MAYBE
proof of input_9nvGC7Jgbc.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 4 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (20) CdtProblem
    (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
        (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (26) CpxTRS
        (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (28) CpxWeightedTrs
            (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (30) CpxTypedWeightedTrs
            (31) CompletionProof [UPPER BOUND(ID), 0 ms]
            (32) CpxTypedWeightedCompleteTrs
                (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (34) CpxTypedWeightedCompleteTrs
                (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (36) CpxRNTS
                (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (38) CpxRNTS
                (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxRNTS
                (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (42) CpxRNTS
                (43) IntTrsBoundProof [UPPER BOUND(ID), 1391 ms]
                (44) CpxRNTS
                (45) IntTrsBoundProof [UPPER BOUND(ID), 575 ms]
                (46) CpxRNTS
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
    (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
    (52) CdtProblem
        (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
(61) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(62) CpxWeightedTrs
    (63) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CpxTypedWeightedTrs
    (65) CompletionProof [UPPER BOUND(ID), 0 ms]
    (66) CpxTypedWeightedCompleteTrs
        (67) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CpxTypedWeightedCompleteTrs
        (69) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (70) CpxRNTS
        (71) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CpxRNTS
        (73) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CpxRNTS
        (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
        (78) CpxRNTS
        (79) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
        (80) CpxRNTS
        (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 1373 ms]
        (84) CpxRNTS
        (85) IntTrsBoundProof [UPPER BOUND(ID), 686 ms]
        (86) CpxRNTS
    (87) CompletionProof [UPPER BOUND(ID), 0 ms]
    (88) CpxTypedWeightedCompleteTrs
        (89) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (90) CpxRNTS


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

(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
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


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

(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S 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(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)

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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S 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(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)

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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
NESTEQL :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c2:c3 -> c:c1
eql :: Nil:Cons -> Nil:Cons
EQL :: Nil:Cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
NUMBER17 :: a -> c4
c4 :: c4
GOAL :: Nil:Cons -> c5
c5 :: c:c1 -> c5
nesteql :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_a5_6 :: a
hole_c56_6 :: c5
hole_b7_6 :: b
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
NESTEQL, eql, EQL, nesteql

They will be analysed ascendingly in the following order:
eql < NESTEQL
EQL < NESTEQL
eql < nesteql

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

(12)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
NESTEQL :: Nil:Cons -> c:c1
Nil :: Nil:Cons
c :: c:c1
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c1 :: c:c1 -> c2:c3 -> c:c1
eql :: Nil:Cons -> Nil:Cons
EQL :: Nil:Cons -> c2:c3
c2 :: c2:c3
c3 :: c2:c3 -> c2:c3
NUMBER17 :: a -> c4
c4 :: c4
GOAL :: Nil:Cons -> c5
c5 :: c:c1 -> c5
nesteql :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c11_6 :: c:c1
hole_Nil:Cons2_6 :: Nil:Cons
hole_c2:c33_6 :: c2:c3
hole_c44_6 :: c4
hole_a5_6 :: a
hole_c56_6 :: c5
hole_b7_6 :: b
gen_c:c18_6 :: Nat -> c:c1
gen_Nil:Cons9_6 :: Nat -> Nil:Cons
gen_c2:c310_6 :: Nat -> c2:c3


Generator Equations:
gen_c:c18_6(0) <=> c
gen_c:c18_6(+(x, 1)) <=> c1(gen_c:c18_6(x), c2)
gen_Nil:Cons9_6(0) <=> Nil
gen_Nil:Cons9_6(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons9_6(x))
gen_c2:c310_6(0) <=> c2
gen_c2:c310_6(+(x, 1)) <=> c3(gen_c2:c310_6(x))


The following defined symbols remain to be analysed:
eql, NESTEQL, EQL, nesteql

They will be analysed ascendingly in the following order:
eql < NESTEQL
EQL < NESTEQL
eql < nesteql

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S 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(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)

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

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

(16)
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
----------------------------------------

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

(18)
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


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

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

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

(20)
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


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

(21) 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)

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

(22)
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


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

(23) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))

The (relative) TRS S consists of the following rules:

   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))

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

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

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1)))
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1)))
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1))

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

(27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
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(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) [1]
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) [1]
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) [0]

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

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

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) [1]
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) [1]
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) [0]

The TRS has the following type information:
NESTEQL :: Cons -> c1
Cons :: a -> b -> Cons
c1 :: c1 -> c3 -> c1
eql :: Cons -> Cons
EQL :: Cons -> c3
c3 :: c3 -> c3

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

(31) 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
   EQL_1

(c) The following functions are completely defined:

   eql_1

Due to the following rules being added:

   eql(v0) -> const1 [0]

And the following fresh constants:    const1, const, const2, const3, const4

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

(32)
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(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) [1]
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) [1]
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) [0]
   eql(v0) -> const1 [0]

The TRS has the following type information:
NESTEQL :: Cons:const1 -> c1
Cons :: a -> b -> Cons:const1
c1 :: c1 -> c3 -> c1
eql :: Cons:const1 -> Cons:const1
EQL :: Cons:const1 -> c3
c3 :: c3 -> c3
const1 :: Cons:const1
const :: c1
const2 :: a
const3 :: b
const4 :: c3

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

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

(34)
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(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) [1]
   NESTEQL(Cons(z0, z1)) -> c1(NESTEQL(const1), EQL(Cons(z0, z1))) [1]
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) [1]
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) [0]
   eql(v0) -> const1 [0]

The TRS has the following type information:
NESTEQL :: Cons:const1 -> c1
Cons :: a -> b -> Cons:const1
c1 :: c1 -> c3 -> c1
eql :: Cons:const1 -> Cons:const1
EQL :: Cons:const1 -> c3
c3 :: c3 -> c3
const1 :: Cons:const1
const :: c1
const2 :: a
const3 :: b
const4 :: c3

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

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

   const1 => 0
   const => 0
   const2 => 0
   const3 => 0
   const4 => 0

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

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

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { EQL }
   { eql }
   { NESTEQL }

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {EQL}, {eql}, {NESTEQL}

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

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

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

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {EQL}, {eql}, {NESTEQL}

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

(43) 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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: z >= 0

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

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

(45) 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: ?

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(0) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: z >= 0

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

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

(47) 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:

   eql(v0) -> null_eql [0]
   NESTEQL(v0) -> null_NESTEQL [0]
   EQL(v0) -> null_EQL [0]

And the following fresh constants:    null_eql, null_NESTEQL, null_EQL, const, const1

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

(48)
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(Cons(z0, z1)) -> c1(NESTEQL(eql(Cons(z0, z1))), EQL(Cons(z0, z1))) [1]
   EQL(Cons(z0, z1)) -> c3(EQL(Cons(z0, z1))) [1]
   eql(Cons(z0, z1)) -> eql(Cons(z0, z1)) [0]
   eql(v0) -> null_eql [0]
   NESTEQL(v0) -> null_NESTEQL [0]
   EQL(v0) -> null_EQL [0]

The TRS has the following type information:
NESTEQL :: Cons:null_eql -> c1:null_NESTEQL
Cons :: a -> b -> Cons:null_eql
c1 :: c1:null_NESTEQL -> c3:null_EQL -> c1:null_NESTEQL
eql :: Cons:null_eql -> Cons:null_eql
EQL :: Cons:null_eql -> c3:null_EQL
c3 :: c3:null_EQL -> c3:null_EQL
null_eql :: Cons:null_eql
null_NESTEQL :: c1:null_NESTEQL
null_EQL :: c3:null_EQL
const :: a
const1 :: b

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

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

   null_eql => 0
   null_NESTEQL => 0
   null_EQL => 0
   const => 0
   const1 => 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   EQL(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   EQL(z) -{ 1 }-> 1 + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NESTEQL(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   NESTEQL(z) -{ 1 }-> 1 + NESTEQL(eql(1 + z0 + z1)) + EQL(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> eql(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eql(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(51) 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)) = [2]
   POL(EQL(x_1)) = 0
   POL(NESTEQL(x_1)) = [2]x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(eql(x_1)) = 0

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

(52)
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


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

(53) 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)))

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

(54)
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


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

(55) 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)))

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

(56)
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


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

(57) 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)))
----------------------------------------

(58)
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


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

(59) 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)))
----------------------------------------

(60)
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


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

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

(62)
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
----------------------------------------

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

(64)
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
----------------------------------------

(65) 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

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

(66)
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
----------------------------------------

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

(68)
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
----------------------------------------

(69) 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

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

(70)
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


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

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

(72)
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


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

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

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

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

(74)
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}

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

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

(76)
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}

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

(77) 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

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

(78)
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]

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

(79) 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

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

(80)
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]

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

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

(82)
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]

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

(83) 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

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

(84)
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]

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

(85) 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: ?

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

(86)
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]

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

(87) 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

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

(88)
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
----------------------------------------

(89) 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

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

(90)
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.