I a) (I b) F (O c)
TAddC' a b T c => TAddC' (I a) (I b) T (I c)
TAddC' a b F c => TAddC' (I a) (O b) F (I c)
TAddC' a b T c => TAddC' (I a) (O b) T (O c)
TAddC' a b F c => TAddC' (O a) (I b) F (I c)
TAddC' a b T c => TAddC' (O a) (I b) T (O c)
TAddC' a b F c => TAddC' (O a) (O b) T (I c)
class TSucc n m | n -> m, m -> n
Finds the unique successor for any normalized binary number
show/hide Instances
TSucc T F
TSucc F (I F)
TSucc (O T) T
(TSucc n m, XI n, XO m) => TSucc (I n) (O m)
TSucc (O (I n)) (I (I n))
TSucc (O (O n)) (I (O n))
tSucc :: TSucc n m => n -> m
tPred :: TSucc n m => m -> n
class TCBinary c a | a -> c
Our set of digits is closed to retain the properties needed for most of the classes herein
show/hide Instances
TCBinary Closure F
TCBinary Closure T
(TCBinary c a, XI a) => TCBinary c (I a)
(TCBinary c a, XO a) => TCBinary c (O a)
class TCBinary Closure a => TBinary a where
We don't want to have to carry the closure parameter around explicitly everywhere, so we shed it here.
Methods
fromTBinary :: Integral b => a -> b
show/hide Instances
TBinary F
TBinary T
(TBinary a, XI a) => TBinary (I a)
(TBinary a, XO a) => TBinary (O a)
fromTBinary :: (TBinary a, Integral b) => a -> b
class TNeg a b | a -> b, b -> a
TNeg obtains the 2s complement of a number and is reversible
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(TNot a b, TSucc b c) => TNeg a c
tNeg :: TNeg a b => a -> b
class TIsNegative n b | n -> b
Returns true if the number is less than zero
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(Trichotomy n s, TEq s Negative b) => TIsNegative n b
tIsNegative :: TIsNegative n b => n -> b
class TIsPositive n b | n -> b
Returns true if the number is greater than zero
show/hide Instances
(Trichotomy n s, TEq s Positive b) => TIsPositive n b
tIsPositive :: TIsPositive n b => n -> b
class TIsZero n b | n -> b
Returns true if the number is equal to zero
show/hide Instances
(Trichotomy n s, TEq s SignZero b) => TIsZero n b
tIsZero :: TIsZero n b => n -> b
class TEven a b | a -> b
Returns true if the lsb of the number is true
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LSB a b c => TEven a b
tEven :: TEven a b => a -> b
class TOdd a b | a -> b
Returns true if the lsb of the number if false
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(LSB a b c, TNot b b') => TOdd a b'
tOdd :: TOdd a b => a -> b
class TAdd a b c | a b -> c, a c -> b, b c -> a
Reversible adder with extra fundeps.
show/hide Instances
(TAdd' a b c, TNeg b b', TAdd' c b' a, TNeg a a', TAdd' c a' b) => TAdd a b c
tAdd :: TAdd a b c => a -> b -> c
tSub :: TAdd a b c => c -> a -> b
class TMul a b c | a b -> c
Multiplication: a * b = c
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TMul a F F
TNeg a b => TMul a T b
(TMul (O a) b c, TAdd' a c d) => TMul a (I b) d
TMul (O a) b c => TMul a (O b) c
tMul :: TMul a b c => a -> b -> c
class TPow a b c | a b -> c
Exponentiation: a^b = c (only defined for non-negative exponents)
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TPow a F (I F)
(TPow a k c, TMul c c d, TMul a d e) => TPow a (I k) e
(TPow a k c, TMul c c d) => TPow a (O k) d
tPow :: TPow a b c => a -> b -> c
class TShift a b c' | a b -> c'
Shift a right b places obtaining c in normal form. | If b is negative then we shift left.
show/hide Instances
(TShift' a b c, TNF c c') => TShift a b c'
tShift :: TShift a b c => a -> b -> c
class TGetBit a b c | a b -> c
get bit #b in a as c in {T,F}
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(TNeg b b', TShift a b' c, LSB c d e) => TGetBit a b d
tGetBit :: TGetBit a b c => a -> b -> c
class TSetBit a b c | a b -> c
set bit #b in a to T, yielding c.
show/hide Instances
(TShift (I F) b c, TOr a c d) => TSetBit a b d
tSetBit :: TSetBit a b c => a -> b -> c
class TChangeBit a b c d | a b c -> d
change bit #b in a to c in {T,F}, yielding d.
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(TSetBit a b d, TUnSetBit a b e, TIf c d e f) => TChangeBit a b c f
tChangeBit :: TChangeBit a b c d => a -> b -> c -> d
class TUnSetBit a b c | a b -> c
set bit #b in a to F, yielding c
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(TShift (O T) b c, TAnd a c d) => TUnSetBit a b d
tUnSetBit :: TUnSetBit a b c => a -> b -> c
class TComplementBit a b c | a b -> c
toggle the value of bit #b in a, yielding c
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(TShift (I F) b c, TXOr' a c d) => TComplementBit a b d
tComplementBit :: TComplementBit a b c => a -> b -> c
class TCountBits a b | a -> b
Count the number of bits set. Since we may have an infinite tail of 1s, we return a negative number in such cases indicating how many bits are NOT set.
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(TIsNegative a t, TCountBits' a b t) => TCountBits a b
tCountBits :: TCountBits a b => a -> b
class TAbs a b | a -> b
Return the absolute value of a
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(TIsNegative a s, TNeg a a', TIf s a' a a'') => TAbs a a''
tAbs :: TAbs a b => a -> b
class TNF a b | a -> b
Shed the additional reduction parameter from TNF'
show/hide Instances
TNF' a b c => TNF a b
tNF :: TNF a b => a -> b
t2n :: TNF (O a) b => a -> b
t2np1 :: TNF (I a) b => a -> b
class TShift' a b c | a b -> c
Shift a right b places obtaining c. If b is negative then we shift left. | TShift' does not yield normal form answers.
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TShift' F F F
TShift' F T F
TShift' T F T
TShift' T T T
(TShift' a b c, TShift' c b d) => TShift' a (O b) d
(TShift' a b c, TShift' c b d) => TShift' a (I b) (O d)
TShift' (I a) T a
TShift' (O a) T a
TShift' (I a) F (I a)
TShift' (O a) F (O a)
class TNF' a b c | a -> b c
Transform a number into normal form, but track whether further reductions may be necessary if this number is extended for efficiency.
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TNF' F F F
TNF' T T F
(TNF' (I a) c b, TIf b (I c) T d) => TNF' (I (I a)) d b
TNF' (I T) T F
TNF' (O F) F F
(TNF' (O a) c b, TIf b (O c) F d) => TNF' (O (O a)) d b
TNF' (I F) (I F) T
TNF' (O a) c b => TNF' (I (O a)) (I c) T
TNF' (I a) c b => TNF' (O (I a)) (O c) T
TNF' (O T) (O T) T
class TAddC' a b c d | a b c -> d
A symmetrical full adder, that does not yield normal form answers.
show/hide Instances
TAddC' F F F F
TAddC' F T F T
TAddC' F T T F
TAddC' T F F T
TAddC' T F T F
TAddC' T T T T
TAddC' F F T (I F)
TAddC' T T F (O T)
TAddC' F (I a) F (I a)
TSucc a b => TAddC' F (I a) T (O b)
TAddC' F (O a) F (O a)
TAddC' F (O a) T (I a)
TAddC' T (I a) F (O a)
TAddC' T (I a) T (I a)
TSucc b a => TAddC' T (O a) F (I b)
TAddC' T (O a) T (O a)
TAddC' (I a) F F (I a)
TSucc a b => TAddC' (I a) F T (O b)
TAddC' (I a) T F (O a)
TAddC' (I a) T T (I a)
TAddC' (O a) F F (O a)
TAddC' (O a) F T (I a)
TSucc b a => TAddC' (O a) T F (I b)
TAddC' (O a) T T (O a)
TAddC' a b T c => TAddC' (I a) (I b) F (O c)
TAddC' a b T c => TAddC' (I a) (I b) T (I c)
TAddC' a b F c => TAddC' (I a) (O b) F (I c)
TAddC' a b T c => TAddC' (I a) (O b) T (O c)
TAddC' a b F c => TAddC' (O a) (I b) F (I c)
TAddC' a b T c => TAddC' (O a) (I b) T (O c)
TAddC' a b F c => TAddC' (O a) (O b) F (O c)
TAddC' a b F c => TAddC' (O a) (O b) T (I c)
class TAdd' a b c | a b -> c
Non-reversible addition. Kept for efficiency purposes.
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(TAddC' a b F d, TNF d d') => TAdd' a b d'
tAdd' :: TAdd' a b c => a -> b -> c
class TSub' a b c | a b -> c
Non-reversible subtraction. Kept for efficiency purposes.
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(TNeg b b', TAdd' a b' c) => TSub' a b c
tSub' :: TSub' a b c => a -> b -> c
class TCountBits' a b t | a t -> b
Count the number of bits set, but track whether the number is positive or negative to simplify casing. Since we may have an infinite tail of 1s, we return a negative number in such cases indicating how many bits are NOT set.
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class TBool d => LSB a d a' | a -> d a', d a' -> a
Extracts the least significant bit of a as d and returns a'. Can also be used to prepend bit d onto a' obtaining a.
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LSB F F F
LSB T T T
LSB (I F) T F
LSB (O T) F T
LSB (I (I n)) T (I n)
LSB (I (O n)) T (O n)
LSB (O (I n)) F (I n)
LSB (O (O n)) F (O n)
tLSB :: LSB a d a' => a -> d -> a'
tBSL :: LSB a d a' => a' -> d -> a
class LSB (I a) T a => XI a
assert 2n+1 != n
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LSB (I a) T a => XI a
class LSB (O a) F a => XO a
assert 2n != n
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LSB (O a) F a => XO a
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