/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 [ComplexityIfPolyImplication, 0 ms]
    (6) CdtProblem
    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 1 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 48 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 6 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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 does not nest defined symbols, 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:

   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   PERFECTP(0) -> c
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c2
   F(0, z0, s(z1), z2) -> c3
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3), F(z0, z3, z2, z3))
S tuples:
   PERFECTP(0) -> c
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
   F(0, z0, 0, z1) -> c2
   F(0, z0, s(z1), z2) -> c3
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   perfectp_1, f_4

Defined Pair Symbols:   PERFECTP_1, F_4

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5_2


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(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PERFECTP(s(z0)) -> c1(F(z0, s(0), s(z0), s(z0)))
Removed 3 trailing nodes:
   F(0, z0, s(z1), z2) -> c3
   PERFECTP(0) -> c
   F(0, z0, 0, z1) -> c2

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3), F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(s(z0), minus(z1, z0), z2, z3), F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   perfectp_1, f_4

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c5_2


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(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:   perfectp_1, f_4

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c5_1


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   perfectp(0) -> false
   perfectp(s(z0)) -> f(z0, s(0), s(z0), s(z0))
   f(0, z0, 0, z1) -> true
   f(0, z0, s(z1), z2) -> false
   f(s(z0), 0, z1, z2) -> f(z0, z2, minus(z1, s(z0)), z2)
   f(s(z0), s(z1), z2, z3) -> if(le(z0, z1), f(s(z0), minus(z1, z0), z2, z3), f(z0, z3, z2, z3))

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c5_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.
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2, x_3, x_4)) = x_1 + x_3
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(minus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
S tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
K tuples:
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c5_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.
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2, x_3, x_4)) = [3]x_1 + x_3
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(minus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [3] + x_1

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(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
S tuples:none
K tuples:
   F(s(z0), s(z1), z2, z3) -> c5(F(z0, z3, z2, z3))
   F(s(z0), 0, z1, z2) -> c4(F(z0, z2, minus(z1, s(z0)), z2))
Defined Rule Symbols:none

Defined Pair Symbols:   F_4

Compound Symbols:   c4_1, c5_1


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(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(16)
BOUNDS(1, 1)

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(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(18)
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:

   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

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

The rewrite sequence

f(s(s(x1_0)), s(y), z, s(y2_0)) ->^+ if(le(s(x1_0), y), f(s(s(x1_0)), minus(y, s(x1_0)), z, s(y2_0)), if(le(x1_0, y2_0), f(s(x1_0), minus(y2_0, x1_0), z, s(y2_0)), f(x1_0, s(y2_0), z, s(y2_0))))

gives rise to a decreasing loop by considering the right hand sides subterm at position [2,2].

The pumping substitution is [x1_0 / s(s(x1_0))].

The result substitution is [y / y2_0].




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


The TRS R consists of the following rules:

   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

S is empty.
Rewrite Strategy: FULL
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, minus(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

S is empty.
Rewrite Strategy: FULL