Runtime Complexity: TRS Innermost pair #487112702

details

property	value
status	complete
benchmark	mul_better.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n140.star.cs.uiowa.edu
space	Frederiksen_Glenstrup

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	5.3186 seconds
cpu usage	17.8141
user time	16.4409
system time	1.37323
max virtual memory	1.8810372E7
max residence set size	3774992.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^3), O(n^3))

output

WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 2 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 198 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 265 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^3, INF)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   mul0(C(x, y), y') -> add0(mul0(y, y'), y')
   mul0(Z, y) -> Z
   add0(C(x, y), y') -> add0(y, C(S, y'))
   add0(Z, y) -> y
   second(C(x, y)) -> y
   isZero(C(x, y)) -> False
   isZero(Z) -> True
   goal(xs, ys) -> mul0(xs, ys)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   mul0(C(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Z, z0) -> Z
   add0(C(z0, z1), z2) -> add0(z1, C(S, z2))
   add0(Z, z0) -> z0
   second(C(z0, z1)) -> z1
   isZero(C(z0, z1)) -> False
   isZero(Z) -> True
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Z, z0) -> c1
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
   ADD0(Z, z0) -> c3
   SECOND(C(z0, z1)) -> c4
   ISZERO(C(z0, z1)) -> c5
   ISZERO(Z) -> c6
   GOAL(z0, z1) -> c7(MUL0(z0, z1))
S tuples:
   MUL0(C(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Z, z0) -> c1
   ADD0(C(z0, z1), z2) -> c2(ADD0(z1, C(S, z2)))
   ADD0(Z, z0) -> c3
   SECOND(C(z0, z1)) -> c4
   ISZERO(C(z0, z1)) -> c5
   ISZERO(Z) -> c6
   GOAL(z0, z1) -> c7(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, second_1, isZero_1, goal_2

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