KILLED
proof of input_JZxXcbKr8w.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (34) CdtProblem
    (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem


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

(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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(X))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(X))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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(g(X)) -> g(f(f(X)))
   f(h(X)) -> h(g(X))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(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:

   f(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]

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

Rewrite Strategy: INNERMOST
----------------------------------------

(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:

   f_1

Due to the following rules being added:

   f(v0) -> const [0]

And the following fresh constants:    const

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

(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:

   f(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> const [0]

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

Rewrite Strategy: INNERMOST
----------------------------------------

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

(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:

   f(g(g(X'))) -> g(f(g(f(f(X'))))) [2]
   f(g(h(X''))) -> g(f(h(g(X'')))) [2]
   f(g(X)) -> g(f(const)) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> const [0]

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

Rewrite Strategy: INNERMOST
----------------------------------------

(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

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(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:

   f(v0) -> null_f [0]

And the following fresh constants:    null_f

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

(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:

   f(g(X)) -> g(f(f(X))) [1]
   f(h(X)) -> h(g(X)) [1]
   f(v0) -> null_f [0]

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

Rewrite Strategy: INNERMOST
----------------------------------------

(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_f => 0

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

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

   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> 1 + f(f(X)) :|: z = 1 + X, X >= 0
   f(z) -{ 1 }-> 1 + (1 + X) :|: z = 1 + X, X >= 0

Only complete derivations are relevant for the runtime complexity.

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

(21) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(g(z0)) -> g(f(f(z0)))
   f(h(z0)) -> h(g(z0))
Tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
S tuples:
   F(g(z0)) -> c(F(f(z0)), F(z0))
   F(h(z0)) -> c1
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c1


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(h(z0)) -> c1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

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

(26)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(27) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(g(h(z0))) -> c(F(h(g(z0))), F(h(z0)))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(32)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

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

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

(34)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2, c_1


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

(35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(g(g(h(x0)))) -> c(F(g(h(g(g(x0))))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(37) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(g(g(g(z0)))) -> c(F(g(f(g(f(f(z0)))))), F(g(g(z0)))) by F(g(g(g(z0)))) -> c(F(g(g(f(f(f(f(z0))))))), F(g(g(z0))))
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2