!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKL M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  4(internal) polynomial datatype and simple functions portable experimentalmik@konecny.aow.czEA polynomial represented by its coefficients it the Chebyshev basis. 8The polynomials are never to be used outside the domain [-1,1]^n. GAll operations are rounded in such a way that the resulting polynomial  is a point-wise upper or lower bound of the exact result. 4Inspect all terms of the polynomial and return the ' degree of the highest degree term.  make a basic x) polynomial for a given variable number 1Convert a polynomial to a string representation, " using the ordinary x^n basis. 8show the polynomial also in its native Chebyshev basis @conversion of polynomials from Chebyshev basis to the X^n basis E(not exact - suffering from rounding in the coefficient conversions) polynomial in Chebyshev basis 7approxition of the equivalent polynomial in X^n basis 2conversion of one Chebyshev term to the X^n basis 7a Chebyshev term represented by the Chebyshev degrees + for each variable in the term 7the polynomial equivalent to the given Chebyshev term * (using integer coefficients) EChebyshev polynomials expressed as associative lists power -> coeff VChebyshev polynomials expressed as lists of integer coefficients for powers 0,1,2...    5(internal) field operations applied to polynomials portable experimentalmik@konecny.aow.czQConvert a polynomial to a lower-order one that is dominated by (resp. dominates) " it closely on the domain [-1,1]. new maximal order ,lower and upper bounds with limited degree Bmultiply a polynomial by a scalar rounding downwards and upwards Kmultiply a polynomial by a scalar interval rounding downwards and upwards %(internal) evaluation of polynomials portable experimentalmik@konecny.aow.czPEvaluate a polynomial at a point, consistently rounding upwards and downwards. 6Evaluate a polynomial at a real number approximation NSubstitute several variables in a polynomial with real number approximations, $ rounding downwards and upwards. *Compose two polynomials, rounding upwards - provided the second polynomial maps [-1,1] into [-1,1]. 3variable to substitute, polynomial to substitute 5(internal) bounds of single and multiple polynomials portable experimentalmik@konecny.aow.cz.Find an upper bound on a polynomial over the  unit domain [-1,1]^n. how hard to try ;Find a close upper bound on an affine polynomial over the  unit domain [-1,1]^n. how hard to try !=Find a close upper bound on a quadratic polynomial over the  unit domain [-1,1]^n. #how hard to try looking for peaks "PApproximate from below and from above the pointwise maximum of two polynomials maximum polynomial degree #@Approximate the function max(0,p(x)) by a polynomial from below  and from above. maximum polynomial degree $ A version of # using a cubic approximation. maximum polynomial degree  !"#$ !"#$ !"#$9(internal) elementary functions applied to polynomials portable experimentalmik@konecny.aow.cz%7Approximate the pointwise square root of a polynomial 6 by another polynomial from below and from above. maximum polynomial degree ?? &7Approximate the pointwise exponential of a polynomial 6 by another polynomial from below and from above. maximum polynomial degree minimum approx Taylor degree '[Approximate the pointwise integer power of a polynomial by another polynomial from above. maximum polynomial degree (=Approximate the pointwise natural logarithm of a polynomial 6 by another polynomial from below and from above. maximum polynomial degree ?? )0Approximate the pointwise sine of a polynomial 6 by another polynomial from below and from above. !True iff sine, False iff cosine maximum polynomial degree minimum approx Taylor degree *2Approximate the pointwise cosine of a polynomial 4 by another polynomial from below and from above  using the tau method  as described in [Mason & Handscomb 2003, p 62]. maximum polynomial degree minimum approx degree %&'()*%&'()*%&'()*,(internal) integration of polynomials etc portable experimentalmik@konecny.aow.cz+DApproximate from below and from above the integral of a polynomial. ;Based on the following formulas for Chebyshev polynomials:   \int T_n(x)dx = . T_{n+1}(x)/2(n+1) - T_{n-1}(x)/2(n-1)   \int T_1(x)dx =  T_2(x)/4 + 1/4  \int T_0(x)dx =  T_1(x) variable to integrate by ,Bmeasure the volume between a polynomial and the zero axis on [-1,1]^n -8Differentiate a polynomial using one of its variables.  variable to differentiate over +,-+,-+,- hierarchical domain partitions portable experimentalmik@konecny.aow.cz./+value splitter function - parameters are: : depth, domain of value, value, variable to split by, 4 point to split at; returns the two split values 011 The root of the tree often represents the whole R^n.  Each node splits the parent's space into two using K a specified variable (ie direction) and an optional splitting point. A By default, a split is taken at the point defined by the method  RA.bisect. 23direction to split in 4point that the split is at 5%the half towards -Infty in split dir 6%the half towards +Infty in split dir 78depth of this node 9domain :value estimate ;<= variable x to split by point in domain of x to split at domain to lookup x in if tree's domain does not have x >:Apply a function to all values, thus creating a new tree. ?:Apply a function to all values, thus creating a new tree. CApply a function to all values, thus creating a list of new trees. @KPerform a given action on all branches of a bisection tree, left to right. 1 (optionally now going below the given depth) AKPerform a given action on all branches of a bisection tree, left to right. 1 (optionally now going below the given depth) BJPerform a given action on all branches of a bisection tree, left to right 3 with the option of further branching the tree. 1 (optionally now going below the given depth) CD9Ensure both trees have equal structure at the top level: = either they are all leaves or they all split at the same - direction with the same splitting point. *Also, unify the domains at the top level. E<Ensure all the trees have equal structure at the top level: = either they are all leaves or they all split at the same - direction with the same splitting point. *Also, unify the domains at the top level. FDCombine two bisection trees using a given value combining function. @ Where necessary, leaves are split so that the resulting tree' s structure I is the union of the two argument tree structures. Such splitting of = values in leaves is performed by the provided functions. $value splitter function for tree 1 $value splitter function for tree 2 )partial function to combine values with GAreturn all values in leafs (except those within some CE subtree) 3 as a list (from the leftmost to the rightmost) HAreturn all values in leafs (except those within some CE subtree) 3 as a list (from the leftmost to the rightmost) #linear ordering on bisection trees IDlookup the smallest subtree whose domain covers the given rectangle #domain to look up within the tree &Update a value on a given sub-domain, . bisecting the tree if necessary to obtain / a better fit for the domain, but not below  a given depth limit. ?With multiple domain dimensions, split the domain according to  .  depth limit domain to update on 6how to update values that intersect the above domain ./0123456789:;<=>?@ABCDEFGHI17289:8934560/.;<CDE=>?@ABFGHI./01 7289:89345623456789:;<=>?@ABCDEFGHI;abstract zipping of domain partitions used for integration portable experimentalmik@konecny.aow.czJ;Transform a bunch of bisection trees over the same domain  by  integrating" them in a very abstract sense. F The trees are unified in their splitting patterns in the process. ? By supplying certain parameters, this function can in fact F perform numerical integration of piece-wise polynomial functions. It can be also viewed as a zipping+folding$ operator over bisection trees that \ generates another bunch of bisection trees, synchronously traversing the original trees J from a certain point on a selected axis outwards in both directions,  carrying some data along.  variable x) (ie axis or direction) to integrate in origin in terms of variable x 'support, ie the domain on which to zip H (automatically extended to include origin when projected to x) !what to do outside the support, 2 possibly being passed values from left/right & when leaving the support should a leaf be split? -integrator for a leaf containing the origin ?integrator over a leaf that sees the origin towards -infinity ?integrator over a leaf that sees the origin towards +infinity input functions output functions 4The number of output functions does not have to be 7 the same as the number of input functions. K<Zip a list of bisection trees in synchrony but do something 2 else inside and not inside a given subdomain. GFurther splitting at default points will be done up to the given depth A in an attempt to separate the subdomain as well as possible. BIf the subdomain is not properly isolated by the splitting at the P maximum depth, splits are made at irregular points to ensure full isolation  of the subdomain.  depth limit  subdomain sd what to do with values inside sd what to do with values outside sd but touching it what to do with values outside sd JKJKJK 8class abstracting imprecise function arithmetic on [-1,1]^n portable experimentalmik@konecny.aow.cz0LMFCheck internal consistency of the function and report problem if any. 9indentification of caller location for easier debugging NOPQ Construct a constant function. RConstruct an affine function.  value at 0 ascent of each base vector S"Multiply a function by a scalar, ) rounding downwards and upwards. T6Multiply a function by an approximation of a scalar, ) rounding downwards and upwards. U6Multiply a function by an approximation of a scalar,  rounding downwards. V6Multiply a function by an approximation of a scalar,  rounding upwards. W,Get the degree of this particular function. 6If the function is a polynomial, this function should  return its degree. X0Decrease the degree of function approximation, 2 rounding pointwise downwards and upwards. YMDecrease the degree of function approximation, rounding pointwise downwards. ZKDecrease the degree of function approximation, rounding pointwise upwards. [GApproximate the integral of p (with 0 at 0) from below and from above. variable to integrate by p(x) \9Approximate the integral of p (with 0 at 0) from below. variable to integrate by p(x) ]9Approximate the integral of p (with 0 at 0) from above. variable to integrate by p(x) ^'Measure the volume between a function . and the zero hyperplane on the domain [-1,1]^n. )axes to include in the measuring domain _)Find an upper bound of the function over [-1,1]^n. `(Find a lower bound of the function over [-1,1]^n. a@Approximate the function max(0,p(x)) from below and from above. max degree for result p(x) bApproximate the function 1/ p(x) from below and from above. max degree for result p(x) cApproximate the function 1/p(x) from below. dApproximate the function 1/p(x) from above. eGApproximate the function max(p_1(x),p_2(x)) from below and from above. max degree for result p_1(x) p_2(x) f8Approximate the function max(p_1(x),p_2(x)) from below. max degree for result p_1(x) p_2(x) g8Approximate the function max(p_1(x),p_2(x)) from above. max degree for result p_1(x) p_2(x) hGApproximate the function min(p_1(x),p_2(x)) from below and from above. max degree for result p_1(x) p_2(x) i8Approximate the function min(p_1(x),p_2(x)) from below. max degree for result p_1(x) p_2(x) j8Approximate the function min(p_1(x),p_2(x)) from above. max degree for result p_1(x) p_2(x) k Approximate  sqrt(p(x)) from below and from above. max degree for result 8how hard to try when approximating exp as a polynomial p(x) l Approximate  exp(p(x)) from below and from above. max degree for result 8how hard to try when approximating exp as a polynomial p(x) m Approximate  log(p(x)) from below and from above. max degree for result 8how hard to try when approximating log as a polynomial p(x) n Approximate  sin(p(x)) from below and from above. max degree for result 8how hard to try when approximating sin as a polynomial p(x) o Approximate  cos(p(x)) from below and from above. max degree for result 8how hard to try when approximating cos as a polynomial p(x) p5Evaluate at a point, rounding upwards and downwards. q)Evaluate at a point, rounding downwards. r)Evaluate at a point, rounding downwards. s=Safely evaluate at a point using a real number approximation + for both the point and the result. t1Partially evaluate at a lower-dimensional point 1 given using a real number approximation. F Approximate the resulting function from below and from above. u1Partially evaluate at a lower-dimensional point 1 given using a real number approximation. 7 Approximate the resulting function from below. v1Partially evaluate at a lower-dimensional point 1 given using a real number approximation. 7 Approximate the resulting function from above. w6Compose two functions, rounding upwards and downwards  provided each f_v ranges within the domain [-1,1]. max degree for result  function f .variables to substitute and for each variable v,  function f_v to substitute for v  that maps [-1,1] into [-1,1] upper and lower bounds of  f[v |-> f_v] x*Compose two functions, rounding downwards  provided each f_v ranges within the domain [-1,1]. max degree for result  function f1 .variables to substitute and for each variable v,  function f_v to substitute for v  that maps [-1,1] into [-1,1] a lower bound of f1.f2 y(Compose two functions, rounding upwards  provided each f_v ranges within the domain [-1,1]. max degree for result  function f1 .variables to substitute and for each variable v,  function f_v to substitute for v  that maps [-1,1] into [-1,1] an upper bound of f1.f2 z1Convert from the interval type to the base type. B (The types are determined by the given example function.) 5this parameter is not used except for type checking {2Convert from the base type to the interval type. B (The types are determined by the given example function.) 5this parameter is not used except for type checking 0LMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{0LMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{0L/MNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{MNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{0polynoms in the Chebyshev basis of the 1st kind portable experimentalmik@konecny.aow.cz ,classes abstracting function approximations portable experimentalmik@konecny.aow.cz|This class extends  by: G making the domain of the function enclosure available for inspection; 8 allowing the construction of basic function enclosures , where the domain has to be specified. };A function enclosure with no information about the function' s values. the domain of the function .how many functions are bundled in this tuple ~9Construct a constant enclosure for a tuple of functions. 8Construct the exact enclosure for a projection function  (ie a variable). -Return the domain of the function enclosure. DSplit the domain into two halves, yoelding two function enclosures. variable (axis) to split on /where exactly to split (this has to be exact) 5Merge function enclosures with neighbouring domains. variable (axis) to glue on Safely integrate a  R^n -> R^m function enclosure E with some initial condition (origin and function at origin). how hard to try function to integrate x = variable to integrate by integration range origin in terms of x; this has to be thin! values at origin Safely integrate a R -> R^m function enclosure. how hard to try unary function to integrate integration range origin values at origin Safely integrate a  R^n -> R^m function enclosure ? intersecting it with a prior enclosure for the result. >The prior enclosure could contains one of more initial value. how hard to try function to integrate variable to integrate by integration domain .a sub-domain with relevant new information - C either about initial value(s) or about derivative 5approximation to result, including initial value(s) Eimproved result and measurement of improvement analogous to the one & returned by pointwise  Safely integrate a R -> R^m function enclosure ? intersecting it with a prior enclosure for the result. >The prior enclosure could contains one of more initial value. how hard to try unary function to integrate integration domain .a sub-domain with relevant new information - C either about initial value(s) or about derivative 5approximation to result, including initial value(s) Eimproved result and measurement of improvement analogous to the one & returned by pointwise  =A class of types that approximate first-order real functions   R^n -> R^m3 using some type of graph enclosures. The domains @ of the functions can be neither specified nor investigated ! by operations in this class. This class extends % so that we could perform point-wise ! operations on the functions. This class is associated with: % two real number types (instances of () for working with parts of the function's domain and range; 3 a type of boxes indexed by variables (instance of ) for working with  parts of the function' s domain. 7Check internal consistency and report problem if any. 9indentification of caller location for easier debugging 5this parameter is not used except for type checking 5this parameter is not used except for type checking @Get the internal degree of quality (usually polynomial degree)  of the approximation. ASet an upper bound on the degree of this function approximation. :This reduces the degree immediately if necessary and also @ affects all operations performed with this value later. 7Get the current uppend bound on the degree associated + with this function approximation. 8Give a close upper bound of the precision of the range 6 at the best approximated point in the domain. 6Find some upper and lower bounds of the function over [-1,1]^n. 8Combine several functions with the same domain into one tuple function. 0Reveal how many functions are bundled together. &Modify a tuple of functions in a way : that does not treat the tuple elements uniformly. IFind close upper and lower bounds of the volume of the entire enclosure. N A negative volume means that the enclosure is certainly inconsistent. BMultiply a function approximation by a real number approximation. HIntersect one enclosure by another but only on a box within its domain. @the subdomain; defined by clipping the range of some variables 3function to improve by intersecting its subdomain Mthe enclosure to be used on the subdomain (but defined on the whole domain) KIntersect two enclosures and measure the global improvement as one number. (Use  defined in module Data.Number.ER.Real.Approx B to measure the improvement using a function enclosure.) Lenclosure intersection and measurement of improvement analogous to the one & returned by pointwise  *Evaluate the function at the given point. >Fix some variables in the function to the given exact values. /A simple and limited composition of functions. NIt is primarily intended to be used for precomposition with affine functions.  enclosure of f Aspecifies the variables to substitute and for each such variable v,  gives an exact enclosure of a function f_v to substitute for v  enclosure of  f[v |-> f_v] 8BEWARE: Enclosure is probably incorrect where values of f_v are outside the domain of v in f. 4Recursively perform a number of bisections and then # glue the bits back together. This way we can ensure that 5 a piece-wise enclosure has a partition that goes " to at least the given depth. required depth of bisection |}~|}~| }~}~ )class abstracting function enclosures on [-1,1]^n portable experimentalmik@konecny.aow.czThis class extends  by: ? assuming that the domain of the function enclosures is always [-1,1]^n for some n; 8 allowing the construction of basic function enclosures ( where the domain has to be known. ;A function enclosure with no information about the function' s values. 9Construct a constant enclosure for a tuple of functions. 7Construct the exact enclosure of an affine function on [-1,1]^n.  values at 0 ascents of each base vector IFind close upper and lower bounds of the volume of the entire enclosure. N A negative volume means that the enclosure is certainly inconsistent. JExplicitly specify the variables to identify the dimension of the domain. KIntersect two enclosures and measure the global improvement as one number. (Use RA.intersectMeasureImprovement defined in module Data.Number.ER.Real.Approx B to measure the improvement using a function enclosure.) JExplicitly specify the variables to identify the dimension of the domain. Lenclosure intersection and measurement of improvement analogous to the one * returned by the pointwise RA.intersectMeasureImprovement Safely integrate a [-1,1] ^n -> R^m function enclosure E with some initial condition (origin and function at origin). how hard to try function to integrate x = variable to integrate by origin in terms of x; this has to be exact! values at origin  +arbitrary precision function enclosures on [-1,1]^n portable experimentalmik@konecny.aow.cz   enclosures translated from [-1,1]^n to another domain portable experimentalmik@konecny.aow.czThe canonical translation of 8 any compact non-empty and non-singleton interval in R " to and from the unit interval [-1,1]. 3This structure holds the two coefficients for both  linear mappings. the interval being mapped %Datatype translating enclosures from [-1,1]^n to any compact  interval in R^n with non-empty interior ; using a bunch of linear maps, one for each dimension. 2separate approximations per domain-box hyper-edge portable experimentalmik@konecny.aow.cz=Use some function approximation type and for each domain box E keep a structure of function approximations of this type indexed E by the hyper-edge structure. For each hyper-edge of the domain, 3 the approximation has this edge as its domain. 'E.g. for a 2D square domain there are: ( one approximation for the whole square + four 1D approximations, one for each edge > eight 0D approximations, one for each endpoint of each edge .a list of approximations over the same domain portable experimentalmik@konecny.aow.cz=A tuple of function approximations allowing one to get from  functions R^n->R to a function  R^n -> R^m. =arbitrary precision piece-wise-something function enclosures portable experimentalmik@konecny.aow.cz*Arbitrary precision piece-wise something 2 (eg linear, polynomial, rational) enclosures  of functions R^n->R^m. 7The type of approximation within segments is specified  by an instance of |. 8The piece-wise construction defines another instance of |.  addressing and modifying leaves portable experimentalmik@konecny.aow.cz 3A representation of a binary tree with a hole that  can be efficiently filled. A path in a binary tree. ) It is used mainly in connection with BisectionTree.BisectionTree. @Assuming that bisection happens at default points as defined by  * and starts from the given root interval. root interval @Lookup a subdomain of a function according to a bisection path. B Return the restrited function as well as a zipper that allows ? an efficient modification of the function on the looked up  subdomain. 3Modify a function in its subdomain as expressed by  the zipper. ;a function on a larger domain and a highlighted subdomain )a function of the highlighted subdomain 3concise names for default function representations non-portable (requires fenv.h) experimentalmik@konecny.aow.cz'definitions useful for testing in ghci non-portable (requires fenv.h) experimentalmik@konecny.aow.cz#     overview of AERN-RnToRm non-portable (requires fenv.h) experimentalmik@konecny.aow.czP !"#$%&'()*+,-./0123456789:;<=>?@A|}~B !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_` a b c d e Q f g h i j k l m n o p q r s t u v w x y z { | } ~   Q o  b k g  Q f o                 !"#$%&'()*+,-./0123456789:;<=>?@ARBCDEAERN-RnToRm-0.3.0.29Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Basic9Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Field8Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Eval:Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Bounds>Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Elementary?Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom.Integration#Data.Number.ER.RnToRm.BisectionTree/Data.Number.ER.RnToRm.BisectionTree.Integration"Data.Number.ER.RnToRm.UnitDom.BaseData.Number.ER.RnToRm.Approx$Data.Number.ER.RnToRm.UnitDom.Approx-Data.Number.ER.RnToRm.UnitDom.Approx.Interval&Data.Number.ER.RnToRm.Approx.DomTransl%Data.Number.ER.RnToRm.Approx.DomEdges"Data.Number.ER.RnToRm.Approx.Tuple&Data.Number.ER.RnToRm.Approx.PieceWise(Data.Number.ER.RnToRm.BisectionTree.Path!Data.Number.ER.RnToRm.DefaultRepr3Data.Number.ER.RnToRm.UnitDom.ChebyshevBase.Polynom!Data.Number.ER.RnToRm.TestingDefsData.Number.ER.RnToRmTermKey ERChebPoly chplCoeffschplConstTermKeychplIsConstTermKey chplTermOrder chplTermArity chplGetDegree chplConstchplVarchplNormaliseUpchplNormaliseDown chplNormalisechplShow chebToXBasis termXtermschebyXCoeffsLists chebyXCoeffs chplAffinechplReduceDegree chplScalechplScaleApproxchplEval chplEvalUp chplEvalDownchebyEvalTsRoundDownUpchebyEvalTsExactchplEvalApproxchplPartialEvalApprox chplComposechplUpperBoundAffinechplUpperBoundAffineCornerschplUpperBoundQuadrchplMax chplNonnegchplNonnegCubicchplSqrtchplExpchplPowchplLogchplSineCosine chplRecip chplIntegratechplVolumeAboveZerochplDifferentiate ValueCombiner ValueSplitterDepth BisectionTreeNodebistrDirbistrPtbistrLObistrHILeaf bistrDepthbistrDombistrValisLeafconstsplit mapWithDom mapLeavesdoBistrdoMap doMapLeaves removeVarssync2syncMany combineWith collectValuescollectDomValueslookupSubtreeDom zipFromOriginzipOnSubdomain ERUnitFnBasecheckgetGranularitysetMinGranularitysetGranularityaffinescale scaleApproxscaleApproxDown scaleApproxUp getDegree reduceDegreereduceDegreeDownreduceDegreeUp integrate integrateDown integrateUpvolumeAboveZero upperBound lowerBoundnonnegrecip recipDownrecipUpmaxmaxDownmaxUpminminDownminUpsqrtexplogsincosevalevalDownevalUp evalApproxpartialEvalApproxpartialEvalApproxDownpartialEvalApproxUpcompose composeDown composeUp raEndpointsraFromEndpoints ERFnDomApprox bottomApproxprojdombisectunBisectintegrateUnaryintegrateMeasureImprovement integrateMeasureImprovementUnary ERFnApprox domra2ranra ranra2domra setMaxDegree getMaxDegreegetBestPrecisiongetRangeApproxtuple getTupleSize applyTupleFnvolumepartialIntersectintersectMeasureImprovement partialEval composeThinbisectUnbisectDepthERUnitFnApprox ERFnContext erfnMaxDegreeerfnCoeffGranularity ERFnInterval erfnUpper erfnLowerNeg erfnGlobalERFnIntervalAny erfnContext DomTransldtrDomdtrFromUnitSlopedtrFromUnitConstdtrToUnitSlopedtrToUnitConstERFnDomTranslApproxerfnUnitApprox erfnDomTranslERFnDomEdgesApproxerfnMainVolume erfnEdges ERFnTuple erfnTuple ERFnPiecewiseFnZipperFnZ_RFnZ_LFnZ_H BisecTreePathBTP_LBTP_RBTP_Hpath2domlookupSubdomain updateFnZFAFAPWPFAPEFAPTFAPDFAPU errorModule chplCheck chplGetVarschplGetGranularitychplSetMinGranularitychplSetGranularitychplReduceDegreeDownchplReduceDegreeUp chplScaleDown chplScaleUpmapMultiLeavescompare updateValAERN-Real-0.9.6Data.Number.ER.Real.DomainBox bestSplitPData.Number.ER.Real.ApproxERApprox ERIntApprox DomainBox dtrIdentity makeDomTransl domToUnit edgesLift1 edgesLift2unifyEdgeVariables addVarToEdges tuplesLift1 tuplesLift2 tuplesSplitpwLift1pwLift2pwbistrZipWithpwSplitfaSplit faCombine bisectDomain$Data.Number.ER.Real.DomainBox.IntMapVarIDBox fapuConst1 fapdConst1 fapdConstU fapdConst01fapd04X0fapd13X0fapd12X1fapdUX0fapdUX1 fapeConst1 fapeConstU fapeConst01fape13X0fape12X1fapeUX0fapeUX1 fapeTestMult fapeMultiVar fapeTestPEval fapeUConst1 fapeUConst13fapeUConst13InitPtfapwUUX0fapwUUX1fapwUX0fapwUX1 fapwUConst1 fapwUConst13fapwUConst13InitPt testIntegrE testIntegrPxfn1fn2fn3showVar defaultVarnewVarID VariableIDlookupfindWithDefaultzipWithDefaultSecondzipWithDefaultzipWith foldWithKeyfoldfilterelemskeys unionWithunion notMembermemberdelete insertWithinsert fromAscList toAscListfromListtoList singletonunaryisNoinfonoinfo differenceintersectionWith mapWithKeymapDomainBoxMappableclassifyPositionunify compatible DomainIntBox