/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^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 0 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^2).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
S tuples:
   F(0) -> c1
   F(1) -> c2
   F(s(z0)) -> c3(F(z0))
   IF(true, s(z0), s(z1)) -> c4
   IF(false, s(z0), s(z1)) -> c5
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, IF_3, G_2

Compound Symbols:   c1, c2, c3_1, c4, c5, c6_1, c7_4


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   IF(false, s(z0), s(z1)) -> c5
   IF(true, s(z0), s(z1)) -> c4
   F(0) -> c1
   F(1) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(G(z0, if(f(z0), c(g(s(z0), z1)), c(z1))), IF(f(z0), c(g(s(z0), z1)), c(z1)), F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_4


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:   f_1, if_3, g_2

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> true
   f(1) -> false
   f(s(z0)) -> f(z0)
   if(true, s(z0), s(z1)) -> s(z0)
   if(false, s(z0), s(z1)) -> s(z1)
   g(z0, c(z1)) -> c(g(z0, z1))
   g(z0, c(z1)) -> g(z0, if(f(z0), c(g(s(z0), z1)), c(z1)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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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.
   G(z0, c(z1)) -> c6(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = 0
   POL(G(x_1, x_2)) = x_1 + x_2
   POL(c(x_1)) = [1] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
K tuples:
   G(z0, c(z1)) -> c6(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = 0
   POL(G(x_1, x_2)) = x_2
   POL(c(x_1)) = [1] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [3] + x_1

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:
   F(s(z0)) -> c3(F(z0))
K tuples:
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0)) -> c3(F(z0))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(F(x_1)) = [2] + x_1
   POL(G(x_1, x_2)) = [2]x_2^2 + x_1*x_2
   POL(c(x_1)) = [2] + x_1
   POL(c3(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [2] + x_1

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(s(z0)) -> c3(F(z0))
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
S tuples:none
K tuples:
   G(z0, c(z1)) -> c6(G(z0, z1))
   G(z0, c(z1)) -> c7(F(z0), G(s(z0), z1))
   F(s(z0)) -> c3(F(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   F_1, G_2

Compound Symbols:   c3_1, c6_1, c7_2


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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.
----------------------------------------

(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:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

S is empty.
Rewrite Strategy: INNERMOST
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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

g(x, c(y)) ->^+ g(x, if(f(x), c(g(s(x), y)), c(y)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].

The pumping substitution is [y / c(y)].

The result substitution is [x / s(x)].




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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^2).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

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^2).


The TRS R consists of the following rules:

   f(0) -> true
   f(1) -> false
   f(s(x)) -> f(x)
   if(true, s(x), s(y)) -> s(x)
   if(false, s(x), s(y)) -> s(y)
   g(x, c(y)) -> c(g(x, y))
   g(x, c(y)) -> g(x, if(f(x), c(g(s(x), y)), c(y)))

S is empty.
Rewrite Strategy: INNERMOST