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ABC conjectureabstractabstract lang completenessactionadditive combinatoricsadele ringadic valuationaffine groupaffine spacesahiv22ahiv22supportalgebraic geometryalgorithmallow denyallows termalmost orthogonalityalmost periodicityanalysisanalytic number theoryanalytic setangular momentumangular velocityantichain operatorantichain tile countappendapprox hom pfrapproximationarcaritharithmetic geometryarithmetic progressionsas tensorasymptoticsasync runtimeauthorizationauthorizeauthorizerautomata theoryautomorphismauxiliaryball extensionball scalingbase changebase permbasicbasicsbasisbasis linearbatched evaluatorbehavioural theoryberlekamp welchbig opbijectivitybilinear formbindingbirthdaybisimulationbivariateblock rel distancebmobn254borel caratheodorybounding mutualbraketbridgebrownian motionbrun titchmarshbuchi congrbuchi congruencebuchi interbundledbundled semanticscaching logging oraclecaching oraclecadlagcadlag modificationcalculationscampanatocan reach viacanonicalcapacitycardinalitycarleson huntcarleson hunt basiccarleson on the real line basiccarleson on the real line continuouscarleson operator realcategory theorycell refchainchangcheck claimchernoffchiral indiceschoueka lemmachurchclass dclass field theoryclassical carlesonclifford algebracma to nmaco psetcoding theorycollisioncollision resistancecombinatoricscombinecommutationcommutative algebracompactcompact hausdorff ringscompact systemcompatibilitycompiled policiescompilercompletenesscompletionscomplexity theorycompositioncomputabilitycomputer scienceconcatconclusionsconcretecondconfluencecongrconsensusconsequencesconsistencyconstantconstant slice distconstant time distconstructionscontinuitycontractioncontractionscontrol approximation effectcontrol approximation effect basiccontrol approximation effect continuousconversionconvolutioncoordinatecore interaction phasecorrectnesscorrugationcostcost accountingcost modelcounter modelcovariancecross productcryptographycurlcutoff admissibilitycyclotomic fieldsdebutdecoderdecoding radiusdedekind mac neille completiondeferred samplingdefsdegree conflictdeltaderivativederivativesdetdet concatdet muller langdetailsdeterministicdifferential geometrydiffie hellmanDiophantine equationsdirichlet kerneldisagreementdiscretedist equivdist of functiondistancedistributiondivdivergence of setsdomaindoob lpdoob meyerdoubling measuredown singletdpidrcdualdual pairdyn systeme2eagereager tableeffective potentialeigenfunctioneisensteineisenstein asymptoticselectric fieldelliptic coefficientsembeddingempotentialendgameenergyenforcementenforcerensembleentanglemententropyentropy pfrequation1323equation677equational logicestimateeuler lagrangeeuler maclaurineuler mascheroni boundseval disteval splitevaluationevaluation binding conflictevaluatorexamplesexceptional setexecutionexistenceexistence theoryexpected costexponential integrabilityexponential mapextractabilityextractorfactorizationfair schedulerFermat's Last Theoremfgfibringfidelityfieldfield strengthfinitary carlesonfinitefinite adele ringfinite coverfinite dimensionalfinite fieldfinitenessfinset termsfirstfirst estimatefive quantaflattenflex approxfoldingfolding polynomialforest complementforest unionforestsforkformal languagesformal logicfoundationsfourierFourier analysisfree semigroupfree statefrey packagefrobeniusfrom mathlibfrom unaryfull beta confluencefull beta eta confluencefull etafunctiongadgetsgauge transformationgen heapgen setsgeneralgeneralizedgenerate seedgeneric liftghost mapgradgradinggrid structuregromovgroupgroup theory stuffguruswamiguruswami sudanhaltinghamiltonianhandler from unaryhandler specsharmonic analysisharmonic waveharnackhas tauhas var adjointhash commitmentheat kernelheat solutionheat solution propertyhelperherbrand quotienthilbert kernelhilbert strong typehilbert94histholder normholder van der corputhom pfrhomotopyhopshprinciplehundred percenthypothesis testingideal gasidentical until badimmersionimplimplicationsimproved pfrindicatorinduction scheme delta1inductive constructionsinequalitiesinf modelinf occinfinite graph ramseyinfinite wireinflationinputinsert and contractinsert and contract natinsert noneinsert someinsertion sortinstanceinstancesintegralintegral closureinteractive protocolsinterleaved codeintervalinverse iterationis dist boundedis extremais plane waveis viableisqbridgeiteriterated laplacianiterationiwaniec kowalski ch1kemdemkey encap mechkinetic energykinetic termknowledge soundnesskoszul signkullbackkummers lemmal2estimatelagrangianlanguagelaplace inversionlaplace runge lenz vectorlarge separationleakagelemmaslevelslex'orderlie traceline in cubiclinear codelinear mv extensionlinearizedlinftylipschitz normlistlocallocal fieldslocal john nirenberglocalisation datalocallyloglog gradientlogging oraclelooplp function toolkitls corelyubashevsky seilermac from prfmachinemagnetic fieldmainmain resultsmain theoremmain thmmapmap unionmarcinkiewicz 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refsvalidationvan der corputvandermondevariancevcdimensionvector bundlesverificationverifierversion spaceversion space latticeweak calderon zygmundweak derivativeweak derivativesweak harnackweak pfrweightedweighted estimateswell typedwell typed okwell typed verificationwfwhole space sobolevwick contractionswick termwicks theorem normalwienerwired runwith abortwitnesseswriter costyukawazero consensuszero count crudezeta boundszf

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Cond KLDiv eq

condKLDiv_eq

Plain-language statement

If $X, Y$ are $G$ -valued random variables, and $Z$ is another random variable defined on the same sample space as $X$ , then $D_{KL}((X|Z)\Vert Y) = D_{KL}(X\Vert Y) + \bbH[X] - \bbH[X|Z].$