/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(NON_POLY, ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(EXP, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 202 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 3 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection
        (13) DecreasingLoopProof [FINISHED, 0 ms]
        (14) BOUNDS(EXP, INF)


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

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


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(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(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(encArg(x_1))

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

(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(EXP, INF).


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(encArg(x_1))

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

(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(EXP, INF).


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(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

+(x, s(y)) ->^+ s(+(x, y))

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

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

The result substitution is [ ].




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

(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(EXP, INF).


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(encArg(x_1))

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

(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(EXP, INF).


The TRS R consists of the following rules:

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_q(x_1) -> q(encArg(x_1))

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

(13) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

g(s(x)) ->^+ pair(+(p(g(x)), q(g(x))), p(g(x)))

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

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

The result substitution is [ ].



The rewrite sequence

g(s(x)) ->^+ pair(+(p(g(x)), q(g(x))), p(g(x)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0].

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

The result substitution is [ ].




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

(14)
BOUNDS(EXP, INF)