MAYBE
proof of input_sgzwH9cmXd.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) 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:

   a(b(x)) -> b(b(a(a(x))))

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:

   a(b(x)) -> b(b(a(a(x))))

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:

   a(b(x)) -> b(b(a(a(x))))

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:

   a(b(x)) -> b(b(a(a(x)))) [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:

   a(b(x)) -> b(b(a(a(x)))) [1]

The TRS has the following type information:
a :: b -> b
b :: b -> b

Rewrite Strategy: INNERMOST
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(9) 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:

   a_1

Due to the following rules being added:

   a(v0) -> const [0]

And the following fresh constants:    const

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(10)
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:

   a(b(x)) -> b(b(a(a(x)))) [1]
   a(v0) -> const [0]

The TRS has the following type information:
a :: b:const -> b:const
b :: b:const -> b:const
const :: b:const

Rewrite Strategy: INNERMOST
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(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(12)
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:

   a(b(b(x'))) -> b(b(a(b(b(a(a(x'))))))) [2]
   a(b(x)) -> b(b(a(const))) [1]
   a(v0) -> const [0]

The TRS has the following type information:
a :: b:const -> b:const
b :: b:const -> b:const
const :: b:const

Rewrite Strategy: INNERMOST
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(13) 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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(14)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   a(z) -{ 1 }-> 1 + (1 + a(0)) :|: x >= 0, z = 1 + x
   a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(x'))))) :|: x' >= 0, z = 1 + (1 + x')


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(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   a(z) -{ 0 }-> 0 :|: z >= 0
   a(z) -{ 1 }-> 1 + (1 + a(0)) :|: z - 1 >= 0
   a(z) -{ 2 }-> 1 + (1 + a(1 + (1 + a(a(z - 2))))) :|: z - 2 >= 0


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(17) 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:

   a(v0) -> null_a [0]

And the following fresh constants:    null_a

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(18)
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:

   a(b(x)) -> b(b(a(a(x)))) [1]
   a(v0) -> null_a [0]

The TRS has the following type information:
a :: b:null_a -> b:null_a
b :: b:null_a -> b:null_a
null_a :: b:null_a

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

   null_a => 0

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(20)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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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:
   a(b(z0)) -> b(b(a(a(z0))))
Tuples:
   A(b(z0)) -> c(A(a(z0)), A(z0))
S tuples:
   A(b(z0)) -> c(A(a(z0)), A(z0))
K tuples:none
Defined Rule Symbols:   a_1

Defined Pair Symbols:   A_1

Compound Symbols:   c_2


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(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A(b(z0)) -> c(A(a(z0)), A(z0)) by 
   A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0)))

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(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a(b(z0)) -> b(b(a(a(z0))))
Tuples:
   A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0)))
S tuples:
   A(b(b(z0))) -> c(A(b(b(a(a(z0))))), A(b(z0)))
K tuples:none
Defined Rule Symbols:   a_1

Defined Pair Symbols:   A_1

Compound Symbols:   c_2