MAYBE
proof of input_6VDEoh4Rc4.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y)))) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y)))) [1]

The TRS has the following type information:
g :: f -> f
f :: f -> f -> f

Rewrite Strategy: INNERMOST
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(9) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   g(v0) -> null_g [0]

And the following fresh constants:    null_g

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(10)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y)))) [1]
   g(v0) -> null_g [0]

The TRS has the following type information:
g :: f:null_g -> f:null_g
f :: f:null_g -> f:null_g -> f:null_g
null_g :: f:null_g

Rewrite Strategy: INNERMOST
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(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   null_g => 0

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + (1 + g(g(x)) + g(g(y))) + (1 + g(g(x)) + g(g(y))) :|: z = 1 + x + y, x >= 0, y >= 0

Only complete derivations are relevant for the runtime complexity.

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(13) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   g_1

Due to the following rules being added:

   g(v0) -> const [0]

And the following fresh constants:    const

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(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(f(x, y)) -> f(f(g(g(x)), g(g(y))), f(g(g(x)), g(g(y)))) [1]
   g(v0) -> const [0]

The TRS has the following type information:
g :: f:const -> f:const
f :: f:const -> f:const -> f:const
const :: f:const

Rewrite Strategy: INNERMOST
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(15) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   g(f(f(x', y'), f(x'', y''))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y'')))))), f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y''))))))) [5]
   g(f(f(x', y'), f(x'', y''))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y'')))))), f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const))) [4]
   g(f(f(x', y'), f(x'', y''))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y'')))))), f(g(const), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y''))))))) [4]
   g(f(f(x', y'), f(x'', y''))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x'')), g(g(y''))), f(g(g(x'')), g(g(y'')))))), f(g(const), g(const))) [3]
   g(f(f(x', y'), f(x4, y4))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const)), f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(f(f(g(g(x4)), g(g(y4))), f(g(g(x4)), g(g(y4))))))) [4]
   g(f(f(x', y'), y)) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const)), f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const))) [3]
   g(f(f(x', y'), f(x5, y5))) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const)), f(g(const), g(f(f(g(g(x5)), g(g(y5))), f(g(g(x5)), g(g(y5))))))) [3]
   g(f(f(x', y'), y)) -> f(f(g(f(f(g(g(x')), g(g(y'))), f(g(g(x')), g(g(y'))))), g(const)), f(g(const), g(const))) [2]
   g(f(f(x2, y2), f(x1, y1))) -> f(f(g(const), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1)))))), f(g(f(f(g(g(x2)), g(g(y2))), f(g(g(x2)), g(g(y2))))), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1))))))) [4]
   g(f(f(x2, y2), f(x1, y1))) -> f(f(g(const), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1)))))), f(g(f(f(g(g(x2)), g(g(y2))), f(g(g(x2)), g(g(y2))))), g(const))) [3]
   g(f(x, f(x1, y1))) -> f(f(g(const), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1)))))), f(g(const), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1))))))) [3]
   g(f(x, f(x1, y1))) -> f(f(g(const), g(f(f(g(g(x1)), g(g(y1))), f(g(g(x1)), g(g(y1)))))), f(g(const), g(const))) [2]
   g(f(f(x3, y3), f(x6, y6))) -> f(f(g(const), g(const)), f(g(f(f(g(g(x3)), g(g(y3))), f(g(g(x3)), g(g(y3))))), g(f(f(g(g(x6)), g(g(y6))), f(g(g(x6)), g(g(y6))))))) [3]
   g(f(f(x3, y3), y)) -> f(f(g(const), g(const)), f(g(f(f(g(g(x3)), g(g(y3))), f(g(g(x3)), g(g(y3))))), g(const))) [2]
   g(f(x, f(x7, y7))) -> f(f(g(const), g(const)), f(g(const), g(f(f(g(g(x7)), g(g(y7))), f(g(g(x7)), g(g(y7))))))) [2]
   g(f(x, y)) -> f(f(g(const), g(const)), f(g(const), g(const))) [1]
   g(v0) -> const [0]

The TRS has the following type information:
g :: f:const -> f:const
f :: f:const -> f:const -> f:const
const :: f:const

Rewrite Strategy: INNERMOST
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(17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   const => 0

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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + (1 + g(0) + g(0)) + (1 + g(0) + g(0)) :|: z = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(0)) + (1 + g(0) + g(1 + (1 + g(g(x7)) + g(g(y7))) + (1 + g(g(x7)) + g(g(y7))))) :|: x7 >= 0, x >= 0, y7 >= 0, z = 1 + x + (1 + x7 + y7)
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(0)) + (1 + g(1 + (1 + g(g(x3)) + g(g(y3))) + (1 + g(g(x3)) + g(g(y3)))) + g(0)) :|: y >= 0, y3 >= 0, x3 >= 0, z = 1 + (1 + x3 + y3) + y
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(0)) + (1 + g(1 + (1 + g(g(x3)) + g(g(y3))) + (1 + g(g(x3)) + g(g(y3)))) + g(1 + (1 + g(g(x6)) + g(g(y6))) + (1 + g(g(x6)) + g(g(y6))))) :|: x6 >= 0, y6 >= 0, y3 >= 0, x3 >= 0, z = 1 + (1 + x3 + y3) + (1 + x6 + y6)
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(0) + g(0)) :|: y1 >= 0, x1 >= 0, x >= 0, z = 1 + x + (1 + x1 + y1)
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) :|: y1 >= 0, x1 >= 0, x >= 0, z = 1 + x + (1 + x1 + y1)
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(1 + (1 + g(g(x2)) + g(g(y2))) + (1 + g(g(x2)) + g(g(y2)))) + g(0)) :|: y1 >= 0, z = 1 + (1 + x2 + y2) + (1 + x1 + y1), x1 >= 0, y2 >= 0, x2 >= 0
   g(z) -{ 4 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(1 + (1 + g(g(x2)) + g(g(y2))) + (1 + g(g(x2)) + g(g(y2)))) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) :|: y1 >= 0, z = 1 + (1 + x2 + y2) + (1 + x1 + y1), x1 >= 0, y2 >= 0, x2 >= 0
   g(z) -{ 2 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(0) + g(0)) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(0) + g(1 + (1 + g(g(x5)) + g(g(y5))) + (1 + g(g(x5)) + g(g(y5))))) :|: y5 >= 0, x5 >= 0, x' >= 0, z = 1 + (1 + x' + y') + (1 + x5 + y5), y' >= 0
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x4)) + g(g(y4))) + (1 + g(g(x4)) + g(g(y4))))) :|: x4 >= 0, z = 1 + (1 + x' + y') + (1 + x4 + y4), x' >= 0, y' >= 0, y4 >= 0
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(0) + g(0)) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(0) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 5 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0


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(19) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 1 }-> 1 + (1 + g(0) + g(0)) + (1 + g(0) + g(0)) :|: z = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(0)) + (1 + g(0) + g(1 + (1 + g(g(x7)) + g(g(y7))) + (1 + g(g(x7)) + g(g(y7))))) :|: x7 >= 0, x >= 0, y7 >= 0, z = 1 + x + (1 + x7 + y7)
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(0)) + (1 + g(1 + (1 + g(g(x3)) + g(g(y3))) + (1 + g(g(x3)) + g(g(y3)))) + g(0)) :|: y >= 0, y3 >= 0, x3 >= 0, z = 1 + (1 + x3 + y3) + y
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(0)) + (1 + g(1 + (1 + g(g(x3)) + g(g(y3))) + (1 + g(g(x3)) + g(g(y3)))) + g(1 + (1 + g(g(x6)) + g(g(y6))) + (1 + g(g(x6)) + g(g(y6))))) :|: x6 >= 0, y6 >= 0, y3 >= 0, x3 >= 0, z = 1 + (1 + x3 + y3) + (1 + x6 + y6)
   g(z) -{ 2 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(0) + g(0)) :|: y1 >= 0, x1 >= 0, x >= 0, z = 1 + x + (1 + x1 + y1)
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) :|: y1 >= 0, x1 >= 0, x >= 0, z = 1 + x + (1 + x1 + y1)
   g(z) -{ 3 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(1 + (1 + g(g(x2)) + g(g(y2))) + (1 + g(g(x2)) + g(g(y2)))) + g(0)) :|: y1 >= 0, z = 1 + (1 + x2 + y2) + (1 + x1 + y1), x1 >= 0, y2 >= 0, x2 >= 0
   g(z) -{ 4 }-> 1 + (1 + g(0) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) + (1 + g(1 + (1 + g(g(x2)) + g(g(y2))) + (1 + g(g(x2)) + g(g(y2)))) + g(1 + (1 + g(g(x1)) + g(g(y1))) + (1 + g(g(x1)) + g(g(y1))))) :|: y1 >= 0, z = 1 + (1 + x2 + y2) + (1 + x1 + y1), x1 >= 0, y2 >= 0, x2 >= 0
   g(z) -{ 2 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(0) + g(0)) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(0) + g(1 + (1 + g(g(x5)) + g(g(y5))) + (1 + g(g(x5)) + g(g(y5))))) :|: y5 >= 0, x5 >= 0, x' >= 0, z = 1 + (1 + x' + y') + (1 + x5 + y5), y' >= 0
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) :|: x' >= 0, y >= 0, y' >= 0, z = 1 + (1 + x' + y') + y
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x4)) + g(g(y4))) + (1 + g(g(x4)) + g(g(y4))))) :|: x4 >= 0, z = 1 + (1 + x' + y') + (1 + x4 + y4), x' >= 0, y' >= 0, y4 >= 0
   g(z) -{ 3 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(0) + g(0)) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(0) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 4 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(0)) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0
   g(z) -{ 5 }-> 1 + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) + (1 + g(1 + (1 + g(g(x')) + g(g(y'))) + (1 + g(g(x')) + g(g(y')))) + g(1 + (1 + g(g(x'')) + g(g(y''))) + (1 + g(g(x'')) + g(g(y''))))) :|: x' >= 0, y' >= 0, y'' >= 0, z = 1 + (1 + x' + y') + (1 + x'' + y''), x'' >= 0


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(21) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
S tuples:
   G(f(z0, z1)) -> c(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2


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(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(f(z0, z1)) -> c(G(g(z0)), G(z0)) by 
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))

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(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
S tuples:
   G(f(z0, z1)) -> c1(G(g(z1)), G(z1))
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c1_2, c2_2, c3_2, c_2


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(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(f(z0, z1)) -> c1(G(g(z1)), G(z1)) by 
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))

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(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
S tuples:
   G(f(z0, z1)) -> c2(G(g(z0)), G(z0))
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c2_2, c3_2, c_2, c1_2


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(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(f(z0, z1)) -> c2(G(g(z0)), G(z0)) by 
   G(f(f(z0, z1), x1)) -> c2(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))

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(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(f(z0, z1), x1)) -> c2(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
S tuples:
   G(f(z0, z1)) -> c3(G(g(z1)), G(z1))
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(f(z0, z1), x1)) -> c2(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c3_2, c_2, c1_2, c2_2


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(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(f(z0, z1)) -> c3(G(g(z1)), G(z1)) by 
   G(f(x0, f(z0, z1))) -> c3(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))

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(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(f(z0, z1)) -> f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))
Tuples:
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(f(z0, z1), x1)) -> c2(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c3(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
S tuples:
   G(f(f(z0, z1), x1)) -> c(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c1(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(f(z0, z1), x1)) -> c2(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
   G(f(x0, f(z0, z1))) -> c3(G(f(f(g(g(z0)), g(g(z1))), f(g(g(z0)), g(g(z1))))), G(f(z0, z1)))
K tuples:none
Defined Rule Symbols:   g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2