Runtime Complexity: TRS Innermost pair #487112716

details

property	value
status	complete
benchmark	mul.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n149.star.cs.uiowa.edu
space	Frederiksen_Glenstrup

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	5.03995 seconds
cpu usage	15.6722
user time	14.4205
system time	1.25165
max virtual memory	1.9009632E7
max residence set size	3799736.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 5 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 189 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), 217 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), 49 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(Cons(x, xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   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(Cons(z0, z1), z2) -> add0(mul0(z1, z2), z2)
   mul0(Nil, z0) -> Nil
   add0(Cons(z0, z1), z2) -> add0(z1, Cons(S, z2))
   add0(Nil, z0) -> z0
   goal(z0, z1) -> mul0(z0, z1)
Tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
S tuples:
   MUL0(Cons(z0, z1), z2) -> c(ADD0(mul0(z1, z2), z2), MUL0(z1, z2))
   MUL0(Nil, z0) -> c1
   ADD0(Cons(z0, z1), z2) -> c2(ADD0(z1, Cons(S, z2)))
   ADD0(Nil, z0) -> c3
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
K tuples:none
Defined Rule Symbols:   mul0_2, add0_2, goal_2

Defined Pair Symbols:   MUL0_2, ADD0_2, GOAL_2

Compound Symbols:   c_2, c1, c2_1, c3, c4_1


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

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c4(MUL0(z0, z1))
Removed 2 trailing nodes:
   ADD0(Nil, z0) -> c3

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