A positive affine transferComplete concrete inputs + interval transfer replay
The problem
Compute sound interval bounds for a sequence of mathematical-integer affine transforms.
The checked result
Sound enclosure: yes; Tight interval bounds: yes.
Endpoint propagation encloses every concrete output. Negative multiplication swaps extrema; multiple transformations compose while preserving containment.
Why the checker accepts it
- Start from every integer in the initial interval.
- Apply each affine transform to the concrete set.
- Apply endpoint arithmetic to the interval, reordering endpoints for negative scales.
- Check every concrete value remains enclosed and the final bounds are tight.
Formal specification
{
"initial_interval": [
-2,
3
],
"transforms": [
[
2,
1
]
]
}Claim and evidence
{
"claim": {
"sound": true,
"tight_interval": true
},
"witness": {
"intervals": [
[
-2,
3
],
[
-3,
7
]
]
}
}Dataset construction
Deterministic finite fixture; full enumeration or witness replay as stated. Structured JSON; field meanings are stated in the specification. Acceptance covers 6 checker units for this record; the unit type is stated in its verification scope.
Complexity and limits
With a concrete interval of k integers and t transformations, this finite replay costs O(kt). Abstract endpoint propagation alone costs O(t).
The abstraction encloses gaps. Machine overflow, branches, and widening are not modeled.
A boundary to investigate
A tight interval can include unattainable interior values; it is not necessarily an exact concrete set. Add control-flow joins, fixed points, widening, and overflow-specific abstractions.