positivity
core extensions #
This file sets up the basic positivity
extensions tagged with the @[positivity]
attribute.
The positivity
extension which identifies expressions of the form ite p a b
,
such that positivity
successfully recognises both a
and b
.
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The positivity
extension which identifies expressions of the form min a b
,
such that positivity
successfully recognises both a
and b
.
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Extension for the max
operator. The max
of two numbers is nonnegative if at least one
is nonnegative, strictly positive if at least one is positive, and nonzero if both are nonzero.
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The positivity
extension which identifies expressions of the form a + b
,
such that positivity
successfully recognises both a
and b
.
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The positivity
extension which identifies expressions of the form a * b
,
such that positivity
successfully recognises both a
and b
.
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The positivity
extension which identifies expressions of the form a / b
,
where a
and b
are integers.
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The positivity
extension which identifies expressions of the form a ^ (0 : ℕ)
.
This extension is run in addition to the general a ^ b
extension (they are overlapping).
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The positivity
extension which identifies expressions of the form a ^ (b : ℕ)
,
such that positivity
successfully recognises both a
and b
.
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Extension for the positivity
tactic: Int.natAbs
is positive when its input is.
Since the output type of Int.natAbs
is ℕ
, the nonnegative case is handled by the default
positivity
tactic.
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Extension for the positivity
tactic: Nat.cast
is always non-negative,
and positive when its input is.
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Extension for the positivity
tactic: Int.cast
is positive (resp. non-negative)
if its input is.
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The positivity
extension which identifies expressions of the form NNRat.num q
,
such that positivity
successfully recognises q
.
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The positivity
extension which identifies expressions of the form Rat.num a
,
such that positivity
successfully recognises a
.
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Extension for posPart
. a⁺
is always nonegative, and positive if a
is.
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Extension for negPart
. a⁻
is always nonegative.