/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 59 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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

(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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

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(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))


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

(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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

The (relative) TRS S consists of the following rules:

   encArg(0(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

The (relative) TRS S consists of the following rules:

   encArg(0(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))

Rewrite Strategy: INNERMOST
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(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

The (relative) TRS S consists of the following rules:

   encArg(0(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))

Rewrite Strategy: INNERMOST
----------------------------------------

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

p(0(x1)) ->^+ 0(s(s(p(x1))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].

The pumping substitution is [x1 / 0(x1)].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following 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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

The (relative) TRS S consists of the following rules:

   encArg(0(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))

Rewrite Strategy: INNERMOST
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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> p(s(g(p(s(s(x1))))))
   g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1)))))))))))
   j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1)))))))))))
   half(0(x1)) -> 0(s(s(half(p(s(p(s(x1))))))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(x1)))))))))

The (relative) TRS S consists of the following rules:

   encArg(0(x_1)) -> 0(encArg(x_1))
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_j(x_1)) -> j(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_rd(x_1)) -> rd(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0(x_1) -> 0(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_j(x_1) -> j(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_rd(x_1) -> rd(encArg(x_1))

Rewrite Strategy: INNERMOST