/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 2 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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


The TRS R consists of the following rules:

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

S is empty.
Rewrite Strategy: FULL
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(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

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

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c1
   A__B -> c2
   A__B -> c3
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c5(A__B)
   MARK(a) -> c6
S tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B)
   A__F(z0, z1, z2) -> c1
   A__B -> c2
   A__B -> c3
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(b) -> c5(A__B)
   MARK(a) -> c6
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1

Defined Pair Symbols:   A__F_3, A__B, MARK_1

Compound Symbols:   c_2, c1, c2, c3, c4_2, c5_1, c6


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(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   MARK(a) -> c6
   A__B -> c2
   A__F(z0, z1, z2) -> c1
   A__B -> c3
   MARK(b) -> c5(A__B)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B)
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
S tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b), A__B)
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c_2, c4_2


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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
S tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
K tuples:none
Defined Rule Symbols:   a__f_3, a__b, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c4_2, c_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
We considered the (Usable) Rules:none
And the Tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1, x_2, x_3)) = x_1 + x_3
   POL(MARK(x_1)) = x_1
   POL(a) = [1]
   POL(a__b) = [1]
   POL(a__f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(mark(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
S tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
Defined Rule Symbols:   a__f_3, a__b, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c4_2, c_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.
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
We considered the (Usable) Rules:none
And the Tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A__F(x_1, x_2, x_3)) = x_1 + x_3
   POL(MARK(x_1)) = x_1
   POL(a) = [1]
   POL(a__b) = [1]
   POL(a__f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(mark(x_1)) = [1] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, z0, z0) -> a__f(z0, a__b, b)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__b -> a
   a__b -> b
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(b) -> a__b
   mark(a) -> a
Tuples:
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
S tuples:none
K tuples:
   A__F(a, z0, z0) -> c(A__F(z0, a__b, b))
   MARK(f(z0, z1, z2)) -> c4(A__F(z0, mark(z1), z2), MARK(z1))
Defined Rule Symbols:   a__f_3, a__b, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c4_2, c_1


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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

(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

S is empty.
Rewrite Strategy: FULL
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(17) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

mark(f(X1, X2, X3)) ->^+ a__f(X1, mark(X2), X3)

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

The pumping substitution is [X2 / f(X1, X2, X3)].

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

S is empty.
Rewrite Strategy: FULL
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   a__f(a, X, X) -> a__f(X, a__b, b)
   a__b -> a
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(b) -> a__b
   mark(a) -> a
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__b -> b

S is empty.
Rewrite Strategy: FULL