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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 69 ms]
    (8) CdtProblem
    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) BOUNDS(1, 1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


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

(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   FIRST(z0, z1) -> c2
   FROM(z0) -> c3
   FROM(z0) -> c4
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))
   ACTIVATE(z0) -> c7
S tuples:
   FIRST(0, z0) -> c
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   FIRST(z0, z1) -> c2
   FROM(z0) -> c3
   FROM(z0) -> c4
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   first_2, from_1, activate_1

Defined Pair Symbols:   FIRST_2, FROM_1, ACTIVATE_1

Compound Symbols:   c, c1_1, c2, c3, c4, c5_1, c6_1, c7


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   FIRST(0, z0) -> c
   FIRST(z0, z1) -> c2
   ACTIVATE(z0) -> c7
   FROM(z0) -> c4
   FROM(z0) -> c3
   ACTIVATE(n__from(z0)) -> c6(FROM(z0))

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
S tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:   first_2, from_1, activate_1

Defined Pair Symbols:   FIRST_2, ACTIVATE_1

Compound Symbols:   c1_1, c5_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   first(0, z0) -> nil
   first(s(z0), cons(z1, z2)) -> cons(z1, n__first(z0, activate(z2)))
   first(z0, z1) -> n__first(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__first(z0, z1)) -> first(z0, z1)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
S tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FIRST_2, ACTIVATE_1

Compound Symbols:   c1_1, c5_1


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ACTIVATE(x_1)) = [1] + x_1
   POL(FIRST(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(n__first(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
S tuples:none
K tuples:
   FIRST(s(z0), cons(z1, z2)) -> c1(ACTIVATE(z2))
   ACTIVATE(n__first(z0, z1)) -> c5(FIRST(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   FIRST_2, ACTIVATE_1

Compound Symbols:   c1_1, c5_1


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

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

(10)
BOUNDS(1, 1)

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

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

(12)
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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

The rewrite sequence

activate(n__first(s(X1_0), cons(Y2_0, Z3_0))) ->^+ cons(Y2_0, n__first(X1_0, activate(Z3_0)))

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

The pumping substitution is [Z3_0 / n__first(s(X1_0), cons(Y2_0, Z3_0))].

The result substitution is [ ].




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

(14)
Complex Obligation (BEST)

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

(15)
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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

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

(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:

   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   from(X) -> cons(X, n__from(s(X)))
   first(X1, X2) -> n__first(X1, X2)
   from(X) -> n__from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST