KILLED
proof of input_6x6MVrGN0x.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 23 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 109 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 46 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 206 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 1327 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 917 ms]
        (38) CpxRNTS
    (39) CompletionProof [UPPER BOUND(ID), 0 ms]
    (40) CpxTypedWeightedCompleteTrs
        (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
(43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(44) CdtProblem
    (45) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (46) CdtProblem
    (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) 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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs))
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs)))
   immatcopy(Nil) -> Nil
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> nestimeql(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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs))
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs)))
   immatcopy(Nil) -> Nil
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> nestimeql(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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs))
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs)))
   immatcopy(Nil) -> Nil
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> nestimeql(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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1]
   nestimeql(Nil) -> number42(Nil) [1]
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1]
   immatcopy(Nil) -> Nil [1]
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1]
   goal(x) -> nestimeql(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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1]
   nestimeql(Nil) -> number42(Nil) [1]
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1]
   immatcopy(Nil) -> Nil [1]
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1]
   goal(x) -> nestimeql(x) [1]

The TRS has the following type information:
immatcopy :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
nestimeql :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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:

   nestimeql_1
   number42_1
   goal_1

(c) The following functions are completely defined:

   immatcopy_1

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1]
   nestimeql(Nil) -> number42(Nil) [1]
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1]
   immatcopy(Nil) -> Nil [1]
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1]
   goal(x) -> nestimeql(x) [1]

The TRS has the following type information:
immatcopy :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
nestimeql :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1]
   nestimeql(Nil) -> number42(Nil) [1]
   nestimeql(Cons(x, xs)) -> nestimeql(Cons(Nil, immatcopy(xs))) [2]
   immatcopy(Nil) -> Nil [1]
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1]
   goal(x) -> nestimeql(x) [1]

The TRS has the following type information:
immatcopy :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
nestimeql :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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:

   Nil => 0

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

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

   goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 1 }-> number42(0) :|: z = 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x


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

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x

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

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

   goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x


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

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

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

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0


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

(19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { number42 }
   { immatcopy }
   { nestimeql }
   { goal }

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

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

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal}

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

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: number42
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 42

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number42}, {immatcopy}, {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: ?, size: O(1) [42]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: number42
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: immatcopy
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {immatcopy}, {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]
  immatcopy: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: immatcopy
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> nestimeql(1 + 0 + immatcopy(xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]
  immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]
  immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: nestimeql
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 42

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]
  immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  nestimeql: runtime: ?, size: O(1) [42]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: nestimeql
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(z) :|: z >= 0
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 2 + xs }-> 1 + 0 + s :|: s >= 0, s <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 3 + xs }-> nestimeql(1 + 0 + s') :|: s' >= 0, s' <= xs, z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 2 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z = 0, x >= 0, 0 = x
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {nestimeql}, {goal}
Previous analysis results are:
  number42: runtime: O(1) [1], size: O(1) [42]
  immatcopy: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  nestimeql: runtime: INF, size: O(1) [42]

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

(39) 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: none

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

(40)
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:

   immatcopy(Cons(x, xs)) -> Cons(Nil, immatcopy(xs)) [1]
   nestimeql(Nil) -> number42(Nil) [1]
   nestimeql(Cons(x, xs)) -> nestimeql(immatcopy(Cons(x, xs))) [1]
   immatcopy(Nil) -> Nil [1]
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))))))))))))))))))))))))))) [1]
   goal(x) -> nestimeql(x) [1]

The TRS has the following type information:
immatcopy :: Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
nestimeql :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil

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

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

   Nil => 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   goal(z) -{ 1 }-> nestimeql(x) :|: x >= 0, z = x
   immatcopy(z) -{ 1 }-> 0 :|: z = 0
   immatcopy(z) -{ 1 }-> 1 + 0 + immatcopy(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nestimeql(z) -{ 1 }-> number42(0) :|: z = 0
   nestimeql(z) -{ 1 }-> nestimeql(immatcopy(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   number42(z) -{ 1 }-> 1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + (1 + 0 + 0))))))))))))))))))))))))))))))))))))))))) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1)))
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> nestimeql(z0)
Tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   IMMATCOPY(Nil) -> c1
   NESTIMEQL(Nil) -> c2(NUMBER42(Nil))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
   NUMBER42(z0) -> c4
   GOAL(z0) -> c5(NESTIMEQL(z0))
S tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   IMMATCOPY(Nil) -> c1
   NESTIMEQL(Nil) -> c2(NUMBER42(Nil))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
   NUMBER42(z0) -> c4
   GOAL(z0) -> c5(NESTIMEQL(z0))
K tuples:none
Defined Rule Symbols:   immatcopy_1, nestimeql_1, number42_1, goal_1

Defined Pair Symbols:   IMMATCOPY_1, NESTIMEQL_1, NUMBER42_1, GOAL_1

Compound Symbols:   c_1, c1, c2_1, c3_2, c4, c5_1


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

(45) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c5(NESTIMEQL(z0))
Removed 3 trailing nodes:
   NUMBER42(z0) -> c4
   NESTIMEQL(Nil) -> c2(NUMBER42(Nil))
   IMMATCOPY(Nil) -> c1

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1)))
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> nestimeql(z0)
Tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
S tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   immatcopy_1, nestimeql_1, number42_1, goal_1

Defined Pair Symbols:   IMMATCOPY_1, NESTIMEQL_1

Compound Symbols:   c_1, c3_2


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

(47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   nestimeql(Nil) -> number42(Nil)
   nestimeql(Cons(z0, z1)) -> nestimeql(immatcopy(Cons(z0, z1)))
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> nestimeql(z0)

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
Tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
S tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   immatcopy_1

Defined Pair Symbols:   IMMATCOPY_1, NESTIMEQL_1

Compound Symbols:   c_1, c3_2


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(immatcopy(Cons(z0, z1))), IMMATCOPY(Cons(z0, z1))) by 
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
Tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1)))
S tuples:
   IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1))
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   immatcopy_1

Defined Pair Symbols:   IMMATCOPY_1, NESTIMEQL_1

Compound Symbols:   c_1, c3_2


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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IMMATCOPY(Cons(z0, z1)) -> c(IMMATCOPY(z1)) by 
   IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
Tuples:
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1)))
   IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1)))
S tuples:
   NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1)))
   IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1)))
K tuples:none
Defined Rule Symbols:   immatcopy_1

Defined Pair Symbols:   NESTIMEQL_1, IMMATCOPY_1

Compound Symbols:   c3_2, c_1


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

(53) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace NESTIMEQL(Cons(z0, z1)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(z1))), IMMATCOPY(Cons(z0, z1))) by 
   NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
Tuples:
   IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1)))
   NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0)))
S tuples:
   IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1)))
   NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0)))
K tuples:none
Defined Rule Symbols:   immatcopy_1

Defined Pair Symbols:   IMMATCOPY_1, NESTIMEQL_1

Compound Symbols:   c_1, c3_2


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IMMATCOPY(Cons(z0, Cons(y0, y1))) -> c(IMMATCOPY(Cons(y0, y1))) by 
   IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   immatcopy(Cons(z0, z1)) -> Cons(Nil, immatcopy(z1))
   immatcopy(Nil) -> Nil
Tuples:
   NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0)))
   IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2))))
S tuples:
   NESTIMEQL(Cons(Nil, y0)) -> c3(NESTIMEQL(Cons(Nil, immatcopy(y0))), IMMATCOPY(Cons(Nil, y0)))
   IMMATCOPY(Cons(z0, Cons(z1, Cons(y1, y2)))) -> c(IMMATCOPY(Cons(z1, Cons(y1, y2))))
K tuples:none
Defined Rule Symbols:   immatcopy_1

Defined Pair Symbols:   NESTIMEQL_1, IMMATCOPY_1

Compound Symbols:   c3_2, c_1