Revert Result Type, Proof of ResultsIndependent, Reduce Brittleness i…

…n FisherYates (#161) By submitting this pull request, I confirm that my contribution is made under the terms of the [MIT license](https://github.com/dafny-lang/dafny/blob/master/LICENSE.txt).
dafny-lang · Mar 6, 2024 · 2697703 · 2697703
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diff --git a/audit.log b/audit.log
@@ -1,8 +1,8 @@
 src/Distributions/Uniform/Correctness.dfy(31,17): UniformFullCorrectness: Declaration has explicit `{:axiom}` attribute. 
 src/Distributions/Uniform/Correctness.dfy(36,17): SampleCoin: Declaration has explicit `{:axiom}` attribute. 
 src/Distributions/Uniform/Model.dfy(19,33): Sample: Declaration has explicit `{:axiom}` attribute. 
+src/Distributions/Uniform/Model.dfy(46,17): IntervalSampleIsMeasurePreserving: Declaration has explicit `{:axiom}` attribute. 
 src/ProbabilisticProgramming/Independence.dfy(30,27): IsIndep: Declaration has explicit `{:axiom}` attribute. 
-src/ProbabilisticProgramming/Independence.dfy(55,6): ResultsIndependent: Definition has `assume {:axiom}` statement in body. 
-src/ProbabilisticProgramming/Independence.dfy(60,17): IsIndepImpliesIsIndepFunction: Declaration has explicit `{:axiom}` attribute. 
-src/ProbabilisticProgramming/Independence.dfy(64,17): MapIsIndep: Declaration has explicit `{:axiom}` attribute. 
+src/ProbabilisticProgramming/Independence.dfy(70,17): IsIndepImpliesIsIndepFunction: Declaration has explicit `{:axiom}` attribute. 
+src/ProbabilisticProgramming/Independence.dfy(74,17): MapIsIndep: Declaration has explicit `{:axiom}` attribute. 
 src/ProbabilisticProgramming/RandomSource.dfy(50,17): ProbIsProbabilityMeasure: Declaration has explicit `{:axiom}` attribute. 
diff --git a/src/Distributions/Uniform/Model.dfy b/src/Distributions/Uniform/Model.dfy
@@ -19,7 +19,7 @@ module Uniform.Model {
   opaque ghost function {:axiom} Sample(n: nat): (h: Monad.Hurd<nat>)
     requires n > 0
     ensures Independence.IsIndep(h)
-    ensures forall s :: h(s).Result? ==> 0 <= h(s).value < n
+    ensures forall s :: 0 <= h(s).value < n
 
   ghost function IntervalSample(a: int, b: int): (f: Monad.Hurd<int>)
     requires a < b
@@ -33,16 +33,18 @@ module Uniform.Model {
 
   lemma SampleBound(n: nat, s: Rand.Bitstream)
     requires n > 0
-    requires Sample(n)(s).Result?
     ensures 0 <= Sample(n)(s).value < n
   {}
 
   lemma IntervalSampleBound(a: int, b: int, s: Rand.Bitstream)
     requires a < b
-    requires IntervalSample(a, b)(s).Result?
     ensures a <= IntervalSample(a, b)(s).value < b
   {
     SampleBound(b-a, s);
   }
 
+  lemma {:axiom} IntervalSampleIsMeasurePreserving(a: int, b: int)
+    requires a < b
+    ensures Measures.IsMeasurePreserving(Rand.eventSpace, Rand.prob, Rand.eventSpace, Rand.prob, s => IntervalSample(a, b)(s).rest)
+
 }
diff --git a/src/ProbabilisticProgramming/Independence.dfy b/src/ProbabilisticProgramming/Independence.dfy
@@ -40,6 +40,7 @@ module Independence {
   )
     requires hIndep: IsIndepFunction(h)
     requires bMeasurable: bSeeds in Rand.eventSpace
+    requires hIsMeasurePreserving: Measures.IsMeasurePreserving(Rand.eventSpace, Rand.prob, Rand.eventSpace, Rand.prob, s => h(s).rest)
     ensures Rand.prob(Monad.BitstreamsWithValueIn(h, aSet) * Monad.BitstreamsWithRestIn(h, bSeeds)) == Rand.prob(Monad.BitstreamsWithValueIn(h, aSet)) * Rand.prob(bSeeds)
   {
     var aSeeds := Monad.BitstreamsWithValueIn(h, aSet);
@@ -52,7 +53,16 @@ module Independence {
       assert Measures.AreIndepEvents(Rand.eventSpace, Rand.prob, aSeeds, restBSeeds);
     }
     assert Rand.prob(restBSeeds) == Rand.prob(bSeeds) by {
-      assume {:axiom} false; // TODO
+      calc {
+        Rand.prob(restBSeeds);
+        Rand.prob(Monad.BitstreamsWithRestIn(h, bSeeds));
+        { assert Monad.BitstreamsWithRestIn(h, bSeeds) == iset s | h(s).rest in bSeeds; }
+        Rand.prob(iset s | h(s).rest in bSeeds);
+        { assert (iset s | h(s).rest in bSeeds) == Measures.PreImage(s => h(s).rest, bSeeds); }
+        Rand.prob(Measures.PreImage(s => h(s).rest, bSeeds));
+        { reveal bMeasurable; reveal hIsMeasurePreserving; }
+        Rand.prob(bSeeds);
+      }
     }
   }
 

diff --git a/src/ProbabilisticProgramming/Monad.dfy b/src/ProbabilisticProgramming/Monad.dfy
@@ -18,20 +18,14 @@ module Monad {
   // The result of a probabilistic computation on a bitstream.
   // It either consists of the computed value and the (unconsumed) rest of the bitstream or indicates nontermination.
   // It differs from Hurd's definition in that the result can be nontermination, which Hurd does not model explicitly.
-  datatype Result<A> =
-    | Result(value: A, rest: Rand.Bitstream)
-    | Diverging
+  datatype Result<A> = Result(value: A, rest: Rand.Bitstream)
   {
     function Map<B>(f: A -> B): Result<B> {
-      match this
-      case Diverging => Diverging
-      case Result(value, rest) => Result(f(value), rest)
+      Result(f(value), rest)
     }
 
     function Bind<B>(f: A -> Hurd<B>): Result<B> {
-      match this
-      case Diverging => Diverging
-      case Result(value, rest) => f(value)(rest)
+      f(value)(rest)
     }
 
     ghost predicate In(s: iset<A>) {
@@ -43,44 +37,30 @@ module Monad {
     }
 
     predicate Satisfies(property: A -> bool) {
-      match this
-      case Diverging => false
-      case Result(value, _) => property(value)
+      property(value)
     }
 
     ghost predicate RestIn(s: iset<Rand.Bitstream>) {
       RestSatisfies(r => r in s)
     }
 
     predicate RestSatisfies(property: Rand.Bitstream -> bool) {
-      match this
-      case Diverging => false
-      case Result(_, rest) => property(rest)
+      property(rest)
     }
 
-    predicate IsFailure() {
-      Diverging?
-    }
-
-    function PropagateFailure<B>(): Result<B>
-      requires Diverging?
-    {
-      Diverging
-    }
-
-    function Extract(): (A, Rand.Bitstream)
-      requires Result?
+    function Extract(): (x: (A, Rand.Bitstream))
+      ensures this == Result(x.0, x.1)
     {
       (this.value, this.rest)
     }
   }
 
   ghost function Values<A>(results: iset<Result<A>>): iset<A> {
-    iset r <- results | r.Result? :: r.value
+    iset r <- results :: r.value
   }
 
   ghost function Rests<A>(results: iset<Result<A>>): iset<Rand.Bitstream> {
-    iset r <- results | r.Result? :: r.rest
+    iset r <- results :: r.rest
   }
 
   ghost function ResultEventSpace<A(!new)>(eventSpace: iset<iset<A>>): iset<iset<Result<A>>> {
@@ -92,11 +72,11 @@ module Monad {
   ghost const natResultEventSpace: iset<iset<Result<nat>>> := ResultEventSpace(Measures.natEventSpace)
 
   ghost function ResultsWithValueIn<A(!new)>(values: iset<A>): iset<Result<A>> {
-    iset result: Result<A> | result.Result? && result.value in values
+    iset result: Result<A> | result.value in values
   }
 
   ghost function ResultsWithRestIn<A(!new)>(rests: iset<Rand.Bitstream>): iset<Result<A>> {
-    iset result: Result<A> | result.Result? && result.rest in rests
+    iset result: Result<A> | result.rest in rests
   }
 
   ghost function BitstreamsWithValueIn<A(!new)>(h: Hurd<A>, aSet: iset<A>): iset<Rand.Bitstream> {
@@ -112,30 +92,13 @@ module Monad {
     (s: Rand.Bitstream) => f(s).Bind(g)
   }
 
-  function BindAlternative<A,B>(f: Hurd<A>, g: A -> Hurd<B>): (h: Hurd<B>)
-    ensures forall s :: h(s) == Bind(f, g)(s)
-  {
-    (s: Rand.Bitstream) =>
-      var (a, s') :- f(s);
-      g(a)(s')
-  }
-
   // Equation (2.42)
   const Coin: Hurd<bool> := s => Result(Rand.Head(s), Rand.Tail(s))
 
   function Composition<A,B,C>(f: A -> Hurd<B>, g: B -> Hurd<C>): A -> Hurd<C> {
     (a: A) => Bind(f(a), g)
   }
 
-  function CompositionAlternative<A(!new),B,C>(f: A -> Hurd<B>, g: B -> Hurd<C>): (h: A -> Hurd<C>)
-    ensures forall a, s :: h(a)(s) == Composition(f, g)(a)(s)
-  {
-    (a: A) =>
-      (s: Rand.Bitstream) =>
-        var (b, s') :- f(a)(s);
-        g(b)(s')
-  }
-
   // Equation (3.3)
   function Return<A>(a: A): Hurd<A> {
     (s: Rand.Bitstream) => Result(a, s)

diff --git a/src/Util/FisherYates/Correctness.dfy b/src/Util/FisherYates/Correctness.dfy
@@ -21,7 +21,7 @@ module FisherYates.Correctness {
     requires i <= |xs|
     requires i <= |p|
   {
-    iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..]
+    iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..]
   }
 
   ghost predicate CorrectnessPredicate<T(!new)>(xs: seq<T>, p: seq<T>, i: nat)
@@ -64,7 +64,7 @@ module FisherYates.Correctness {
   {
     var e := iset s | Model.Shuffle(xs)(s).Equals(p);
     var i := 0;
-    var e' := iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..];
+    var e' := iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..];
     assert e == e';
     assert |xs| == |p| by {
       Model.PermutationsPreserveCardinality(xs, p);
@@ -81,7 +81,7 @@ module FisherYates.Correctness {
     requires multiset(p[i..]) == multiset(xs[i..])
     ensures CorrectnessPredicate(xs, p, i)
   {
-    var e := iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..];
+    var e := iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..];
     if |xs[i..]| <= 1 {
       CorrectnessFisherYatesUniqueElementsGeneralLeq1(xs, p, i);
     } else {
@@ -100,16 +100,16 @@ module FisherYates.Correctness {
     ensures CorrectnessPredicate(xs, p, i)
   {
     Model.PermutationsPreserveCardinality(p[i..], xs[i..]);
-    var e := iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..];
+    var e := iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..];
     assert e == Measures.SampleSpace() by {
       forall s
         ensures s in e
       {
         calc {
           s in e;
-          Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..];
+          Model.Shuffle(xs, i)(s).value[i..] == p[i..];
           { assert Model.Shuffle(xs, i)(s) == Monad.Return(xs)(s); }
-          Monad.Return(xs)(s).Result? && Monad.Return(xs)(s).value[i..] == p[i..];
+          Monad.Return(xs)(s).value[i..] == p[i..];
           { assert Monad.Return(xs)(s).value == xs; }
           xs[i..] == p[i..];
           if |xs[i..]| == 0 then [] == p[i..] else [xs[i]] == p[i..];
@@ -147,9 +147,12 @@ module FisherYates.Correctness {
     ensures CorrectnessPredicate(xs, p, i)
   {
     Model.PermutationsPreserveCardinality(p[i..], xs[i..]);
-    var e := iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..];
+    var e := iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..];
     assert |xs| > i + 1;
     var h := Uniform.Model.IntervalSample(i, |xs|);
+    assert hIsMeasurePreserving: Measures.IsMeasurePreserving(Rand.eventSpace, Rand.prob, Rand.eventSpace, Rand.prob, s => h(s).rest) by {
+      Uniform.Model.IntervalSampleIsMeasurePreserving(i, |xs|);
+    }
     assert HIsIndependent: Independence.IsIndepFunction(h) by {
       Uniform.Correctness.IntervalSampleIsIndep(i, |xs|);
       Independence.IsIndepImpliesIsIndepFunction(h);
@@ -238,33 +241,68 @@ module FisherYates.Correctness {
     assert DecomposeE: e == Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e') by {
       DecomposeE(xs, ys, p, i, j, h, A, e, e');
     }
+
     assert CorrectnessPredicate(xs, p, i) by {
       reveal DecomposeE;
       reveal HIsIndependent;
       reveal BitStreamsInA;
-      assert e' in Rand.eventSpace && Rand.prob(e') == 1.0 / (NatArith.FactorialTraditional(|xs|-(i+1)) as real) by {
-        assert e' in Rand.eventSpace by {
-          assert CorrectnessPredicate(ys, p, i+1) by { reveal InductionHypothesis; }
-          assert e' == CorrectnessConstructEvent(ys, p, i+1);
-        }
-        calc {
-          Rand.prob(e');
-          { assert CorrectnessPredicate(ys, p, i+1) by { reveal InductionHypothesis; }
-            assert e' == CorrectnessConstructEvent(ys, p, i+1); }
-          1.0 / (NatArith.FactorialTraditional(|ys|-(i+1)) as real);
-          { assert |xs| == |ys|;
-            assert |ys|-(i+1) == |xs|-(i+1);
-            assert NatArith.FactorialTraditional(|ys|-(i+1)) == NatArith.FactorialTraditional(|xs|-(i+1));
-            assert (NatArith.FactorialTraditional(|ys|-(i+1)) as real) == (NatArith.FactorialTraditional(|xs|-(i+1)) as real); }
-          1.0 / (NatArith.FactorialTraditional(|xs|-(i+1)) as real);
-        }
+      reveal InductionHypothesis;
+      reveal hIsMeasurePreserving;
+      CorrectnessFisherYatesUniqueElementsGeneralGreater1Helper(xs, ys, p, i, j, h, A, e, e');
+    }
+
+  }
+
+  lemma CorrectnessFisherYatesUniqueElementsGeneralGreater1Helper<T(!new)>(xs: seq<T>, ys: seq<T>, p: seq<T>, i: nat, j: nat, h: Monad.Hurd<int>, A: iset<int>, e: iset<Rand.Bitstream>, e': iset<Rand.Bitstream>)
+    decreases |xs| - i
+    requires i <= |xs|
+    requires i <= |p|
+    requires forall a, b | i <= a < b < |xs| :: xs[a] != xs[b]
+    requires |xs| == |p|
+    requires multiset(p[i..]) == multiset(xs[i..])
+    requires |xs[i..]| > 1
+    requires i <= j < |xs| && xs[j] == p[i]
+    requires |xs| == |ys|
+    requires ys == Model.Swap(xs, i, j)
+    requires e == CorrectnessConstructEvent(xs, p, i)
+    requires e' == CorrectnessConstructEvent(ys, p, i+1)
+    requires DecomposeE: e == Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e')
+    requires HIsIndependent: Independence.IsIndepFunction(h)
+    requires BitStreamsInA: Monad.BitstreamsWithValueIn(h, A) == (iset s | Uniform.Model.IntervalSample(i, |xs|)(s).Equals(j))
+    requires InductionHypothesis: CorrectnessPredicate(ys, p, i+1)
+    requires hIsMeasurePreserving: Measures.IsMeasurePreserving(Rand.eventSpace, Rand.prob, Rand.eventSpace, Rand.prob, s => h(s).rest)
+    ensures CorrectnessPredicate(xs, p, i)
+  {
+    reveal DecomposeE;
+    reveal HIsIndependent;
+    reveal BitStreamsInA;
+    assert e' in Rand.eventSpace && Rand.prob(e') == 1.0 / (NatArith.FactorialTraditional(|xs|-(i+1)) as real) by {
+      assert e' in Rand.eventSpace by {
+        assert CorrectnessPredicate(ys, p, i+1) by { reveal InductionHypothesis; }
+        assert e' == CorrectnessConstructEvent(ys, p, i+1);
       }
-      ProbabilityOfE(xs, ys, p, i, j, h, A, e, e');
+      calc {
+        Rand.prob(e');
+        { assert CorrectnessPredicate(ys, p, i+1) by { reveal InductionHypothesis; }
+          assert e' == CorrectnessConstructEvent(ys, p, i+1); }
+        1.0 / (NatArith.FactorialTraditional(|ys|-(i+1)) as real);
+        { assert |xs| == |ys|;
+          assert |ys|-(i+1) == |xs|-(i+1);
+          assert NatArith.FactorialTraditional(|ys|-(i+1)) == NatArith.FactorialTraditional(|xs|-(i+1));
+          assert (NatArith.FactorialTraditional(|ys|-(i+1)) as real) == (NatArith.FactorialTraditional(|xs|-(i+1)) as real); }
+        1.0 / (NatArith.FactorialTraditional(|xs|-(i+1)) as real);
+      }
+    }
+    reveal hIsMeasurePreserving;
+    assert e in Rand.eventSpace by {
       EInEventSpace(xs, p, h, A, e, e');
     }
-
+    assert Rand.prob(e) == 1.0 / (NatArith.FactorialTraditional(|xs|-i) as real) by {
+      ProbabilityOfE(xs, ys, p, i, j, h, A, e, e');
+    }
   }
 
+
   lemma BitStreamsInA<T(!new)>(xs: seq<T>, p: seq<T>, i: nat, j: nat, h: Monad.Hurd<int>, A: iset<int>)
     requires i <= |xs|
     requires i <= |p|
@@ -302,8 +340,8 @@ module FisherYates.Correctness {
     requires A == iset{j}
     requires h == Uniform.Model.IntervalSample(i, |xs|)
     requires ys == Model.Swap(xs, i, j)
-    requires e == iset s | Model.Shuffle(xs, i)(s).Result? && Model.Shuffle(xs, i)(s).value[i..] == p[i..]
-    requires e' == iset s | Model.Shuffle(ys, i+1)(s).Result? && Model.Shuffle(ys, i+1)(s).value[i+1..] == p[i+1..]
+    requires e == iset s | Model.Shuffle(xs, i)(s).value[i..] == p[i..]
+    requires e' == iset s | Model.Shuffle(ys, i+1)(s).value[i+1..] == p[i+1..]
     ensures e == Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e')
   {
     assert forall s :: s in e <==> s in Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e') by {
@@ -313,7 +351,6 @@ module FisherYates.Correctness {
         if s in e {
           var zs := Model.Shuffle(xs, i)(s).value;
           assert zs[i..] == p[i..];
-          assert h(s).Result?;
           var k := Uniform.Model.IntervalSample(i, |xs|)(s).value;
           Uniform.Model.IntervalSampleBound(i, |xs|, s);
           var s' := Uniform.Model.IntervalSample(i, |xs|)(s).rest;
@@ -332,7 +369,6 @@ module FisherYates.Correctness {
             assert k in A;
           }
           assert s in Monad.BitstreamsWithRestIn(h, e') by {
-            assert Model.Shuffle(ys, i+1)(s').Result?;
             assert Model.Shuffle(ys, i+1)(s').value[i+1..] == p[i+1..];
           }
           assert s in Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e');
@@ -491,6 +527,7 @@ module FisherYates.Correctness {
     requires |xs| == |ys|
     requires DecomposeE: e == Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e')
     requires HIsIndependent: Independence.IsIndepFunction(h)
+    requires hIsMeasurePreserving: Measures.IsMeasurePreserving(Rand.eventSpace, Rand.prob, Rand.eventSpace, Rand.prob, s => h(s).rest)
     requires InductionHypothesis: Rand.prob(e') == 1.0 / (NatArith.FactorialTraditional(|xs|-(i+1)) as real)
     requires BitStreamsInA: Monad.BitstreamsWithValueIn(h, A) == (iset s | Uniform.Model.IntervalSample(i, |xs|)(s).Equals(j))
     ensures
@@ -500,7 +537,7 @@ module FisherYates.Correctness {
       Rand.prob(e);
       { reveal DecomposeE; }
       Rand.prob(Monad.BitstreamsWithValueIn(h, A) * Monad.BitstreamsWithRestIn(h, e'));
-      { reveal HIsIndependent; reveal InductionHypothesis; Independence.ResultsIndependent(h, A, e'); }
+      { reveal HIsIndependent; reveal InductionHypothesis; reveal hIsMeasurePreserving; Independence.ResultsIndependent(h, A, e'); }
       Rand.prob(Monad.BitstreamsWithValueIn(h, A)) * Rand.prob(e');
       { assert Rand.prob(Monad.BitstreamsWithValueIn(h, A)) == Rand.prob(iset s | Uniform.Model.IntervalSample(i, |xs|)(s).Equals(j)) by { reveal BitStreamsInA; } }
       Rand.prob(iset s | Uniform.Model.IntervalSample(i, |xs|)(s).Equals(j)) * Rand.prob(e');

diff --git a/src/Util/FisherYates/Implementation.dfy b/src/Util/FisherYates/Implementation.dfy
@@ -27,7 +27,10 @@ module FisherYates.Implementation {
           assert prevASeq == a[..]; // ghost
           Swap(a, i, j);
         }
+      } else {
+        assert prevASeq == a[..]; // ghost
       }
+
     }
 
     method Swap<T>(a: array<T>, i: nat, j: nat)