/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 52 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 5 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c2
   FROM(z0) -> c3
   S(z0) -> c4
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
   ACTIVATE(z0) -> c7
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   FROM(z0) -> c2
   FROM(z0) -> c3
   S(z0) -> c4
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, FROM_1, S_1, ACTIVATE_1

Compound Symbols:   c, c1_2, c2, c3, c4, c5_2, c6_2, c7


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   FROM(z0) -> c3
   FROM(z0) -> c2
   S(z0) -> c4
   2ND(cons1(z0, cons(z1, z2))) -> c
   ACTIVATE(z0) -> c7

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
S tuples:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(FROM(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, ACTIVATE_1

Compound Symbols:   c1_2, c5_2, c6_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   2ND_1, ACTIVATE_1

Compound Symbols:   c1_1, c5_1, c6_1


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(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   2ND(cons(z0, z1)) -> c1(ACTIVATE(z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1, s_1, activate_1

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, activate(z1)))
   from(z0) -> cons(z0, n__from(n__s(z0)))
   from(z0) -> n__from(z0)
   s(z0) -> n__s(z0)
   activate(n__from(z0)) -> from(activate(z0))
   activate(n__s(z0)) -> s(activate(z0))
   activate(z0) -> z0

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(n__from(x_1)) = x_1
   POL(n__s(x_1)) = [1] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
K tuples:
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(n__s(x_1)) = x_1

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
S tuples:none
K tuples:
   ACTIVATE(n__s(z0)) -> c6(ACTIVATE(z0))
   ACTIVATE(n__from(z0)) -> c5(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   ACTIVATE_1

Compound Symbols:   c5_1, c6_1


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(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(16)
BOUNDS(1, 1)

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(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

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

The rewrite sequence

activate(n__s(X)) ->^+ s(activate(X))

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

The pumping substitution is [X / n__s(X)].

The result substitution is [ ].




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(20)
Complex Obligation (BEST)

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(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST
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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(23)
BOUNDS(n^1, INF)

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(24)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
   from(X) -> cons(X, n__from(n__s(X)))
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST