KILLED
proof of input_034lQ4nNXx.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
(15) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxWeightedTrs
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxTypedWeightedTrs
    (19) CompletionProof [UPPER BOUND(ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
        (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS


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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:

   f(x) -> f(f(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:

   f(x) -> f(f(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:

   f(x) -> f(f(x))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(z0) -> c(F(f(z0)), F(z0))
S tuples:
   F(z0) -> c(F(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0) -> c(F(f(z0)), F(z0)) by 
   F(z0) -> c(F(f(f(z0))), F(z0))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(z0) -> c(F(f(f(z0))), F(z0))
S tuples:
   F(z0) -> c(F(f(f(z0))), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0) -> c(F(f(f(z0))), F(z0)) by 
   F(x0) -> c(F(f(f(f(x0)))), F(x0))

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(x0) -> c(F(f(f(f(x0)))), F(x0))
S tuples:
   F(x0) -> c(F(f(f(f(x0)))), F(x0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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(11) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(x0) -> c(F(f(f(f(x0)))), F(x0)) by F(z0) -> c(F(f(f(f(f(z0))))), F(z0))
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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(z0) -> c(F(f(f(f(f(z0))))), F(z0))
S tuples:
   F(z0) -> c(F(f(f(f(f(z0))))), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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(13) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) by F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0))
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(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0))
S tuples:
   F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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(15) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
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(16)
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:

   f(x) -> f(f(x)) [1]

Rewrite Strategy: INNERMOST
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(18)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x) -> f(f(x)) [1]

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

Rewrite Strategy: INNERMOST
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(19) 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:
none

And the following fresh constants:    const

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(20)
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:

   f(x) -> f(f(x)) [1]

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

Rewrite Strategy: INNERMOST
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(21) 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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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> f(f(x)) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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(23) 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:

   f_1

Due to the following rules being added:
none

And the following fresh constants:    const

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(24)
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:

   f(x) -> f(f(x)) [1]

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

Rewrite Strategy: INNERMOST
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(25) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(26)
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:

   f(x) -> f(f(f(x))) [2]

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

Rewrite Strategy: INNERMOST
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(27) 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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(28)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> f(f(f(x))) :|: x >= 0, z = x


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(29) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(30)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> f(f(f(z))) :|: z >= 0