MAYBE
proof of input_ePmXzAqTqo.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 1307 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 617 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) 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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs)))
   eql(Nil) -> Nil
   eql(Cons(x, xs)) -> eql(Cons(x, xs))
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs)))
   eql(Nil) -> Nil
   eql(Cons(x, xs)) -> eql(Cons(x, xs))
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs)))
   eql(Nil) -> Nil
   eql(Cons(x, xs)) -> eql(Cons(x, xs))
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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]
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1]
   eql(Nil) -> Nil [1]
   eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1]
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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]
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1]
   eql(Nil) -> Nil [1]
   eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1]
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(x) [1]

The TRS has the following type information:
nesteql :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
eql :: Nil:Cons -> Nil:Cons
number17 :: a -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons

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:

   nesteql_1
   number17_1
   goal_1

(c) The following functions are completely defined:

   eql_1

Due to the following rules being added:
none

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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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]
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1]
   eql(Nil) -> Nil [1]
   eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1]
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(x) [1]

The TRS has the following type information:
nesteql :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
eql :: Nil:Cons -> Nil:Cons
number17 :: a -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
const :: a

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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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]
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [2]
   eql(Nil) -> Nil [1]
   eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1]
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(x) [1]

The TRS has the following type information:
nesteql :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
eql :: Nil:Cons -> Nil:Cons
number17 :: a -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
const :: a

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
   const => 0

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(x) :|: x >= 0, z = x
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: n >= 0, z = n


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

(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:

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0


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

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

   { number17 }
   { eql }
   { nesteql }
   { goal }

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal}

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

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: number17
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 17

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {number17}, {eql}, {nesteql}, {goal}
Previous analysis results are:
  number17: runtime: ?, size: O(1) [17]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: number17
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {eql}, {nesteql}, {goal}
Previous analysis results are:
  number17: runtime: O(1) [1], size: O(1) [17]

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

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {eql}, {nesteql}, {goal}
Previous analysis results are:
  number17: runtime: O(1) [1], size: O(1) [17]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: eql
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {eql}, {nesteql}, {goal}
Previous analysis results are:
  number17: runtime: O(1) [1], size: O(1) [17]
  eql: runtime: ?, size: O(1) [0]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(z) :|: z >= 0
   nesteql(z) -{ 2 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: z >= 0

Function symbols to be analyzed: {eql}, {nesteql}, {goal}
Previous analysis results are:
  number17: runtime: O(1) [1], size: O(1) [17]
  eql: runtime: INF, size: O(1) [0]

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

(31) 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

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

(32)
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:

   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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]
   nesteql(Cons(x, xs)) -> nesteql(eql(Cons(x, xs))) [1]
   eql(Nil) -> Nil [1]
   eql(Cons(x, xs)) -> eql(Cons(x, xs)) [1]
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nesteql(x) [1]

The TRS has the following type information:
nesteql :: Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
eql :: Nil:Cons -> Nil:Cons
number17 :: a -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
const :: a

Rewrite Strategy: INNERMOST
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(33) 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
   const => 0

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

   eql(z) -{ 1 }-> eql(1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
   eql(z) -{ 1 }-> 0 :|: z = 0
   goal(z) -{ 1 }-> nesteql(x) :|: x >= 0, z = x
   nesteql(z) -{ 1 }-> nesteql(eql(1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0
   nesteql(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 + 0)))))))))))))))) :|: z = 0
   number17(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 + 0)))))))))))))))) :|: n >= 0, z = n

Only complete derivations are relevant for the runtime complexity.

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(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1)))
   eql(Nil) -> Nil
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nesteql(z0)
Tuples:
   NESTEQL(Nil) -> c
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Nil) -> c2
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTEQL(z0))
S tuples:
   NESTEQL(Nil) -> c
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Nil) -> c2
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NUMBER17(z0) -> c4
   GOAL(z0) -> c5(NESTEQL(z0))
K tuples:none
Defined Rule Symbols:   nesteql_1, eql_1, number17_1, goal_1

Defined Pair Symbols:   NESTEQL_1, EQL_1, NUMBER17_1, GOAL_1

Compound Symbols:   c, c1_2, c2, c3_1, c4, c5_1


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(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c5(NESTEQL(z0))
Removed 3 trailing nodes:
   NESTEQL(Nil) -> c
   NUMBER17(z0) -> c4
   EQL(Nil) -> c2

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(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1)))
   eql(Nil) -> Nil
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nesteql(z0)
Tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
S tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   nesteql_1, eql_1, number17_1, goal_1

Defined Pair Symbols:   NESTEQL_1, EQL_1

Compound Symbols:   c1_2, c3_1


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(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   nesteql(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nesteql(Cons(z0, z1)) -> nesteql(eql(Cons(z0, z1)))
   eql(Nil) -> Nil
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nesteql(z0)

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(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
S tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   NESTEQL_1, EQL_1

Compound Symbols:   c1_2, c3_1


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(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
We considered the (Usable) Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
And the Tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(EQL(x_1)) = 0
   POL(NESTEQL(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(eql(x_1)) = 0

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(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
S tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   NESTEQL_1, EQL_1

Compound Symbols:   c1_2, c3_1


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(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by 
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))

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(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
S tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   EQL_1, NESTEQL_1

Compound Symbols:   c3_1, c1_2


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(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by 
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))

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(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
S tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   EQL_1, NESTEQL_1

Compound Symbols:   c3_1, c1_2


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(47) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
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(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
S tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   EQL_1, NESTEQL_1

Compound Symbols:   c3_1, c1_2


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(49) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) by NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
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(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))
Tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
S tuples:
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
K tuples:
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
Defined Rule Symbols:   eql_1

Defined Pair Symbols:   EQL_1, NESTEQL_1

Compound Symbols:   c3_1, c1_2