Derivational Complexity: TRS Innermost pair #487108968

details

property	value
status	complete
benchmark	PEANO_nokinds_C.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	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	292.782 seconds
cpu usage	1141.35
user time	1127.99
system time	13.3613
max virtual memory	3.7912332E7
max residence set size	1.4953928E7

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 502 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 451 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), 163 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 154 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 163 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 203 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 2299 ms]
        (26) 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:

   active(U11(tt, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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(tt) -> tt
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(0) -> 0
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_U11(x_1, x_2)) -> U11(encArg(x_1), encArg(x_2))
   encArg(cons_U21(x_1, x_2, x_3)) -> U21(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))

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actions all output return to Derivational Complexity: TRS Innermost