WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1043 ms]
(2) CpxRelTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(6) BEST
    (7) proven lower bound
        (8) LowerBoundPropagationProof [FINISHED, 0 ms]
        (9) BOUNDS(n^1, INF)
    (10) TRS for Loop Detection


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(0)
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:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

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

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

Rewrite Strategy: INNERMOST
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(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(4)
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:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

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

The rewrite sequence

PLUS(z0, s(s(z1))) ->^+ c14(PLUS(s(z0), z1))

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

The pumping substitution is [z1 / s(s(z1))].

The result substitution is [z0 / s(z0)].




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(6)
Complex Obligation (BEST)

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(7)
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:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

Rewrite Strategy: INNERMOST
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(8) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(9)
BOUNDS(n^1, INF)

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(10)
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:

   DOUBLE(z0) -> c1(PERMUTE(z0, z0, a))
   PERMUTE(z0, z1, a) -> c2(PERMUTE(isZero(z0), z0, b), ISZERO(z0))
   PERMUTE(false, z0, b) -> c3(PERMUTE(ack(z0, z0), p(z0), c), ACK(z0, z0))
   PERMUTE(false, z0, b) -> c4(PERMUTE(ack(z0, z0), p(z0), c), P(z0))
   PERMUTE(true, z0, b) -> c5
   PERMUTE(z0, z1, c) -> c6(PERMUTE(z1, z0, a))
   P(0) -> c7
   P(s(z0)) -> c8
   ACK(0, z0) -> c9(PLUS(z0, s(0)))
   ACK(s(z0), 0) -> c10(ACK(z0, s(0)))
   ACK(s(z0), s(z1)) -> c11(ACK(z0, ack(s(z0), z1)), ACK(s(z0), z1))
   PLUS(0, z0) -> c12
   PLUS(s(z0), z1) -> c13(PLUS(z0, s(z1)))
   PLUS(z0, s(s(z1))) -> c14(PLUS(s(z0), z1))
   PLUS(z0, s(0)) -> c15
   PLUS(z0, 0) -> c16
   ISZERO(0) -> c17
   ISZERO(s(z0)) -> c18

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

   double(z0) -> permute(z0, z0, a)
   permute(z0, z1, a) -> permute(isZero(z0), z0, b)
   permute(false, z0, b) -> permute(ack(z0, z0), p(z0), c)
   permute(true, z0, b) -> 0
   permute(z0, z1, c) -> s(s(permute(z1, z0, a)))
   p(0) -> 0
   p(s(z0)) -> z0
   ack(0, z0) -> plus(z0, s(0))
   ack(s(z0), 0) -> ack(z0, s(0))
   ack(s(z0), s(z1)) -> ack(z0, ack(s(z0), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   plus(z0, s(s(z1))) -> s(plus(s(z0), z1))
   plus(z0, s(0)) -> s(z0)
   plus(z0, 0) -> z0
   isZero(0) -> true
   isZero(s(z0)) -> false

Rewrite Strategy: INNERMOST