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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 329 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 1 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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(encArg(x_1))


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(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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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

   encArg(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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

   encArg(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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

   encArg(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(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

foldC(t, s(n)) ->^+ f(foldC(t, n), C)

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

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

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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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

   encArg(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(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(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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

   encArg(A) -> A
   encArg(B) -> B
   encArg(C) -> C
   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(triple(x_1, x_2, x_3)) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_foldB(x_1, x_2)) -> foldB(encArg(x_1), encArg(x_2))
   encArg(cons_foldC(x_1, x_2)) -> foldC(encArg(x_1), encArg(x_2))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_f'(x_1, x_2)) -> f'(encArg(x_1), encArg(x_2))
   encArg(cons_f''(x_1)) -> f''(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_A -> A
   encode_B -> B
   encode_C -> C
   encode_foldB(x_1, x_2) -> foldB(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_foldC(x_1, x_2) -> foldC(encArg(x_1), encArg(x_2))
   encode_f'(x_1, x_2) -> f'(encArg(x_1), encArg(x_2))
   encode_triple(x_1, x_2, x_3) -> triple(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f''(x_1) -> f''(encArg(x_1))

Rewrite Strategy: FULL