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Prove independence of BernoulliExpNeg.SampleLe1 #139

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Jan 31, 2024
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Prove independence of BernoulliExpNeg.SampleLe1 #139

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19 changes: 9 additions & 10 deletions audit.log
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Original file line number Diff line number Diff line change
@@ -1,16 +1,15 @@
src/DafnyVMC.dfy(23,6): UniformPowerOfTwoSample: Definition has `assume {:axiom}` statement in body.
src/Distributions/Bernoulli/Correctness.dfy(59,17): BernoulliCorrectnessCaseFalse: Declaration has explicit `{:axiom}` attribute.
src/Distributions/BernoulliExpNeg/Correctness.dfy(228,6): CorrectnessLe1: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(251,17): SampleLe1IsIndep: Declaration has explicit `{:axiom}` attribute.
src/Distributions/BernoulliExpNeg/Correctness.dfy(303,17): Le1SeriesConvergesToExpNeg: Declaration has explicit `{:axiom}` attribute.
src/Distributions/BernoulliExpNeg/Correctness.dfy(368,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(369,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(376,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(377,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(381,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(382,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(572,8): Le1LoopCorrectnessEq: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(573,8): Le1LoopCorrectnessEq: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(336,17): Le1SeriesConvergesToExpNeg: Declaration has explicit `{:axiom}` attribute.
src/Distributions/BernoulliExpNeg/Correctness.dfy(401,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(402,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(409,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(410,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(414,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(415,6): Le1LoopCutDecomposeProb: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(605,8): Le1LoopCorrectnessEq: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Correctness.dfy(606,8): Le1LoopCorrectnessEq: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Model.dfy(112,10): Le1LoopTerminatesAlmostSurely: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Model.dfy(148,4): Le1LoopDivergenceProbabilityBound: Definition has `assume {:axiom}` statement in body.
src/Distributions/BernoulliExpNeg/Model.dfy(161,8): Le1LoopDivergenceProbabilityBound: Definition has `assume {:axiom}` statement in body.
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35 changes: 34 additions & 1 deletion src/Distributions/BernoulliExpNeg/Correctness.dfy
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Expand Up @@ -248,10 +248,43 @@ module BernoulliExpNeg.Correctness {
}
}

lemma {:axiom} SampleLe1IsIndep(gamma: Rationals.Rational)
lemma SampleLe1IsIndep(gamma: Rationals.Rational)
requires 0 <= gamma.numer <= gamma.denom
ensures Independence.IsIndep(Model.SampleLe1(gamma))
{
Le1LoopIsIndep(gamma);
Independence.MapIsIndep(Model.Le1Loop(gamma)((true, 0)), (ak: (bool, nat)) => ak.1 % 2 == 1);
}

lemma Le1LoopIsIndep(gamma: Rationals.Rational)
requires 0 <= gamma.numer <= gamma.denom
ensures Independence.IsIndep(Model.Le1Loop(gamma)((true, 0)))
{
Model.Le1LoopTerminatesAlmostSurely(gamma);
forall ak: (bool, nat) ensures Independence.IsIndep(Model.Le1LoopIter(gamma)(ak)) {
Le1LoopIterIsIndep(gamma, ak);
}
Loops.WhileIsIndep(
Model.Le1LoopCondition,
Model.Le1LoopIter(gamma),
(true, 0)
);
assert Independence.IsIndep(Model.Le1Loop(gamma)((true, 0))) by {
reveal Model.Le1Loop();
}
}

lemma Le1LoopIterIsIndep(gamma: Rationals.Rational, ak: (bool, nat))
requires 0 <= gamma.numer <= gamma.denom
ensures Independence.IsIndep(Model.Le1LoopIter(gamma)(ak))
{
var k': nat := ak.1 + 1;
Bernoulli.Correctness.SampleIsIndep(gamma.numer, k' * gamma.denom);
Independence.MapIsIndep(
Bernoulli.Model.Sample(gamma.numer, k' * gamma.denom),
a => (a, k')
);
}

// Proves the correctness of `Model.Le1LoopIter`.
// Correctness means that when run on an initial value of `(true, k)` it produces
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