Derivational Complexity: TRS Innermost pair #487108074

details

property	value
status	complete
benchmark	tpa04.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n138.star.cs.uiowa.edu
space	Secret_06_TRS

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	292.298 seconds
cpu usage	1131.33
user time	1117.92
system time	13.4096
max virtual memory	3.7660764E7
max residence set size	1.5166012E7

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), ?)

output

WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 210 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 4 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 266 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 65 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 77 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 54 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 127 ms]
        (24) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2))
   encArg(cons_max(x_1, x_2)) -> max(encArg(x_1), encArg(x_2))
   encArg(cons_twice(x_1)) -> twice(encArg(x_1))
   encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_max(x_1, x_2) -> max(encArg(x_1), encArg(x_2))
   encode_twice(x_1) -> twice(encArg(x_1))
   encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

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