exp-pairs 0.1.4.1 → 0.1.5.0

raw patch · 21 files changed

+4170/−175 lines, 21 filesPVP: major bump suggested

API removals or changes: PVP suggests a major version bump

API changes (from Hackage documentation)

Files

• CHANGELOG.md +6/−0
• Math/ExpPairs.hs +56/−12
• Math/ExpPairs/Ivic.hs +55/−48
• Math/ExpPairs/Kratzel.hs +259/−54
• Math/ExpPairs/LinearForm.hs +15/−12
• Math/ExpPairs/Matrix3.hs +11/−2
• Math/ExpPairs/MenzerNowak.hs +6/−9
• Math/ExpPairs/Pair.hs +5/−1
• Math/ExpPairs/PrettyProcess.hs +1/−1
• Math/ExpPairs/Process.hs +17/−13
• Math/ExpPairs/ProcessMatrix.hs +3/−0
• Math/ExpPairs/RatioInf.hs +9/−0
• exp-pairs.cabal +3/−2
• tests/Instances.hs +13/−3
• tests/Ivic.hs +55/−18
• tests/Process.hs +29/−0
• tests/Tests.hs +2/−0
• tests/etalon-mOnS.txt +791/−0
• tests/etalon-tauab.txt +974/−0
• tests/etalon-tauabc.txt +966/−0
• tests/etalon-zetaOnS.txt +894/−0

@@ -1,6 +1,12 @@ Changes ======= +Version 0.1.5.0+----------------++Convenient combinators for linear forms and constraints.+New experimental functions in Math.ExpPairs.Kratzel: tauabcd and tauA for general multidimensional divisor problems.+ Version 0.1.4.0 ---------------- 

@@ -14,7 +14,9 @@ A set of useful applications can be found in "Math.ExpPairs.Ivic", "Math.ExpPairs.Kratzel" and "Math.ExpPairs.MenzerNowak". -}-{-# LANGUAGE CPP  #-}+{-# LANGUAGE CPP             #-}+{-# LANGUAGE PatternSynonyms #-}+{-# LANGUAGE TemplateHaskell #-}  module Math.ExpPairs   ( optimize@@ -24,14 +26,19 @@   , optimalPath   , simulateOptimize   , simulateOptimize'-  , LinearForm (..)+  , LinearForm   , RationalForm (..)-  , IneqType (..)-  , Constraint (..)+  , IneqType+  , Constraint   , InitPair   , Path   , RatioInf (..)   , RationalInf+  , pattern K+  , pattern L+  , pattern M+  , (>.), (>=.), (<.), (<=.)+  , scaleLF   ) where  import Control.Arrow hiding ((<+>))@@ -39,18 +46,47 @@ import Data.Ord      (comparing) import Data.List     (minimumBy) import Data.Monoid+import Data.Ratio import Text.PrettyPrint.Leijen hiding ((<$>), (<>)) import qualified Text.PrettyPrint.Leijen as PP import Text.Printf+import Data.Function.Memoize (deriveMemoizable)  import Math.ExpPairs.LinearForm import Math.ExpPairs.Process import Math.ExpPairs.Pair import Math.ExpPairs.RatioInf +-- | For a given @c@ returns linear form @c * k@+pattern K n = LinearForm n 0 0+-- | For a given @c@ returns linear form @c * l@+pattern L n = LinearForm 0 n 0+-- | For a given @c@ returns linear form @c * m@+pattern M n = LinearForm 0 0 n++-- | Build a constraint, which states that the value of the first linear form is greater than the value of the second one.+(>.) :: Num t => LinearForm t -> LinearForm t -> Constraint t+lf1 >. lf2  = Constraint (lf1 - lf2) Strict+infix 5 >.++-- | Build a constraint, which states that the value of the first linear form is greater or equal to the value of the second one.+(>=.) :: Num t => LinearForm t -> LinearForm t -> Constraint t+lf1 >=. lf2 = Constraint (lf1 - lf2) NonStrict+infix 5 >=.++-- | Build a constraint, which states that the value of the first linear form is less than the value of the second one.+(<.) :: Num t => LinearForm t -> LinearForm t -> Constraint t+lf1 <. lf2  = Constraint (lf2 - lf1) Strict+infix 5 <.++-- | Build a constraint, which states that the value of the first linear form is less or equal to the value of the second one.+(<=.) :: Num t => LinearForm t -> LinearForm t -> Constraint t+lf1 <=. lf2 = Constraint (lf2 - lf1) NonStrict+infix 5 <=.+ evalFunctional :: [InitPair] -> [InitPair] -> [RationalForm Rational] -> [Constraint Rational] -> Path -> (RationalInf, InitPair) evalFunctional corners interiors rfs cons path = case rs of-  [] -> (InfPlus, undefined)+  [] -> (InfPlus, error "evalFunctional: cannot find any exponential pair, which satisfies constraints")   _  -> minimumBy (comparing fst) rs   where     applyPath  = map (evalPath path . initPairToProjValue &&& id)@@ -58,9 +94,10 @@     interiors' = applyPath interiors      predicate (p, _) = all (checkConstraint p) cons-    qs = if all predicate corners'-          then corners'-          else filter predicate interiors'+    qs+      | all predicate corners' = corners'+      | any predicate corners' = filter predicate interiors'+      | otherwise              = []      rs = map (first $ \p -> maximum (map (evalRF p) rfs)) qs @@ -80,11 +117,18 @@   }   deriving (Show) +deriveMemoizable ''OptimizeResult+ instance Pretty OptimizeResult where-  pretty (OptimizeResult r' ip p) = pretty' r' PP.<$> pretty ip </> pretty p where-    pretty' r@(Finite rr) = text (printf "%.6f" (fromRational rr :: Double)) <+> equals <+> pretty r-    pretty' r = pretty r+  pretty (OptimizeResult r' ip p) = pretty1 r' PP.<$>+    (parens (pretty (k%m) PP.<> comma PP.<> pretty (l%m)) <+> equals <+> pretty p </> pretty ip)+    where+      pretty1 r@(Finite rr) = text (printf "%.6f" (fromRational rr :: Double)) <+> equals <+> pretty r+      pretty1 r = pretty r +      (k, l, m) = evalPath p $ initPairToProjValue ip++ instance Eq OptimizeResult where   (==) = (==) `on` optimalValue @@ -127,5 +171,5 @@       pathba  = path <> baPath       branchB@(OptimizeResult r2' _ _) = optimize' rfs cons (OptimizeResult r1 ip1 pathba) -    consBuilder rr (RationalForm num den) = Constraint (substituteLF (num, den, 1) (LinearForm (-1) (toRational rr) 0)) Strict+    consBuilder rr (num :/: den) = (substituteLF (num, den, 1) (L (toRational rr) - K 1)) >. 0 

@@ -21,6 +21,7 @@   , findMinAbscissa   , mBigOnHalf   , reverseMBigOnHalf+  , kolpakova2011   ) where  import Data.Ratio@@ -35,8 +36,8 @@ zetaOnS s   | s >= 1  = simulateOptimize 0   | s >= 1%2 = optimize-    [RationalForm (LinearForm 1 1 (-s)) 2]-    [Constraint (LinearForm (-1) 1 (-s)) NonStrict]+    [K 1 + L 1 - M s :/: 2]+    [L 1 >=. K 1 + M s]   | otherwise = optRes {optimalValue = r} where     optRes = zetaOnS (1-s)     r = Finite (1%2 - s) + optimalValue optRes@@ -48,10 +49,10 @@ reverseZetaOnS :: Rational -> OptimizeResult reverseZetaOnS mu   | mu >= 1%2   = simulateOptimize 0-  | mu > zetaOnHalf = optimize [RationalForm (LinearForm 1 (-1) 1) 1] [Constraint (LinearForm 0 (-2) (1+2*mu)) NonStrict]+  | mu > zetaOnHalf = optimize [K 1 - L 1 + M 1 :/: 1] [M (1 + 2 * mu) >=. L 2]   | mu == zetaOnHalf = simulateOptimize (1 % 2)   | otherwise = optRes {optimalValue = negate $ optimalValue optRes} where-  optRes = optimize [RationalForm (LinearForm 1 (-1) 0) 1] [Constraint (LinearForm 1 0 (-mu)) NonStrict, Constraint (LinearForm (-1) 1 (-1%2)) NonStrict]+  optRes = optimize [K 1 - L 1 :/: 1] [K 1 >=. M mu, L 2 >=. K 2 + 1]  lemma82_f :: Rational -> Rational lemma82_f s@@ -77,42 +78,58 @@     beta1  = -12/(1+2*s)     x1 = optRes {optimalValue = Finite $ (1-alpha1)/muS - beta1} -    --alpha2 = 4*(1-s)*(k+l)/((2*m+4*l)*s-m+2*k-2*l)-    --beta2  = -4*(m+2*k+2*l)/((2*m+4*l)*s-m+2*k-2*l)-    --ratio = (1-alpha2)/muS - beta2-    --numer = numerator ratio-    --denom = denominator ratio-    numer = LinearForm-      (-4*s + (-8*muS + 2))-      (-8*s + (-8*muS + 6))-      (-2*s + (-4*muS + 1))-    denom = LinearForm-      (2*muS)-      (4*muS*s - 2*muS)-      (2*muS*s - muS)+    -- alpha2 = 4*(1-s)*(k+l)/((2*m+4*l)*s-m+2*k-2*l)+    -- beta2  = -4*(m+2*k+2*l)/((2*m+4*l)*s-m+2*k-2*l)+    -- numer % denom = (1-alpha2)/muS - beta2+    t     = scaleLF s (L 4 + 2) - 1 + K 2 - L 2+    numer = t - scaleLF (4 * (1-s)) (K 1 + L 1) + scaleLF (4 * muS) (K 2 + L 2 + 1)+    denom = scaleLF muS t -    cons = if s >= 2%3 then [] else [Constraint-      (LinearForm (4*s-2) (8*s-6) (2*s-1)) NonStrict-      ]+    cons = if s >= 2%3 then [] else [scaleLF s (K 4 + L 8 + 2) >=. K 2 + L 6 + 1] -    x2' = optimize [RationalForm numer denom] cons+    x2' = optimize [- numer :/: denom] cons     x2 = x2' {optimalValue = negate $ optimalValue x2'} +data Choice = Least | Median | Greatest++binarySearch :: (Rational -> Bool) -> Choice -> Rational -> Rational -> Rational -> Rational+binarySearch predicate choice precision = go+  where+    go a b+      | b - a < precision = case choice of+                            Least    -> a+                            Median   -> c+                            Greatest -> b+      | predicate c = go a c+      | otherwise   = go c b+      where+        c = (numerator a + numerator b) % (denominator a + denominator b)++mOnSTwoThird :: RationalInf+mOnSTwoThird = optimalValue $ mOnS $ 2 % 3+ -- | Try to reverse 'mOnS': for a given precision and m compute minimal possible σ. -- Implementation is usual try-and-divide search, so performance is very poor. -- Sometimes, when 'mOnS' gets especially lucky exponent pair, 'reverseMOnS' can miss--- real σ and returns bigger value.+-- real σ and returns significantly bigger value.+--+-- For integer m>=4 this function corresponds to the multidimensional Dirichlet problem+-- and returns σ from error term O(x^{σ+ε}). See Ch. 13 in Ivić2003. reverseMOnS :: Rational -> RationalInf -> Rational-reverseMOnS prec m = reverseMOnS' from to where-  from = 1 % 2-  to   = 1-  reverseMOnS' a b-    | b - a < prec = c-    | optimalValue (mOnS c) > m = reverseMOnS' a c-    | otherwise = reverseMOnS' c b-    where-      c = (numerator a + numerator b) % (denominator a + denominator b)+reverseMOnS _ InfPlus = 1+reverseMOnS _ (Finite m)+  | m <= 4 = 1 % 2+  | m <= 8 = 3 % 4 - recip m+reverseMOnS prec m+  | m < mOnSTwoThird = go (5 % 8) (2 % 3)+  | otherwise        = go (2 % 3) 1+  where+    go = binarySearch (\c -> optimalValue (mOnS c) > m) Greatest prec +-- | An estimate of the symmetric multidimensional divisor function from Kolpakova, 2011.+kolpakova2011 :: Integer -> Double+kolpakova2011 k = 1 - 1/3 * 2**(2/3) * (4.45 * fromInteger k)**(-2/3)+ -- | Check whether ∫_1^T   Π_i |ζ(n_i*σ+it)|^m_i dt ≪ T^(1+ε) for a given list of pairs [(n_1, m_1), ...] and fixed σ. checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool checkAbscissa xs s = sum rs < Finite 1 where@@ -122,15 +139,7 @@ -- | Find for a given precision and list of pairs [(n_1, m_1), ...] the minimal σ -- such that ∫_1^T   Π_i|ζ(n_i*σ+it)|^m_i dt ≪ T^(1+ε). findMinAbscissa :: Rational -> [(Rational, Rational)] -> Rational-findMinAbscissa prec xs = searchMinAbscissa' from to where-  from = 1 % 2 / minimum (map fst xs)-  to   = 1 % 1-  searchMinAbscissa' a b-    | b - a < prec = b-    | checkAbscissa xs c = searchMinAbscissa' a c-    | otherwise = searchMinAbscissa' c b-    where-      c = (numerator a + numerator b) % (denominator a + denominator b)+findMinAbscissa prec xs = binarySearch (checkAbscissa xs) Greatest prec (1 % 2 / minimum (map fst xs)) 1  -- | Compute minimal M(A) such that ∫_1^T |ζ(1/2+it)|^A dt ≪ T^(M(A)+ε). -- See Ch. 8 in Ivić2003. Further justification will be published elsewhere.@@ -138,16 +147,16 @@ mBigOnHalf a   | a < 4     = simulateOptimize 1   | a < 12    = simulateOptimize $ 1+(a-4)/8-  | a > 41614060315296730740083860226662 % 2636743270445733804969041895717 = simulateOptimize $ 1 + 32*(a-6)/205+  | a > 41614060315296730740083860226662 % 2636743270445733804969041895717 = simulateOptimize $ 1 + (a - 6) * zetaOnHalf   | otherwise = if Finite x >= optimalValue optRes     then simulateOptimize x     else optRes where-      optRes = optimize [RationalForm (LinearForm 1 1 0) (LinearForm 1 0 0)]-        [Constraint (LinearForm (4-a) 4 2) NonStrict]+      optRes = optimize [K 1 + L 1 :/: K 1]+        [K (4 - a) + L 4 + 2 >=. 0]       x = 1 + 32*(a-6)/205 -- Constant 41614060315296730740083860226662 % 2636743270445733804969041895717 -- is produced by--- optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (-64) (-77) 64) Strict]+-- optimize [K 4 + L 4 + 2 :/: K 1] [64 >. K 64 + L 77]  -- | Try to reverse 'mBigOnHalf': for a given M(A) find maximal possible A. -- Sometimes, when 'mBigOnHalf' gets especially lucky exponent pair, 'reverseMBigOnHalf' can miss@@ -158,7 +167,5 @@   | otherwise = if Finite a <= optimalValue optRes     then simulateOptimize a     else optRes where-    a = (m-1)*205/32 + 6-    optRes = optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (1-m) 1 0) NonStrict]--+    a = (m - 1) / zetaOnHalf + 6+    optRes = optimize [K 4 + L 4 + 2 :/: K 1] [K (1 - m) + L 1 >=. 0]

@@ -24,20 +24,32 @@ He also provided a set of theorems to estimate Θ(a, b) and Θ(a, b, c).  -}++{-# LANGUAGE TemplateHaskell #-}+ module Math.ExpPairs.Kratzel   ( TauabTheorem (..)   , tauab   , TauabcTheorem (..)   , tauabc+  , TauabcdTheorem (..)+  , tauabcd+  , Theorem (..)+  , TauAResult (..)+  , tauA   ) where -import Control.Arrow-import Data.Ratio ((%))+import Control.Arrow hiding ((<+>))+import Data.Function+import Data.Maybe+import Data.Ratio import Data.Ord   (comparing)-import Data.List  (minimumBy)+import Data.List  (minimumBy, sort) import Text.PrettyPrint.Leijen+import Data.Function.Memoize    (memoize, deriveMemoizable)  import Math.ExpPairs+import Math.ExpPairs.Ivic  -- |Special type to specify the theorem of Krätzel1988, -- which provided the best estimate of Θ(a, b)@@ -58,32 +70,36 @@ divideResult :: Real a => a -> (b, OptimizeResult) -> (b, OptimizeResult) divideResult d = second (\o -> o {optimalValue = optimalValue o / Finite (toRational d)}) +tauab' :: Rational -> Rational -> (TauabTheorem, OptimizeResult)+tauab' a b = minimumBy (comparing snd) [kr511a, kr511b, kr512a, kr512b]+  where+    kr511a = (Kr511a, optimize+      [K 2 + L 2 - 1 :/: M (a+b)]+      [L (2 * a) >=. K (2 * b) + M a])+    kr511b = (Kr511b, optimize+      [K 1 :/: K b - L a + M a]+      [K (2 * b) + M a >. L (2 * a)])+    kr512a = (Kr512a, simulateOptimize r)+      where+        r = if 11*a >= 8*b then 19/29/(a+b) else 1%1+    kr512b = if 11*a >= 8*b then kr512a else (Kr512b, optimize+      [L 8 - K 11 - 4 :/: L (29 * a) - K (29 * b) + M (4*b-20*a)]+      [ L (2 * a) >=. K (2 * b) + M a+      , 4 >. K 29+      , K 29 + L 29 >. 24+      ])+ -- |Compute Θ(a, b) for given a and b. tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult)-tauab a' b'-  | d /= 1 = divideResult d $ tauab (a'`div` d) (b' `div` d) where-      d = gcd a' b'-tauab a' b' = minimumBy (comparing (optimalValue . snd)) [kr511a, kr511b, kr512a, kr512b] where-  a = toRational a'-  b = toRational b'-  kr511a = (Kr511a, optimize-    [RationalForm (LinearForm 2 2 (-1)) (LinearForm 0 0 (a+b))]-    [Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict])-  kr511b = (Kr511b, optimize-    [RationalForm (LinearForm 1 0 0) (LinearForm b (-a) a)]-    [Constraint (LinearForm (2*b) (-2*a) a) Strict])-  kr512a = (Kr512a, simulateOptimize r) where-    r = if 11*a >= 8*b then 19/29/(a+b) else 1%1-  kr512b = if 11*a >= 8*b then kr512a else (Kr512b, optimize-    [-      RationalForm (LinearForm (-11) 8 (-4)) (LinearForm (-29*b) (29*a) (4*b-20*a))-    ]-    [-      Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict,-      Constraint (LinearForm (-29) 0 4) Strict,-      Constraint (LinearForm 29 29 (-24)) Strict-    ])+tauab a b+  | d /= 1 = divideResult d $ tauab (a `div` d) (b `div` d)+    where+      d = a `gcd` b+tauab a b = tauab' a' b'+  where+    [a', b'] = sort $ map toRational [a, b] + -- |Special type to specify the theorem of Krätzel1988, -- which provided the best estimate of Θ(a, b, c) data TauabcTheorem@@ -113,35 +129,224 @@   pretty (Tauab t) = pretty t   pretty t         = pretty (show t) +tauabc' :: Rational -> Rational -> Rational -> (TauabcTheorem, OptimizeResult)+tauabc' a b c = minimumBy (comparing snd) [kr61, kr62, kr63, kr64, kr65, kr66]+  where+    abc = a + b + c+    kr61+      | c<a+b = (Kr61, simulateOptimize $ 2/abc)+      | optimalValue optRes < Finite (recip c) = (Kr61, simulateOptimize $ 1/c)+      | otherwise = (Tauab th, optRes)+      where+        (th, optRes) = tauab' a b+    kr62 = (Kr62, optimize+      [K 2 + L 2 :/: M (a + b + c)]+      [ L a >=. K (b + c)+      , M (a + b + c) >=. K (2 * c) + L (2 * c)+      ])+    kr63 = (Kr63, optimize+      [K 4 + L 2 + 3 :/: K (2 * abc) + M (3 * abc)]+      [scaleLF (2 * a) (K 1 + L 1 + 1) >=. scaleLF (b + c) (K 2 + 1)])+    kr64 = (Kr64, simulateOptimize r) where+      r = recip abc * minimum (abc:[2-4*(k-1)%(3*2^k-4) | k<-[1..maxk], (3*2^k-2*k-4)%1 * a >= 2 * (b+c), (3*2^k-8)%1 * (a+b) >= (3*2^k-4*k+4)%1 * c])+      maxk = 4 `max` floor (logBase 2 (fromRational $ b+c) :: Double)+    kr65 = (Kr65, simulateOptimize r) where+      r = if 7*a>=2*(b+c) && 4*(a+b)>=5*c then 3%2/abc else 1%1+    kr66 = (Kr66, simulateOptimize r) where+      r = if 18*a>=7*(b+c) && 2*(a+b)>=3*c then 25%17/abc else 1%1+ -- |Compute Θ(a, b, c) for given a, b and c. tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult)-tauabc a' b' c'-  | d /= 1 = divideResult d $ tauabc (a'`div` d) (b' `div` d) (c' `div` d) where-      d = gcd (gcd a' b') c' tauabc 1 1 1 = (Kolesnik, simulateOptimize $ 43%96)-tauabc a' b' c' = minimumBy (comparing (optimalValue . snd)) [kr61, kr62, kr63, kr64, kr65, kr66] where-  a = toRational a'-  b = toRational b'-  c = toRational c'-  kr61-    | c<a+b = (Kr61, simulateOptimize $ 2/(a+b+c))-    | optimalValue optRes < Finite (recip c) = (Kr61, simulateOptimize $ 1/c)-    | otherwise = (Tauab th, optRes)+tauabc a b c+  | d /= 1 = divideResult d $ tauabc (a `div` d) (b `div` d) (c `div` d)     where-      (th, optRes) = tauab a' b'-  kr62 = (Kr62, optimize-    [RationalForm (LinearForm 2 2 0) (LinearForm 0 0 (a+b+c))]-    [-      Constraint (LinearForm (-b-c) a 0) NonStrict,-      Constraint (LinearForm (-2*c) (-2*c) (a+b+c)) NonStrict-    ])-  kr63 = (Kr63, optimize-    [RationalForm (LinearForm 4 2 3) (LinearForm (2*(a+b+c)) 0 (3*(a+b+c)))]-    [Constraint (LinearForm (2*(a-b-c)) (2*a) (2*a-b-c)) NonStrict])-  kr64 = (Kr64, simulateOptimize r) where-    r = recip (a+b+c) * minimum ((a+b+c):[2-4*(k-1)%(3*2^k-4) | k<-[1..maxk], (3*2^k-2*k-4)%1 * a >= 2 * (b+c), (3*2^k-8)%1 * (a+b) >= (3*2^k-4*k+4)%1 * c])-    maxk = 4 `max` floor (logBase 2 (fromRational $ b+c) :: Double)-  kr65 = (Kr65, simulateOptimize r) where-    r = if 7*a>=2*(b+c) && 4*(a+b)>=5*c then 3%2/(a+b+c) else 1%1-  kr66 = (Kr66, simulateOptimize r) where-    r = if 18*a>=7*(b+c) && 2*(a+b)>=3*c then 25%17/(a+b+c) else 1%1+      d = a `gcd` b `gcd` c+tauabc a b c = tauabc' a' b' c'+  where+    [a', b', c'] = sort $ map toRational [a, b, c]+++-- |Special type to specify the theorem of Krätzel1988,+-- which provided the best estimate of Θ(a, b, c, d)+data TauabcdTheorem+  = HeathBrown+  | Tauabc TauabcTheorem+  | Kr611+  | Kr1992_2+  | Kr1992_31+  | Kr1992_32+  | Kr2010_1a+  | Kr2010_1b+  | Kr2010_2+  | Kr2010_3+  | CaoZhai+  deriving (Eq, Ord, Show)++instance Pretty TauabcdTheorem where+  pretty (Tauabc t) = pretty t+  pretty t          = pretty (show t)++tauabcd' :: Rational -> Rational -> Rational -> Rational -> (TauabcdTheorem, OptimizeResult)+tauabcd' a1 a2 a3 a4 = minimumBy (comparing snd) [kr611, kr1992_2, kr1992_31, kr1992_32, kr2010_1a, kr2010_1b, kr2010_2, kr2010_3]+  where+    a12 = a1 + a2+    a123 = a1 + a2 + a3+    a1234 = a1 + a2 + a3 + a4++    kr611+      | optimalValue optRes3 < Finite (recip a4) = (Kr611, optimize [form] cons)+      | otherwise = (Tauabc th3, optRes3)+      where+        (th3, optRes3) = tauabc' a1 a2 a3+        form = K 2 + L 2 + 1 :/: M (a1 + a2 + a3 + a4)                              -- (6.46)+        cons =+          [ scaleLF a1 (L 2 - 1) >. K (2 * a4)                                      -- (6.41)+          , scaleLF a3 (scaleLF a2 (K 2 + L 2) + M a4) <. M ((a1 + a2) * (a2 + a4)) -- (6.42)+          , scaleLF a1 (K 2 + L 2 + 1) >=. M (a2 + a3)+          , M (a1 + a2) >=. scaleLF a3 (K 2 + L 2 - 1)+          ]++    kr1992_2 = (Kr1992_2, optimize [form] cons)+      where+        form = K 3 + L 1 + 4 :/: scaleLF a1234 (K 1 + 2)+        cons =+          [ scaleLF a4 (K 6 + L 2 + 8) <. scaleLF a1234 (K 2 + 4)+          , scaleLF a1234 (K 2 + 2) <=. scaleLF a1 (K 6 + L 2 + 8)+          ]++    kr1992_31 = (Kr1992_31, optimize [form] cons1)+      where+        form = K 1 + L 1 + 2 :/: K a1 + L a1 + M a1234+        cons0 =+          [ scaleLF a4 (K 1 + L 1 + 2) <. K a1 + L a1 + M a1234+          , scaleLF a1 (K 2 + L 2 + 2) <=. scaleLF (a2 + a3) (K 2 + 1)+          ]+        cons1 = cons0 +++          [ L a1 <=. K a2+          , scaleLF a1 (K 1 + L 1 + 1) >=. K (a2 + a3)+          ]++    kr1992_32 = (Kr1992_32, simulateOptimize $ if cond then val else 1)+      where+        k = 32 % 205+        l = k + 1 % 2+        val = (k + l + 2) / ((k + l) * a1 + a1234)+        cond = (k + l - 2) * a4 < (k + l) * a1 + a1234+          && 2 * (k + l + 1) * a1 <= (2 * k + l) * (a2 + a3)+          && l * a1 >= k * a2+          && (l - k) * (2 * k + 1) * a3 <= (2 * l - 2 * k - 1) * (k + l + 1) * a1 + (2 * k * (k - l + 1) + 1) * a2++    kr2010_1a = (Kr2010_1a, simulateOptimize $ if cond then val else 1)+      where+        val = 45 % 19 / a1234+        cond = 5 * a1 >= a1234 && 9 * a123 >= 34 * a1 && 9 * a12 >= 20 * a1++    kr2010_1b = (Kr2010_1b, simulateOptimize $ if cond then val else 1)+      where+        val = 45 / (15 * a1 + 16 * a1234)+        cond = 5 * a1 <= a1234 && a1234 <= 15%2 * a1 && 2 * a123 >= 9 * a1 && 2 * a12 >= 5 * a1++    kr2010_2 = (Kr2010_2, simulateOptimize $ if cond then val else 1)+      where+        val = 35 / (11 * a4 + 16 * a123)+        cond = 2 * a123 >= 3 * a4 && 34 * a1 >= 9 * a123 && 7 * a12 >= 10 * a3++    kr2010_3 = (Kr2010_3, simulateOptimize $ if cond then val else 1)+      where+        val = 235 / (406 * a1 + 81 * a4)+        cond = a1 == a2 && 2 * a1 == a3 && 29 * a1 >= 11 * a4+++-- |Compute Θ(a, b, c, d) for given a, b, c and d.+tauabcd :: Integer -> Integer -> Integer -> Integer -> (TauabcdTheorem, OptimizeResult)+tauabcd 1 1 1 1 = (HeathBrown, simulateOptimize $ 1%2)+tauabcd a1 a2 a3 a4+  | d /= 1 = divideResult d $ tauabcd (a1 `div` d) (a2 `div` d) (a3 `div` d) (a4 `div` d)+    where+      d = a1 `gcd` a2 `gcd` a3 `gcd` a4+tauabcd a1 a2 a3 a4 = tauabcd' a1' a2' a3' a4'+  where+    [a1', a2', a3', a4'] = sort $ map toRational [a1, a2, a3, a4]+++-- |Special type to specify the theorem of Krätzel1988,+-- which provided the best estimate of Θ(a1, a2...)+data Theorem+  = NoTheorem+  | Ivic+  | Ab   TauabTheorem+  | Abc  TauabcTheorem+  | Abcd TauabcdTheorem+  deriving (Eq, Ord, Show)++instance Pretty Theorem where+  pretty NoTheorem = text ""+  pretty Ivic = text "Ivic"+  pretty (Ab t) = pretty t+  pretty (Abc t) = pretty t+  pretty (Abcd t) = pretty t++-- |Special type to specify the theorem of Krätzel1988,+-- which provided the best estimate of Θ(a1, a2...)+data TauAResult+  = Node Theorem OptimizeResult+  | Combination TauAResult TauAResult Rational+  deriving (Show)++instance Pretty TauAResult where+  pretty (Node th o) = pretty th <+> pretty o+  pretty (Combination t1 t2 r) = pretty t1 <+> pretty t2 <+> pretty r+++deriveMemoizable ''Theorem+deriveMemoizable ''TauabTheorem+deriveMemoizable ''TauabcTheorem+deriveMemoizable ''TauabcdTheorem+deriveMemoizable ''TauAResult++extractValue :: TauAResult -> Rational+extractValue (Node _ o) = toRational $ optimalValue o+extractValue (Combination _ _ r1) = r1++instance Eq TauAResult where+  (==) = (==) `on` extractValue++instance Ord TauAResult where+  compare = compare `on` extractValue++-- | Compute Θ(a1, a2...) for given list [a1, a2...].+tauA :: [Integer] -> TauAResult+tauA = go' . sort+  where+    fi :: Integer -> Rational+    fi = fromIntegral++    go' = memoize go++    go :: [Integer] -> TauAResult+    go [] = Node NoTheorem (simulateOptimize 0)+    go [_] = Node NoTheorem (simulateOptimize 0)+    go [a, b]    = (\(t, o) -> Node (Ab t) o) $ tauab a b+    go [a, b, c] = (\(t, o) -> Node (Abc t) o) $ tauabc a b c+    go as@[a,b,c,d] = (\(t, o) -> Node (Abcd t) o) (tauabcd a b c d) `min` go608 as+    go as@(a:_)+      | all (== a) as+      = Node Ivic $ simulateOptimize $ reverseMOnS 1e-6 (fromIntegral $ length as) / fi a+    go as = go608' as++    go608' = memoize go608++    go608 as = minimum $ mapMaybe f [1 .. length as - 1]+      where+        f q = if (alphaV `max` betaV) < 1 / fi (last as)+          then Just $ Combination alpha beta ret+          else Nothing+          where+            alpha = go' $ take q as+            alphaV = extractValue alpha+            beta  = go' $ drop q as+            betaV = extractValue beta+            a0 = fi $ head as+            aq = fi $ as !! q+            ret = (1 - a0 * aq * alphaV * betaV) / (a0 + aq - a0 * aq * (alphaV + betaV))

@@ -12,6 +12,7 @@ -} module Math.ExpPairs.LinearForm   ( LinearForm (..)+  , scaleLF   , evalLF   , substituteLF   , RationalForm (..)@@ -33,7 +34,7 @@  import Math.ExpPairs.RatioInf --- |Define an affine linear form of two variables: a*k + b*l + c*m.+-- |Define an affine linear form of three variables: a*k + b*l + c*m. -- First argument of 'LinearForm' stands for a, second for b -- and third for c. Linear forms form a monoid by addition. data LinearForm t = LinearForm !t !t !t@@ -62,6 +63,7 @@   mempty  = 0   mappend = (+) +-- | Multiply a linear form by a given coefficient. scaleLF :: (Num t, Eq t) => t -> LinearForm t -> LinearForm t scaleLF 0 = const 0 scaleLF s = fmap (* s)@@ -77,26 +79,27 @@ substituteLF (k, l, m) (LinearForm a b c) = scaleLF a k + scaleLF b l + scaleLF c m  -- | Define a rational form of two variables, equal to the ratio of two 'LinearForm'.-data RationalForm t = RationalForm (LinearForm t) (LinearForm t)+data RationalForm t = (LinearForm t) :/: (LinearForm t)   deriving (Eq, Show, Functor, Foldable, Generic)+infix 5 :/:  instance (Num t, Eq t, Pretty t) => Pretty (RationalForm t) where-  pretty (RationalForm l1 l2) = parens (pretty l1) </> parens (pretty l2)+  pretty (l1 :/: l2) = parens (pretty l1) </> parens (pretty l2)  instance NFData t => NFData (RationalForm t) where   rnf = rnf . toList  instance Num t => Num (RationalForm t) where-  (+) = error "Addition of RationalForm is undefined"-  (*) = error "Multiplication of RationalForm is undefined"-  negate (RationalForm a b) = RationalForm (negate a) b-  abs = error "Absolute value of RationalForm is undefined"-  signum = error "Signum of RationalForm is undefined"-  fromInteger n = RationalForm (fromInteger n) 1+  (+)              = error "Addition of RationalForm is undefined"+  (*)              = error "Multiplication of RationalForm is undefined"+  negate (a :/: b) = negate a :/: b+  abs              = error "Absolute value of RationalForm is undefined"+  signum           = error "Signum of RationalForm is undefined"+  fromInteger n    = fromInteger n :/: 1  instance Num t => Fractional (RationalForm t) where-  fromRational r = RationalForm (fromInteger $ numerator r) (fromInteger $ denominator r)-  recip (RationalForm a b) = RationalForm b a+  fromRational r = fromInteger (numerator r) :/: fromInteger (denominator r)+  recip (a :/: b) = b :/: a  mapTriple :: (a -> b) -> (a, a, a) -> (b, b, b) mapTriple f (x, y, z) = (f x, f y, f z)@@ -105,7 +108,7 @@ -- |Evaluate a rational form (a*k + b*l + c*m) \/ (a'*k + b'*l + c'*m) -- for given k, l and m. evalRF :: (Real t, Num t) => (Integer, Integer, Integer) -> RationalForm t -> RationalInf-evalRF (k, l, m) (RationalForm num den) = if denom==0 then InfPlus else Finite (numer / denom) where+evalRF (k, l, m) (num :/: den) = if denom==0 then InfPlus else Finite (numer / denom) where   klm = mapTriple fromInteger (k, l, m)   numer = toRational $ evalLF klm num   denom = toRational $ evalLF klm den

@@ -1,4 +1,3 @@-{-# LANGUAGE RecordWildCards, DeriveFunctor, DeriveFoldable, DeriveGeneric #-} {-| Module      : Math.ExpPairs.Matrix3 Description : Implements matrices of order 3@@ -11,6 +10,13 @@ Provides types and functions for matrices and vectors of order 3. Can be used instead of "Data.Matrix" to reduce overhead and simplify code. -}++{-# LANGUAGE DeriveFunctor   #-}+{-# LANGUAGE DeriveFoldable  #-}+{-# LANGUAGE DeriveGeneric   #-}+{-# LANGUAGE RecordWildCards #-}+{-# LANGUAGE TemplateHaskell #-}+ module Math.ExpPairs.Matrix3   ( Matrix3 (..)   , fromList@@ -29,6 +35,7 @@ import Data.List      (transpose) import GHC.Generics   (Generic (..)) import Text.PrettyPrint.Leijen+import Data.Function.Memoize (deriveMemoizable)  -- |Matrix of order 3. Instances of 'Num' and 'Fractional' -- are given in terms of the multiplicative group of matrices,@@ -83,7 +90,7 @@    negate = fmap negate -  abs = undefined+  abs = error "abs of Matrix3 is undefined"    signum = diag . signum . det @@ -299,3 +306,5 @@   a31 * a1 + a32 * a2 + a33 * a3   ) {-# INLINE multCol #-}++deriveMemoizable ''Matrix3

@@ -21,17 +21,14 @@   ( menzerNowak   ) where -import Data.Ratio    ((%))- import Math.ExpPairs  -- |Compute Θ(a, b) for given a and b. menzerNowak :: Integer -> Integer -> OptimizeResult menzerNowak a' b' = optimize-  [-    RationalForm (LinearForm 1 1 0) (LinearForm (a+b) 0 (a+b)),-    RationalForm (LinearForm 1 0 0) (LinearForm (a+b) (-a) a)-  ]-  [] where-    a = a'%1-    b = b'%1+  [ K 1 + L 1 :/: K (a + b) + M (a + b)+  , K 1       :/: K (a + b) - L a + M a+  ] []+  where+    a = fromInteger a'+    b = fromInteger b'

@@ -14,6 +14,7 @@ -} {-# LANGUAGE DeriveGeneric        #-} {-# LANGUAGE FlexibleInstances    #-}+{-# LANGUAGE TemplateHaskell      #-} {-# LANGUAGE TypeSynonymInstances #-}  {-# OPTIONS_GHC -fno-warn-orphans #-}@@ -31,6 +32,7 @@ import Data.Ratio import GHC.Generics (Generic (..)) import Text.PrettyPrint.Leijen+import Data.Function.Memoize  -- |Vertices of the triangle of initial exponent pairs. data Triangle@@ -63,7 +65,9 @@   -- Exactly   -- 'Mix' a b = a * 'Corput16' + b * 'HuxW87b1' + (1-a-b) * 'Hux05'   | Mix !t !t-  deriving (Eq, Show, Generic)+  deriving (Eq, Ord, Show, Generic)++deriveMemoizable ''InitPair'  -- |Exponent pair built from rational fractions of -- 'Corput16', 'HuxW87b1' and 'Hux05'

@@ -110,7 +110,7 @@      bcs = takeWhile (not . null . snd) $ iterate bcf ([head xs], tail xs) -    bcf (_, [])    = undefined+    bcf (_, [])    = error "prettify': unexpected second argument of bcf"     bcf (zs, y:ys) = (zs++[y], ys)      f (bs, cs) = PPWL (Sequence bsP csP) (bsW + csW) where

@@ -9,7 +9,11 @@  Provides types for sequences of /A/- and /B/-processes of van der Corput. A good account on this topic can be found in /Graham S. W.,  Kolesnik G. A./ Van Der Corput's Method of Exponential Sums, Cambridge University Press, 1991, especially Ch. 5. -}-{-# LANGUAGE DeriveGeneric, CPP #-}++{-# LANGUAGE CPP             #-}+{-# LANGUAGE DeriveGeneric   #-}+{-# LANGUAGE TemplateHaskell #-}+ module Math.ExpPairs.Process   ( Process ()   , Path (Path)@@ -22,19 +26,18 @@ import GHC.Generics             (Generic) import Data.Monoid import Text.PrettyPrint.Leijen hiding ((<>))+import Data.Function.Memoize (deriveMemoizable)  import Math.ExpPairs.ProcessMatrix import Math.ExpPairs.PrettyProcess  -- | Holds a list of 'Process' and a matrix of projective--- transformation, which they define. It also provides a fancy 'Show'--- instance. E. g.,------ > show (mconcat $ replicate 10 aPath) == "A^10"---+-- transformation, which they define. data Path = Path !ProcessMatrix ![Process]   deriving (Eq, Show, Generic) +deriveMemoizable ''Path+ instance Monoid Path where   mempty  = Path mempty mempty   mappend (Path m1 p1) (Path m2 p2) = Path (m1 <> m2) (p1 <> p2)@@ -52,13 +55,14 @@     reads' xs = (mempty, xs)  instance Ord Path where-  (Path _ q1) <= (Path _ q2) = cmp q1 q2 where-    cmp (A:p1)  (A:p2)  = cmp p1 p2-    cmp (BA:p1) (BA:p2) = cmp p2 p1-    cmp (A:_)   (BA:_)  = True-    cmp (BA:_)  (A:_)   = False-    cmp []      _       = True-    cmp _       []      = False+  compare (Path _ x) (Path _ y) = cmp x y where+    cmp     []      []  = EQ+    cmp ( A:u) ( A:v) = cmp u v+    cmp (BA:u) (BA:v) = cmp v u+    cmp ( A:_)     _  = LT+    cmp (BA:_)     _  = GT+    cmp     _  ( A:_) = GT+    cmp     _  (BA:_) = LT  -- | Path consisting of a single process 'A'. aPath :: Path

@@ -40,8 +40,11 @@  deriveMemoizable ''Process +-- | Sequence of processes, represented as a matrix 3x3. newtype ProcessMatrix = ProcessMatrix (Matrix3 Integer)   deriving (Eq, Num, Show, Pretty)++deriveMemoizable ''ProcessMatrix  instance Monoid ProcessMatrix where   mempty = 1

@@ -9,6 +9,11 @@  Provides types and necessary instances for rational numbers, extended with infinite values. Just use 'RationalInf' instead of 'Rational' from "Data.Ratio". -}++{-# LANGUAGE TemplateHaskell #-}++{-# OPTIONS_GHC -fno-warn-orphans #-}+ module Math.ExpPairs.RatioInf   ( RatioInf (..)   , RationalInf@@ -16,6 +21,7 @@  import Data.Ratio (Ratio, numerator, denominator) import Text.PrettyPrint.Leijen+import Data.Function.Memoize (deriveMemoizable)  -- |Extends a rational type with positive and negative -- infinities.@@ -27,6 +33,9 @@   -- |Positive infinity   | InfPlus   deriving (Eq, Ord, Show)++deriveMemoizable ''Ratio+deriveMemoizable ''RatioInf  -- |Arbitrary-precision rational numbers with positive and negative -- infinities.

@@ -1,5 +1,5 @@ name:                exp-pairs-version:             0.1.4.1+version:             0.1.5.0 synopsis:            Linear programming over exponent pairs description:         Package implements an algorithm to minimize rational objective function over the set of exponent pairs homepage:            https://github.com/Bodigrim/exp-pairs@@ -9,7 +9,7 @@ maintainer:          andrew.lelechenko@gmail.com category:            Math build-type:          Simple-extra-source-files:  CHANGELOG.md+extra-source-files:  CHANGELOG.md, tests/*.txt cabal-version:       >=1.10  source-repository head@@ -47,6 +47,7 @@                        Matrix3,                        MenzerNowak,                        Pair,+                       Process,                        PrettyProcess,                        RatioInf   build-depends:       base >=4 && <5,

@@ -6,9 +6,13 @@ import Test.SmallCheck.Series import Control.Applicative import Control.Monad+#if __GLASGOW_HASKELL__ < 710+import Data.Foldable+#endif import GHC.Generics          (Generic (..))  import Math.ExpPairs.LinearForm+import Math.ExpPairs.Process import Math.ExpPairs.ProcessMatrix import Math.ExpPairs.Pair (InitPair' (..)) import Math.ExpPairs.Matrix3 as M3 (Matrix3, fromList)@@ -21,11 +25,11 @@   series = cons3 LinearForm  instance Arbitrary a => Arbitrary (RationalForm a) where-  arbitrary = RationalForm <$> arbitrary <*> arbitrary+  arbitrary = (:/:) <$> arbitrary <*> arbitrary   shrink = genericShrink  instance (Monad m, Serial m a) => Serial m (RationalForm a) where-  series = cons2 RationalForm+  series = cons2 (:/:)  instance Arbitrary a => Arbitrary (Constraint a) where   arbitrary = Constraint <$> arbitrary <*> arbitrary@@ -55,7 +59,7 @@  instance (Ord t, Fractional t, Arbitrary t) => Arbitrary (Ratio01 t) where   arbitrary = Ratio01 <$> (arbitrary `suchThat` (\x -> 0 <= x && x <= 1))-  shrink = genericShrink+  shrink (Ratio01 y) = Ratio01 <$> filter (\x -> 0 <= x && x <= 1) (shrink y)  instance (Ord t, Fractional t, Serial m t) => Serial m (Ratio01 t) where   series = Ratio01 <$> (series `suchThatSerial` (\x -> 0 <= x && x <= 1))@@ -163,3 +167,9 @@ instance (Ord t, Serial m t) => Serial m (Sorted (t, t, t, t, t, t)) where   series = Sorted <$> (series `suchThatSerial` (\(a, b, c, d, e, f) -> a <= b && b <= c && c <= d && d <= e && e <= f)) ++instance Arbitrary Path where+  arbitrary = foldMap (\x -> if x then aPath else baPath) <$> (arbitrary :: Gen [Bool])++instance Monad m => Serial m Path where+  series = foldMap (\x -> if x then aPath else baPath) <$> (series :: Monad m => Series m [Bool])

@@ -6,9 +6,11 @@  import Test.Tasty import Test.Tasty.SmallCheck as SC-import Test.Tasty.QuickCheck as QC+import Test.Tasty.QuickCheck as QC hiding (Positive) import Test.Tasty.HUnit +import Debug.Trace+ import Instances import Etalon (testEtalon) @@ -56,26 +58,45 @@ testMOnSInf (Ratio01 a') = a < 1 || (optimalValue . mOnS) a == InfPlus where   a = fromMinus3To3 a' -testZetaReverse :: Ratio01 Rational -> Bool-testZetaReverse (Ratio01 s') = abs (s - t) <= 5 % 1000 where-  s = s' / 2+testZetaReverse1 :: Ratio01 Rational -> Bool+testZetaReverse1 (Ratio01 s') = if t <= s + 2e-2 && s <= t + 2e-3 then True else trace (show $ fromRational $ s-t) False where+  s = fromHalfToOne s'   zs = zetaOnS s   t = toRational $ optimalValue $ reverseZetaOnS $ toRational $ optimalValue zs --- Convexity tests - they fail and it is OK-testZetaConvex :: Sorted (Ratio01 Rational, Ratio01 Rational, Ratio01 Rational) -> Bool-testZetaConvex (Sorted (Ratio01 a, Ratio01 b, Ratio01 c)) = a == b || b == c || zb <= k * Finite b + l where-  [za, zb, zc] = map (optimalValue . zetaOnS) [a, b, c]-  k = (za - zc) / Finite (a - c)-  l = za - k * Finite a+testZetaReverse2 :: Ratio01 Rational -> Bool+testZetaReverse2 (Ratio01 s') = if t <= s + 1e-10 && s <= t + 4e-3 then True else trace (show $ fromRational $ s-t) False where+  s = s' * 32 / 205+  zs = reverseZetaOnS s+  t = toRational $ optimalValue $ zetaOnS $ toRational $ optimalValue zs --- Ivic, Th. 8.1, p. 205-testMConvex :: Sorted (Ratio01 Rational, Ratio01 Rational, Ratio01 Rational) -> Bool-testMConvex (Sorted (Ratio01 a', Ratio01 b', Ratio01 c')) = a==b || b==c || za==InfPlus || zc==InfPlus-  || zb>= za*zc*Finite(c-a)/(zc*Finite(c-b) + za*Finite(b-a)) where-    [a,b,c] = map fromHalfToOne [a', b', c']-    [za, zb, zc] = map (optimalValue . mOnS) [a,b,c] :: [RationalInf]+testMOnSReverse1 :: Ratio01 Rational -> Bool+testMOnSReverse1 (Ratio01 s') =+  if t <= s + 4e-2 && s <= t + 1e-3 then True else trace (show $ fromRational $ s-t) False+  where+    s = fromHalfToOne s'+    zs = mOnS s+    t = toRational $ reverseMOnS 1e-3 $ optimalValue zs +testMOnSReverse2 :: Ratio01 Rational -> Bool+testMOnSReverse2 (Ratio01 s') = s' == 0 || if recip t <= recip s + 1e-3 && recip s <= recip t + 1e-3 then True else trace (show $ fromRational $ recip s - recip t) False where+  s = 4 * recip s'+  zs = reverseMOnS 1e-3 (Finite s)+  t = toRational $ optimalValue $ mOnS $ toRational zs++testMBigOnHalfReverse1 :: Positive Rational -> Bool+testMBigOnHalfReverse1 (Positive s') = if recip t <= recip s + 2e-3 && recip s <= recip t + 1e-10 then True else trace (show $ fromRational $ recip s - recip t) False where+  s = s' + 4+  zs = mBigOnHalf s+  t = toRational $ optimalValue $ reverseMBigOnHalf $ toRational $ optimalValue zs++testMBigOnHalfReverse2 :: Positive Rational -> Bool+testMBigOnHalfReverse2 (Positive s') = if recip t <= recip s + 2e-3 && recip s <= recip t + 1e-10 then True else trace (show $ fromRational $ recip s - recip t) False where+  s = s' + 1+  zs = reverseMBigOnHalf s+  t = toRational $ optimalValue $ mBigOnHalf $ toRational $ optimalValue zs++ etalonZetaOnS :: Integer -> Integer -> Integer -> Integer -> Bool etalonZetaOnS a b c d = Finite (c%d) >= optimalValue (zetaOnS $ a%b) @@ -100,8 +121,24 @@   , adjustOption (\(SC.SmallCheckDepth n) -> SC.SmallCheckDepth (n `div` 2)) $       SC.testProperty "mOnS strict monotonic" testMOnS2   , QC.testProperty "mOnS strict monotonic" testMOnS2-  , SC.testProperty "zetaOnS reverse" testZetaReverse-  , QC.testProperty "zetaOnS reverse" testZetaReverse++  , SC.testProperty "reverseZetaOnS . zetaOnS == id" testZetaReverse1+  , QC.testProperty "reverseZetaOnS . zetaOnS == id" testZetaReverse1+  , SC.testProperty "zetaOnS . reverseZetaOnS == id" testZetaReverse2+  , QC.testProperty "zetaOnS . reverseZetaOnS == id" testZetaReverse2++  , SC.testProperty "reverseMOnS . mOnS == id" testMOnSReverse1+  , adjustOption (\(QC.QuickCheckTests n) -> QC.QuickCheckTests (n `min` 100)) $+      QC.testProperty "reverseMOnS . mOnS == id" testMOnSReverse1+  , SC.testProperty "mOnS . reverseMOnS == id" testMOnSReverse2+  , adjustOption (\(QC.QuickCheckTests n) -> QC.QuickCheckTests (n `min` 100)) $+      QC.testProperty "mOnS . reverseMOnS == id" testMOnSReverse2++  , SC.testProperty "reverseMBigOnHalf . mBigOnHalf == id" testMBigOnHalfReverse1+  , QC.testProperty "reverseMBigOnHalf . mBigOnHalf == id" testMBigOnHalfReverse1+  , SC.testProperty "mBigOnHalf . reverseMBigOnHalf == id" testMBigOnHalfReverse2+  , QC.testProperty "mBigOnHalf . reverseMBigOnHalf == id" testMBigOnHalfReverse2+   , SC.testProperty "zetaOnS symmetry" testZetaOnSsym   , QC.testProperty "zetaOnS symmetry" testZetaOnSsym   , SC.testProperty "zetaOnS above s=1" testZetaOnSZero

@@ -0,0 +1,29 @@+module Process where++import Math.ExpPairs.Process++import Test.Tasty+import Test.Tasty.SmallCheck as SC+import Test.Tasty.QuickCheck as QC hiding (Positive)++import Instances ()++testReadShow :: Path -> Bool+testReadShow p@(Path _ xs) = read (concatMap show xs) == p++testOrd :: Path -> Path -> Bool+testOrd p1 p2 = compare p1 p2 == compare (x1 / z1) (x2 / z2)+  && compare (y2 / z2) (y1 / z1) == compare (x1 / z1) (x2 / z2)+  where+    (x1, y1, z1) = evalPath p1 (1, 4, 6)+    (x2, y2, z2) = evalPath p2 (1, 4, 6)++testSuite :: TestTree+testSuite = testGroup "Process"+  [ adjustOption (\(SC.SmallCheckDepth n) -> SC.SmallCheckDepth (n `min` 13)) $+      SC.testProperty "read . show == id" testReadShow+  , QC.testProperty "read . show == id" testReadShow+  , adjustOption (\(SC.SmallCheckDepth n) -> SC.SmallCheckDepth (n `min` 8)) $+      SC.testProperty "Ord of Processes" testOrd+  , QC.testProperty "Ord of Processes" testOrd+  ]

@@ -3,6 +3,7 @@ import qualified RatioInf (testSuite) import qualified Pair (testSuite) import qualified PrettyProcess (testSuite)+import qualified Process (testSuite)  import qualified Ivic (testSuite) import qualified Kratzel (testSuite)@@ -20,6 +21,7 @@   , RatioInf.testSuite   , Pair.testSuite   , PrettyProcess.testSuite+  , Process.testSuite   , Ivic.testSuite   , Kratzel.testSuite   , MenzerNowak.testSuite

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