fact
stringlengths 3
3.37k
| type
stringclasses 3
values | library
stringclasses 3
values | imports
listlengths 0
65
| filename
stringlengths 31
124
| symbolic_name
stringlengths 1
66
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
AlltokenSpec(i: int) requires Valid() decreases |allTokens| requires 0 <= i < |allTokens| ensures allTokens == allTokens[..i] + allTokens[i].allTokens
|
lemma
|
Source
|
[
"System"
] |
Source/DafnyCore/AST/DafnyInterface.dfy
|
AlltokenSpec
| |
TokenNextIsIPlus1(middle: Token, i: int) requires Valid() requires 0 <= i < |allTokens| requires allTokens[i] == middle requires middle.Next != null ensures 0 <= i + 1 < |allTokens| && allTokens[i+1] == middle.Next
|
lemma
|
Source
|
[
"System"
] |
Source/DafnyCore/AST/DafnyInterface.dfy
|
TokenNextIsIPlus1
| |
TokenNextIsNullImpliesLastToken(middle: Token, i: int) requires middle.Valid() && this.Valid() requires 0 <= i < |allTokens| requires middle == allTokens[i] requires middle.Next == null ensures i == |allTokens|-1
|
lemma
|
Source
|
[
"System"
] |
Source/DafnyCore/AST/DafnyInterface.dfy
|
TokenNextIsNullImpliesLastToken
| |
Valid()
|
predicate
|
Source
|
[
"System"
] |
Source/DafnyCore/AST/DafnyInterface.dfy
|
Valid
| |
EndsWithNewline(input: CsString)
|
predicate
|
Source
|
[
"System"
] |
Source/DafnyCore/AST/DafnyInterface.dfy
|
EndsWithNewline
| |
GetNewLeadingTrivia(token: Token) : CsString
|
function
|
Source
|
[
"MicrosoftDafny",
"System"
] |
Source/DafnyCore/AST/Formatting.dfy
|
GetNewLeadingTrivia
| |
GetNewTrailingTrivia(token: Token) : CsString
|
function
|
Source
|
[
"MicrosoftDafny",
"System"
] |
Source/DafnyCore/AST/Formatting.dfy
|
GetNewTrailingTrivia
| |
IsAllocated(x: T) ensures allocated(x)
|
lemma
|
Source
|
[
"MicrosoftDafny",
"System"
] |
Source/DafnyCore/AST/Formatting.dfy
|
IsAllocated
| |
ContainsTransitive(other: CsString, needle: CsString) requires Contains(other) && other.Contains(needle) ensures Contains(needle) } lemma ContainsTransitive() ensures forall s1: CsString, s2: CsString, s3: CsString | s1.Contains(s2) && s2.Contains(s3) :: s1.Contains(s3)
|
lemma
|
Source
|
[] |
Source/DafnyCore/AST/System.dfy
|
ContainsTransitive
| |
BelowIsNonStrictLess(a: Box, b: Box) ensures Below(a, b) <==> Less(a, b) || a == b
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
BelowIsNonStrictLess
| |
LessIrreflexive(a: Box) ensures !Less(a, a)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
LessIrreflexive
| |
LessAsymmetric(a: Box, b: Box) requires Less(a, b) && Less(b, a) ensures false
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
LessAsymmetric
| |
LessBelowAsymmetric(a: Box, b: Box) requires Less(a, b) && Below(b, a) ensures false
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
LessBelowAsymmetric
| |
NotLess(a: Box, b: Box) ensures !Less(a, b) <==> Below(b, a)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
NotLess
| |
LessTransitive(a: Box, b: Box, c: Box) requires Less(a, b) && Less(b, c) ensures Less(a, c)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
LessTransitive
| |
LessBelowTransitive(a: Box, b: Box, c: Box) requires || (Less(a, b) && Below(b, c)) || (Below(a, b) && Less(b, c)) ensures Less(a, c)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
LessBelowTransitive
| |
Connected(a: Box, b: Box) ensures a != b ==> Less(a, b) || Less(b, a)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
Connected
| |
Less(a: Box, b: Box)
|
predicate
|
Source
|
[] |
Source/DafnyCore/Prelude/Boxes.dfy
|
Less
| |
Length() : nat
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Length
| |
Append(xs: List<X>) : List<X>
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Append
| |
At(i: nat) : X
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
At
| |
Take(n: nat) : List<X>
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Take
| |
Drop(n: nat) : List<X>
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Drop
| |
Split(n: nat) : (split: (List<X>, List<X>))
|
function
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Split
| |
LengthAppend(xs: List<X>) ensures Append(xs).Length() == Length() + xs.Length()
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
LengthAppend
| |
HeadTailAt() requires Cons? ensures head == At(0) ensures forall i :: 0 <= i < tail.Length() ==> tail.At(i) == At(i + 1)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
HeadTailAt
| |
ContainsAt(x: X) requires Contains(x) ensures i < Length() && At(i) == x
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
ContainsAt
| |
AtContains(i: nat, x: X) requires i < Length() && At(i) == x ensures Contains(x)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AtContains
| |
ConsAt(x: X, i: nat) requires i < Cons(x, this).Length() ensures Cons(x, this).At(i) == if i == 0 then x else At(i - 1)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
ConsAt
| |
AppendAt(xs: List<X>, i: nat) requires i < Append(xs).Length() ensures Append(xs).At(i) == if i < Length() then At(i) else (LengthAppend(xs); xs.At(i - Length()))
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AppendAt
| |
AboutDrop(n: nat) requires n < Length() ensures Drop(n).Cons?
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AboutDrop
| |
AppendTake(xs: List<X>) ensures (LengthAppend(xs); Append(xs).Take(Length()) == this)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AppendTake
| |
TakeFromAppend(xs: List<X>, n: nat) requires n <= Length() + xs.Length() ensures (LengthAppend(xs); Append(xs).Take(n) == if n <= Length() then Take(n) else Append(xs.Take(n - Length())))
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
TakeFromAppend
| |
AppendDrop(xs: List<X>) ensures (LengthAppend(xs); Append(xs).Drop(Length()) == xs)
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AppendDrop
| |
DropFromAppend(xs: List<X>, n: nat) requires n <= Length() + xs.Length() ensures (LengthAppend(xs); Append(xs).Drop(n) == if n <= Length() then Drop(n).Append(xs) else xs.Drop(n - Length()))
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
DropFromAppend
| |
AppendTakeDrop(i: nat) requires i <= Length() ensures Take(i).Append(Drop(i)) == this
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
AppendTakeDrop
| |
LengthTakeDrop(i: nat) requires i <= Length() ensures Take(i).Length() == i && Drop(i).Length() == Length() - i
|
lemma
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
LengthTakeDrop
| |
Contains(x: X)
|
predicate
|
Source
|
[] |
Source/DafnyCore/Prelude/Lists.dfy
|
Contains
| |
BoolInt(b: bool) : nat
|
function
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
BoolInt
| |
Update(m: List<Box>, o: Box, n: nat) : List<Box>
|
function
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
Update
| |
RemoveOne(a: Multiset, x: Box) : Multiset
|
function
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
RemoveOne
| |
MultiplicityWrapper(a: Multiset, o: Box) : int
|
function
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityWrapper
| |
TailIncreasing(s: List<Box>) requires Increasing(s) && s.Cons? ensures Increasing(s.tail)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
TailIncreasing
| |
MultiplicityUpdateSame(m: Multiset, o: Box, n: int) requires 0 <= n ensures (UpdatePreservesIncreasing(m, o, n); Multiplicity(Update(m, o, n), o) == n)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityUpdateSame
| |
MultiplicityUpdateDifferent(m: Multiset, o: Box, n: int, p: Box) requires 0 <= n && o != p ensures (UpdatePreservesIncreasing(m, o, n); Multiplicity(Update(m, o, n), p) == Multiplicity(m, p))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityUpdateDifferent
| |
ElementsOfUpdate(m: List<Box>, o: Box, n: nat, m': List<Box>) requires m' == Update(m, o, n) ensures forall i :: 0 <= i < m'.Length() ==> m'.At(i) == o || m.Contains(m'.At(i))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ElementsOfUpdate
| |
HeadIsLeast(x: Box, m: List<Box>) requires Increasing(m) requires m == Nil || Below(x, m.head) ensures forall i :: 0 <= i < m.Length() ==> Below(x, m.At(i))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
HeadIsLeast
| |
ConsUpdatePreservesIncreasing(x: Box, m: List<Box>, o: Box, n: nat) requires Increasing(m) requires Increasing(Update(m, o, n)) requires forall i :: 0 <= i < m.Length() ==> Below(x, m.At(i)) requires Below(x, o) ensures Increasing(Cons(x, Update(m, o, n)))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ConsUpdatePreservesIncreasing
| |
UpdatePreservesIncreasing(m: List<Box>, o: Box, n: nat) requires Increasing(m) ensures Increasing(Update(m, o, n))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
UpdatePreservesIncreasing
| |
PrependPreservesIncreasingAll(o: Box, m: List<Box>) requires Increasing(m) requires forall i :: 0 <= i < m.Length() ==> Below(o, m.At(i)) ensures Increasing(Cons(o, m))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
PrependPreservesIncreasingAll
| |
PrependPreservesIncreasing(o: Box, m: List<Box>) requires Increasing(m) requires m == Nil || Below(o, m.head) ensures Increasing(Cons(o, m))
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
PrependPreservesIncreasing
| |
MultiplicityCard(m: Multiset, x: Box) ensures 0 <= Multiplicity(m, x) <= Card(m)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityCard
| |
ZeroMultiplicityForLargerElements(s: Multiset, x: Box) requires s == Nil || Less(x, s.head) ensures Multiplicity(s, x) == 0
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ZeroMultiplicityForLargerElements
| |
ZeroMultiplicityForNonElements(s: Multiset, o: Box) requires !s.Contains(o) ensures Multiplicity(s, o) == 0
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ZeroMultiplicityForNonElements
| |
NonZeroMultiplicityForElements(s: Multiset, o: Box) ensures s.Contains(o) <==> Multiplicity(s, o) != 0
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
NonZeroMultiplicityForElements
| |
UnionOneContains(a: Multiset, x: Box, o: Box) ensures UnionOne(a, x).Contains(o) <==> o == x || a.Contains(o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
UnionOneContains
| |
MultiplicityDelta(a: Multiset, x: Box, y: Box) ensures Multiplicity(UnionOne(a, x), y) == BoolInt(x == y) + Multiplicity(a, y)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityDelta
| |
RemoveOnePreservesBelow(a: Multiset, x: Box, o: Box) requires a.Contains(x) requires forall i :: 0 <= i < a.Length() ==> Below(o, a.At(i)) ensures forall i :: 0 <= i < RemoveOne(a, x).Length() ==> Below(o, RemoveOne(a, x).At(i)) decreases a
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
RemoveOnePreservesBelow
| |
MultiplicityRemoveOne(a: Multiset, x: Box, o: Box) requires a.Contains(x) ensures Multiplicity(RemoveOne(a, x), o) == Multiplicity(a, o) - BoolInt(x == o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityRemoveOne
| |
ZeroMinMultiplicity(a: Multiset, o: Box) ensures Math.Min(0, Multiplicity(a, o)) == 0 == Math.Min(Multiplicity(a, o), 0)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ZeroMinMultiplicity
| |
RevealMin() ensures forall a, b :: Math.Min(a, b) == if a < b then a else b
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
RevealMin
| |
MinDistribution(a: int, b: int, c: int) ensures Math.Min(a - c, b - c) == Math.Min(a, b) - c
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MinDistribution
| |
ClipNonPos(i: int) requires i <= 0 ensures Math.Clip(i) == 0
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ClipNonPos
| |
ClipDistribution(a: nat, b: nat) ensures Math.Clip(a) + b == Math.Clip(a + b)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ClipDistribution
| |
ClipMultiplicity(a: Multiset, o: Box) ensures Math.Clip(Multiplicity(a, o)) == Multiplicity(a, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ClipMultiplicity
| |
ExhibitSubsetDifference(a: Multiset, b: Multiset) requires !Subset(a, b) ensures Multiplicity(a, o) > Multiplicity(b, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ExhibitSubsetDifference
| |
ExhibitDifference(a: Multiset, b: Multiset) requires a != b ensures Multiplicity(a, o) != Multiplicity(b, o) decreases a != Nil, a.Cons? && b.Cons? && !Less(a.head, b.head), a
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
ExhibitDifference
| |
MultiplicityFromSetIsBinary(s: Sets.Set, o: Box) ensures 0 <= Multiplicity(FromSet(s), o) <= 1
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityFromSetIsBinary
| |
MultiplicityFromSeqAppend(a: Sequences.Seq, b: Sequences.Seq, o: Box) ensures Multiplicity(FromSeq(Sequences.Append(a, b)), o) == Multiplicity(FromSeq(a), o) + Multiplicity(FromSeq(b), o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityFromSeqAppend
| |
MultiplicityFromSeqContainment(s: Sequences.Seq, v: Box, x: Box) requires s.Contains(v) ensures Multiplicity(Singleton(v), x) <= Multiplicity(FromSeq(s), x)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityFromSeqContainment
| |
MultiplicityFromSeqCons(s: Sequences.Seq, x: Box) requires s.Cons? ensures Multiplicity(FromSeq(s), x) == Multiplicity(FromSeq(s.tail), x) + BoolInt(s.head == x)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
MultiplicityFromSeqCons
| |
SequenceAndMultisetContainment(s: Sequences.Seq, o: Box) ensures Sequences.Contains(s, o) <==> FromSeq(s).Contains(o)
|
lemma
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
SequenceAndMultisetContainment
| |
Increasing(m: List<Box>)
|
predicate
|
Source
|
[
"Boxes",
"Lists",
"Math",
"Sets",
"Sequences"
] |
Source/DafnyCore/Prelude/Multisets.dfy
|
Increasing
| |
TailContains(s: Seq, val: Box) requires s.Cons? ensures Contains(s, val) <==> s.head == val || Contains(s.tail, val)
|
lemma
|
Source
|
[
"Lists",
"Boxes"
] |
Source/DafnyCore/Prelude/Sequences.dfy
|
TailContains
| |
DatatypeEquality(s0: Seq, s1: Seq) requires s0.Length() == s1.Length() requires forall j :: 0 <= j < s0.Length() ==> s0.At(j) == s1.At(j) ensures s0 == s1
|
lemma
|
Source
|
[
"Lists",
"Boxes"
] |
Source/DafnyCore/Prelude/Sequences.dfy
|
DatatypeEquality
| |
UnionOneRaw(s: List<Box>, o: Box) : List<Box>
|
function
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
UnionOneRaw
| |
IntersectionRaw(a: List<Box>, b: List<Box>) : List<Box>
|
function
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
IntersectionRaw
| |
DifferenceRaw(a: List<Box>, b: List<Box>) : List<Box>
|
function
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
DifferenceRaw
| |
TailStrictlyIncreasing(s: List<Box>) requires StrictlyIncreasing(s) && s.Cons? ensures StrictlyIncreasing(s.tail)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
TailStrictlyIncreasing
| |
UnionOneRawSame(s: List<Box>, o: Box) requires StrictlyIncreasing(s) && s.Contains(o) ensures s == UnionOneRaw(s, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
UnionOneRawSame
| |
UnionOnePreservesStrictlyIncreasing(s: List<Box>, o: Box) requires StrictlyIncreasing(s) ensures StrictlyIncreasing(UnionOneRaw(s, o))
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
UnionOnePreservesStrictlyIncreasing
| |
PrependIncreasing(o: Box, s: List<Box>) requires StrictlyIncreasing(s) requires s != Nil && Less(o, s.head) ensures StrictlyIncreasing(Cons(o, s))
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
PrependIncreasing
| |
UnionCommutative(a: Set, b: Set) ensures Union(a, b) == Union(b, a)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
UnionCommutative
| |
DisjointDifferenceAux(a: Set, b: Set) requires Disjoint(a, b) ensures Difference(Union(a, b), a) == b
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
DisjointDifferenceAux
| |
IntersectionRawElements(a: List<Box>, b: List<Box>, o: Box) ensures In(IntersectionRaw(a, b), o) <==> In(a, o) && In(b, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
IntersectionRawElements
| |
IntersectionRawPreservesStrictlyIncreasing(a: List<Box>, b: List<Box>) requires StrictlyIncreasing(a) && StrictlyIncreasing(b) ensures StrictlyIncreasing(IntersectionRaw(a, b))
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
IntersectionRawPreservesStrictlyIncreasing
| |
IntersectionCommutative(a: Set, b: Set) ensures Intersection(a, b) == Intersection(b, a)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
IntersectionCommutative
| |
CardUnion(a: Set, b: Set) requires a != Nil ensures (TailStrictlyIncreasing(a); var m, n := Card(Union(a, b)), Card(Union(a.tail, b)); m == if In(b, a.head) then n else n + 1)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
CardUnion
| |
DifferenceRawElements(a: List<Box>, b: List<Box>, o: Box) ensures In(DifferenceRaw(a, b), o) <==> In(a, o) && !In(b, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
DifferenceRawElements
| |
DifferenceRawPreservesStrictlyIncreasing(a: List<Box>, b: List<Box>) requires StrictlyIncreasing(a) && StrictlyIncreasing(b) ensures StrictlyIncreasing(DifferenceRaw(a, b))
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
DifferenceRawPreservesStrictlyIncreasing
| |
CardOfDisjoint(a: Set, b: Set) requires Disjoint(a, b) ensures Card(Union(a, b)) == Card(a) + Card(b)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
CardOfDisjoint
| |
HeadIsSmallest(s: List<Box>, o: Box) requires StrictlyIncreasing(s) && s.Contains(o) ensures Below(s.head, o)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
HeadIsSmallest
| |
NothingIsSmallerThanHead(s: List<Box>, o: Box) requires StrictlyIncreasing(s) && s != Nil && Less(o, s.head) ensures !s.Contains(o)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
NothingIsSmallerThanHead
| |
TailDoesNotContainHead(s: List<Box>) requires StrictlyIncreasing(s) && s != Nil ensures !s.tail.Contains(s.head)
|
lemma
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
TailDoesNotContainHead
| |
StrictlyIncreasing(s: List<Box>)
|
predicate
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
StrictlyIncreasing
| |
In(s: List<Box>, o: Box)
|
predicate
|
Source
|
[
"Boxes",
"Lists"
] |
Source/DafnyCore/Prelude/Sets.dfy
|
In
| |
SeqToHeight(s: seq<T>, f: T --> nat) : (r: nat)
|
function
|
Source
|
[
"Std.Wrappers",
"Std"
] |
Source/DafnyCore/Backends/Dafny/AST.dfy
|
SeqToHeight
| |
name() : Name
|
function
|
Source
|
[
"Std.Wrappers",
"Std"
] |
Source/DafnyCore/Backends/Dafny/AST.dfy
|
name
| |
attributes() : seq<Attribute>
|
function
|
Source
|
[
"Std.Wrappers",
"Std"
] |
Source/DafnyCore/Backends/Dafny/AST.dfy
|
attributes
| |
Replace(mapping: map<Type, Type>) : Type
|
function
|
Source
|
[
"Std.Wrappers",
"Std"
] |
Source/DafnyCore/Backends/Dafny/AST.dfy
|
Replace
|
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