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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 252 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [FINISHED, 0 ms]
(8) BOUNDS(EXP, INF)


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(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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(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(true) -> true
   encArg(false) -> false
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_fibo(x_1)) -> fibo(encArg(x_1))
   encArg(cons_fib(x_1)) -> fib(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_fibo(x_1) -> fibo(encArg(x_1))
   encode_fib(x_1) -> fib(encArg(x_1))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))


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(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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_fibo(x_1)) -> fibo(encArg(x_1))
   encArg(cons_fib(x_1)) -> fib(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_fibo(x_1) -> fibo(encArg(x_1))
   encode_fib(x_1) -> fib(encArg(x_1))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_fibo(x_1)) -> fibo(encArg(x_1))
   encArg(cons_fib(x_1)) -> fib(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_fibo(x_1) -> fibo(encArg(x_1))
   encode_fib(x_1) -> fib(encArg(x_1))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

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(EXP, INF).


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2))
   encArg(cons_fibo(x_1)) -> fibo(encArg(x_1))
   encArg(cons_fib(x_1)) -> fib(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3, x_4, x_5)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))
   encArg(cons_sum(x_1, x_2)) -> sum(encArg(x_1), encArg(x_2))
   encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_true -> true
   encode_false -> false
   encode_fibo(x_1) -> fibo(encArg(x_1))
   encode_fib(x_1) -> fib(encArg(x_1))
   encode_sum(x_1, x_2) -> sum(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3, x_4, x_5) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5))

Rewrite Strategy: INNERMOST
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(7) DecreasingLoopProof (FINISHED)
The following loop(s) give(s) rise to the lower bound EXP:

The rewrite sequence

fibo(s(s(x))) ->^+ sum(fibo(s(x)), fibo(x))

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

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

The result substitution is [ ].



The rewrite sequence

fibo(s(s(x))) ->^+ sum(fibo(s(x)), fibo(x))

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

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

The result substitution is [ ].




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(8)
BOUNDS(EXP, INF)