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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (full) 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), 223 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


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


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

(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(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))


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

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


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

   encArg(0) -> 0
   encArg(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

   encArg(0) -> 0
   encArg(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

   encArg(0) -> 0
   encArg(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))

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

The rewrite sequence

g(h, s(s(x))) ->^+ s(g(h, x))

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

The pumping substitution is [x / s(s(x))].

The result substitution is [ ].




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(8)
Complex Obligation (BEST)

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

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


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

   encArg(0) -> 0
   encArg(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))

Rewrite Strategy: FULL
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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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   g(x, 0) -> 0
   g(d, s(x)) -> s(s(g(d, x)))
   g(h, s(0)) -> 0
   g(h, s(s(x))) -> s(g(h, x))
   double(x) -> g(d, x)
   half(x) -> g(h, x)
   f(s(x), y) -> f(half(s(x)), double(y))
   f(s(0), y) -> y
   id(x) -> f(x, s(0))

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

   encArg(0) -> 0
   encArg(d) -> d
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(h) -> h
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_double(x_1)) -> double(encArg(x_1))
   encArg(cons_half(x_1)) -> half(encArg(x_1))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_d -> d
   encode_s(x_1) -> s(encArg(x_1))
   encode_h -> h
   encode_double(x_1) -> double(encArg(x_1))
   encode_half(x_1) -> half(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_id(x_1) -> id(encArg(x_1))

Rewrite Strategy: FULL