/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^2))
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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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^2).

(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)), 60 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 38 ms]
    (14) CdtProblem
    (15) CdtKnowledgeProof [FINISHED, 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


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   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:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(activate(z1))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   HEAD(cons(z0, z1)) -> c2
   2ND(cons(z0, z1)) -> c3(HEAD(activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   TAKE(z0, z1) -> c6
   SEL(0, cons(z0, z1)) -> c7
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c9(FROM(z0))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
   ACTIVATE(z0) -> c11
S tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   HEAD(cons(z0, z1)) -> c2
   2ND(cons(z0, z1)) -> c3(HEAD(activate(z1)), ACTIVATE(z1))
   TAKE(0, z0) -> c4
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   TAKE(z0, z1) -> c6
   SEL(0, cons(z0, z1)) -> c7
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c9(FROM(z0))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
   ACTIVATE(z0) -> c11
K tuples:none
Defined Rule Symbols:   from_1, head_1, 2nd_1, take_2, sel_2, activate_1

Defined Pair Symbols:   FROM_1, HEAD_1, 2ND_1, TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4, c5_1, c6, c7, c8_2, c9_1, c10_1, c11


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   SEL(0, cons(z0, z1)) -> c7
   TAKE(0, z0) -> c4
   FROM(z0) -> c1
   ACTIVATE(n__from(z0)) -> c9(FROM(z0))
   HEAD(cons(z0, z1)) -> c2
   ACTIVATE(z0) -> c11
   FROM(z0) -> c
   TAKE(z0, z1) -> c6

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(activate(z1))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   2ND(cons(z0, z1)) -> c3(HEAD(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
S tuples:
   2ND(cons(z0, z1)) -> c3(HEAD(activate(z1)), ACTIVATE(z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
K tuples:none
Defined Rule Symbols:   from_1, head_1, 2nd_1, take_2, sel_2, activate_1

Defined Pair Symbols:   2ND_1, TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c3_2, c5_1, c8_2, c10_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(activate(z1))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
   2ND(cons(z0, z1)) -> c3(ACTIVATE(z1))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
   2ND(cons(z0, z1)) -> c3(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:   from_1, head_1, 2nd_1, take_2, sel_2, activate_1

Defined Pair Symbols:   TAKE_2, SEL_2, ACTIVATE_1, 2ND_1

Compound Symbols:   c5_1, c8_2, c10_1, c3_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   2ND(cons(z0, z1)) -> c3(ACTIVATE(z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(activate(z1))
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
K tuples:none
Defined Rule Symbols:   from_1, head_1, 2nd_1, take_2, sel_2, activate_1

Defined Pair Symbols:   TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c5_1, c8_2, c10_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   head(cons(z0, z1)) -> z0
   2nd(cons(z0, z1)) -> head(activate(z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
K tuples:none
Defined Rule Symbols:   activate_1, from_1, take_2

Defined Pair Symbols:   TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c5_1, c8_2, c10_1


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
We considered the (Usable) Rules:none
And the Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ACTIVATE(x_1)) = 0
   POL(SEL(x_1, x_2)) = x_1
   POL(TAKE(x_1, x_2)) = 0
   POL(activate(x_1)) = [1]
   POL(c10(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(from(x_1)) = [1] + x_1
   POL(n__from(x_1)) = [1] + x_1
   POL(n__take(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(take(x_1, x_2)) = [1] + x_1 + x_2

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
S tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
K tuples:
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
Defined Rule Symbols:   activate_1, from_1, take_2

Defined Pair Symbols:   TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c5_1, c8_2, c10_1


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
We considered the (Usable) Rules:
   take(z0, z1) -> n__take(z0, z1)
   from(z0) -> cons(z0, n__from(s(z0)))
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   activate(n__take(z0, z1)) -> take(z0, z1)
   take(0, z0) -> nil
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
And the Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(ACTIVATE(x_1)) = x_1
   POL(SEL(x_1, x_2)) = x_1*x_2 + x_1^2
   POL(TAKE(x_1, x_2)) = x_1 + x_2
   POL(activate(x_1)) = [1] + x_1
   POL(c10(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(from(x_1)) = 0
   POL(n__from(x_1)) = 0
   POL(n__take(x_1, x_2)) = x_1 + x_2
   POL(nil) = [2]
   POL(s(x_1)) = [1] + x_1
   POL(take(x_1, x_2)) = [1] + x_1 + x_2

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(n__take(z0, z1)) -> take(z0, z1)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   take(0, z0) -> nil
   take(s(z0), cons(z1, z2)) -> cons(z1, n__take(z0, activate(z2)))
   take(z0, z1) -> n__take(z0, z1)
Tuples:
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
S tuples:
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
K tuples:
   SEL(s(z0), cons(z1, z2)) -> c8(SEL(z0, activate(z2)), ACTIVATE(z2))
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
Defined Rule Symbols:   activate_1, from_1, take_2

Defined Pair Symbols:   TAKE_2, SEL_2, ACTIVATE_1

Compound Symbols:   c5_1, c8_2, c10_1


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

(15) CdtKnowledgeProof (FINISHED)
The following tuples could be moved from S to K by knowledge propagation:
   ACTIVATE(n__take(z0, z1)) -> c10(TAKE(z0, z1))
   TAKE(s(z0), cons(z1, z2)) -> c5(ACTIVATE(z2))
Now S is empty
----------------------------------------

(16)
BOUNDS(1, 1)

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

(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^2).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   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

sel(s(N), cons(X, XS)) ->^+ sel(N, XS)

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

The pumping substitution is [N / s(N), XS / cons(X, XS)].

The result substitution is [ ].




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

(20)
Complex Obligation (BEST)

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

(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^2).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> X

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

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

(23)
BOUNDS(n^1, INF)

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

(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^2).


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   head(cons(X, XS)) -> X
   2nd(cons(X, XS)) -> head(activate(XS))
   take(0, XS) -> nil
   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   from(X) -> n__from(X)
   take(X1, X2) -> n__take(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__take(X1, X2)) -> take(X1, X2)
   activate(X) -> X

S is empty.
Rewrite Strategy: INNERMOST