WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 1588 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:

   FUNCTION(iszero, 0, z0, z1) -> c
   FUNCTION(iszero, s(z0), z1, z2) -> c1
   FUNCTION(p, 0, z0, z1) -> c2
   FUNCTION(p, s(0), z0, z1) -> c3
   FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0))
   FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1))
   FUNCTION(if, true, z0, z1) -> c6
   FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1))
   FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1))
   FUNCTION(third, z0, z1, z2) -> c9

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

   function(iszero, 0, z0, z1) -> true
   function(iszero, s(z0), z1, z2) -> false
   function(p, 0, z0, z1) -> 0
   function(p, s(0), z0, z1) -> 0
   function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0))
   function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2)
   function(if, true, z0, z1) -> z1
   function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))
   function(third, z0, z1, z2) -> z2

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:

   FUNCTION(iszero, 0, z0, z1) -> c
   FUNCTION(iszero, s(z0), z1, z2) -> c1
   FUNCTION(p, 0, z0, z1) -> c2
   FUNCTION(p, s(0), z0, z1) -> c3
   FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0))
   FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1))
   FUNCTION(if, true, z0, z1) -> c6
   FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1))
   FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1))
   FUNCTION(third, z0, z1, z2) -> c9

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

   function(iszero, 0, z0, z1) -> true
   function(iszero, s(z0), z1, z2) -> false
   function(p, 0, z0, z1) -> 0
   function(p, s(0), z0, z1) -> 0
   function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0))
   function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2)
   function(if, true, z0, z1) -> z1
   function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))
   function(third, z0, z1, z2) -> z2

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:

   FUNCTION(iszero, 0, z0, z1) -> c
   FUNCTION(iszero, s(z0), z1, z2) -> c1
   FUNCTION(p, 0, z0, z1) -> c2
   FUNCTION(p, s(0), z0, z1) -> c3
   FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0))
   FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1))
   FUNCTION(if, true, z0, z1) -> c6
   FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1))
   FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1))
   FUNCTION(third, z0, z1, z2) -> c9

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

   function(iszero, 0, z0, z1) -> true
   function(iszero, s(z0), z1, z2) -> false
   function(p, 0, z0, z1) -> 0
   function(p, s(0), z0, z1) -> 0
   function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0))
   function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2)
   function(if, true, z0, z1) -> z1
   function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))
   function(third, z0, z1, z2) -> z2

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

FUNCTION(p, s(s(z0)), z1, z2) ->^+ c4(FUNCTION(p, s(z0), z0, z0))

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

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

The result substitution is [z1 / z0, z2 / 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:

   FUNCTION(iszero, 0, z0, z1) -> c
   FUNCTION(iszero, s(z0), z1, z2) -> c1
   FUNCTION(p, 0, z0, z1) -> c2
   FUNCTION(p, s(0), z0, z1) -> c3
   FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0))
   FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1))
   FUNCTION(if, true, z0, z1) -> c6
   FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1))
   FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1))
   FUNCTION(third, z0, z1, z2) -> c9

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

   function(iszero, 0, z0, z1) -> true
   function(iszero, s(z0), z1, z2) -> false
   function(p, 0, z0, z1) -> 0
   function(p, s(0), z0, z1) -> 0
   function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0))
   function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2)
   function(if, true, z0, z1) -> z1
   function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))
   function(third, z0, z1, z2) -> z2

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:

   FUNCTION(iszero, 0, z0, z1) -> c
   FUNCTION(iszero, s(z0), z1, z2) -> c1
   FUNCTION(p, 0, z0, z1) -> c2
   FUNCTION(p, s(0), z0, z1) -> c3
   FUNCTION(p, s(s(z0)), z1, z2) -> c4(FUNCTION(p, s(z0), z0, z0))
   FUNCTION(plus, z0, z1, z2) -> c5(FUNCTION(if, function(iszero, z1, z1, z1), z1, z2), FUNCTION(iszero, z1, z1, z1))
   FUNCTION(if, true, z0, z1) -> c6
   FUNCTION(if, false, z0, z1) -> c7(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(third, z0, z1, z1))
   FUNCTION(if, false, z0, z1) -> c8(FUNCTION(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1)), FUNCTION(p, z0, z0, z1))
   FUNCTION(third, z0, z1, z2) -> c9

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

   function(iszero, 0, z0, z1) -> true
   function(iszero, s(z0), z1, z2) -> false
   function(p, 0, z0, z1) -> 0
   function(p, s(0), z0, z1) -> 0
   function(p, s(s(z0)), z1, z2) -> s(function(p, s(z0), z0, z0))
   function(plus, z0, z1, z2) -> function(if, function(iszero, z1, z1, z1), z1, z2)
   function(if, true, z0, z1) -> z1
   function(if, false, z0, z1) -> function(plus, function(third, z0, z1, z1), function(p, z0, z0, z1), s(z1))
   function(third, z0, z1, z2) -> z2

Rewrite Strategy: INNERMOST