KILLED
proof of input_BpGv9vjClw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 1 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 183 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 11 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 134 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (48) CdtProblem
    (49) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 21 ms]
    (70) CdtProblem
    (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) 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:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> L

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:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0', zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> L

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:

   incr(nil) -> nil
   incr(cons(X, L)) -> cons(s(X), incr(L))
   adx(nil) -> nil
   adx(cons(X, L)) -> incr(cons(X, adx(L)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(X, L)) -> X
   tail(cons(X, L)) -> L

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:

   incr(nil) -> nil [1]
   incr(cons(X, L)) -> cons(s(X), incr(L)) [1]
   adx(nil) -> nil [1]
   adx(cons(X, L)) -> incr(cons(X, adx(L))) [1]
   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   head(cons(X, L)) -> X [1]
   tail(cons(X, L)) -> L [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:

   incr(nil) -> nil [1]
   incr(cons(X, L)) -> cons(s(X), incr(L)) [1]
   adx(nil) -> nil [1]
   adx(cons(X, L)) -> incr(cons(X, adx(L))) [1]
   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   head(cons(X, L)) -> X [1]
   tail(cons(X, L)) -> L [1]

The TRS has the following type information:
incr :: nil:cons -> nil:cons
nil :: nil:cons
cons :: s:0 -> nil:cons -> nil:cons
s :: s:0 -> s:0
adx :: nil:cons -> nil:cons
nats :: nil:cons
zeros :: nil:cons
0 :: s:0
head :: nil:cons -> s:0
tail :: 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:

   nats
   head_1
   tail_1

(c) The following functions are completely defined:

   zeros
   adx_1
   incr_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:

   incr(nil) -> nil [1]
   incr(cons(X, L)) -> cons(s(X), incr(L)) [1]
   adx(nil) -> nil [1]
   adx(cons(X, L)) -> incr(cons(X, adx(L))) [1]
   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   head(cons(X, L)) -> X [1]
   tail(cons(X, L)) -> L [1]

The TRS has the following type information:
incr :: nil:cons -> nil:cons
nil :: nil:cons
cons :: s:0 -> nil:cons -> nil:cons
s :: s:0 -> s:0
adx :: nil:cons -> nil:cons
nats :: nil:cons
zeros :: nil:cons
0 :: s:0
head :: nil:cons -> s:0
tail :: nil:cons -> nil:cons

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:

   incr(nil) -> nil [1]
   incr(cons(X, L)) -> cons(s(X), incr(L)) [1]
   adx(nil) -> nil [1]
   adx(cons(X, nil)) -> incr(cons(X, nil)) [2]
   adx(cons(X, cons(X', L'))) -> incr(cons(X, incr(cons(X', adx(L'))))) [2]
   nats -> adx(cons(0, zeros)) [2]
   zeros -> cons(0, zeros) [1]
   head(cons(X, L)) -> X [1]
   tail(cons(X, L)) -> L [1]

The TRS has the following type information:
incr :: nil:cons -> nil:cons
nil :: nil:cons
cons :: s:0 -> nil:cons -> nil:cons
s :: s:0 -> s:0
adx :: nil:cons -> nil:cons
nats :: nil:cons
zeros :: nil:cons
0 :: s:0
head :: nil:cons -> s:0
tail :: nil:cons -> nil:cons

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
   0 => 0

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + X + 0) :|: z = 1 + X + 0, X >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 


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

(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:

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 


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

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

   { incr }
   { head }
   { zeros }
   { tail }
   { adx }
   { nats }

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats}

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

(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:

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: incr
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {head}, {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: ?, size: O(n^1) [2*z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 2 }-> incr(1 + (z - 1) + 0) :|: z - 1 >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(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:

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {head}, {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  head: runtime: ?, size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  head: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  head: runtime: O(1) [1], size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  zeros: runtime: ?, size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   adx(z) -{ 3 + z }-> s' :|: s' >= 0, s' <= 2 * (1 + (z - 1) + 0), z - 1 >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(L'))) :|: L' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + L')
   adx(z) -{ 1 }-> 0 :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 2 + L }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * L, z = 1 + X + L, X >= 0, L >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {zeros}, {tail}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  head: runtime: O(1) [1], size: O(n^1) [z]
  zeros: runtime: INF, size: O(1) [0]

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

(37) 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:

   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   incr(v0) -> null_incr [0]
   adx(v0) -> null_adx [0]

And the following fresh constants:    null_head, null_tail, null_incr, null_adx

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

(38)
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:

   incr(nil) -> nil [1]
   incr(cons(X, L)) -> cons(s(X), incr(L)) [1]
   adx(nil) -> nil [1]
   adx(cons(X, L)) -> incr(cons(X, adx(L))) [1]
   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   head(cons(X, L)) -> X [1]
   tail(cons(X, L)) -> L [1]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]
   incr(v0) -> null_incr [0]
   adx(v0) -> null_adx [0]

The TRS has the following type information:
incr :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx
nil :: nil:cons:null_tail:null_incr:null_adx
cons :: s:0:null_head -> nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx
s :: s:0:null_head -> s:0:null_head
adx :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx
nats :: nil:cons:null_tail:null_incr:null_adx
zeros :: nil:cons:null_tail:null_incr:null_adx
0 :: s:0:null_head
head :: nil:cons:null_tail:null_incr:null_adx -> s:0:null_head
tail :: nil:cons:null_tail:null_incr:null_adx -> nil:cons:null_tail:null_incr:null_adx
null_head :: s:0:null_head
null_tail :: nil:cons:null_tail:null_incr:null_adx
null_incr :: nil:cons:null_tail:null_incr:null_adx
null_adx :: nil:cons:null_tail:null_incr:null_adx

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

(39) 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
   0 => 0
   null_head => 0
   null_tail => 0
   null_incr => 0
   null_adx => 0

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

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

   adx(z) -{ 1 }-> incr(1 + X + adx(L)) :|: z = 1 + X + L, X >= 0, L >= 0
   adx(z) -{ 1 }-> 0 :|: z = 0
   adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   head(z) -{ 1 }-> X :|: z = 1 + X + L, X >= 0, L >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 1 }-> 0 :|: z = 0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(L) :|: z = 1 + X + L, X >= 0, L >= 0
   nats -{ 1 }-> adx(zeros) :|: 
   tail(z) -{ 1 }-> L :|: z = 1 + X + L, X >= 0, L >= 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   incr(nil) -> nil
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   INCR(nil) -> c
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(nil) -> c2
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros), ZEROS)
   ZEROS -> c5(ZEROS)
   HEAD(cons(z0, z1)) -> c6
   TAIL(cons(z0, z1)) -> c7
S tuples:
   INCR(nil) -> c
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(nil) -> c2
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros), ZEROS)
   ZEROS -> c5(ZEROS)
   HEAD(cons(z0, z1)) -> c6
   TAIL(cons(z0, z1)) -> c7
K tuples:none
Defined Rule Symbols:   incr_1, adx_1, nats, zeros, head_1, tail_1

Defined Pair Symbols:   INCR_1, ADX_1, NATS, ZEROS, HEAD_1, TAIL_1

Compound Symbols:   c, c1_1, c2, c3_2, c4_2, c5_1, c6, c7


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   HEAD(cons(z0, z1)) -> c6
   ADX(nil) -> c2
   TAIL(cons(z0, z1)) -> c7
   INCR(nil) -> c

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   incr(nil) -> nil
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros), ZEROS)
   ZEROS -> c5(ZEROS)
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros), ZEROS)
   ZEROS -> c5(ZEROS)
K tuples:none
Defined Rule Symbols:   incr_1, adx_1, nats, zeros, head_1, tail_1

Defined Pair Symbols:   INCR_1, ADX_1, NATS, ZEROS

Compound Symbols:   c1_1, c3_2, c4_2, c5_1


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

(45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   incr(nil) -> nil
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
   NATS -> c(ZEROS)
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
   NATS -> c(ZEROS)
K tuples:none
Defined Rule Symbols:   incr_1, adx_1, nats, zeros, head_1, tail_1

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(47) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   NATS -> c(ZEROS)

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   incr(nil) -> nil
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
K tuples:none
Defined Rule Symbols:   incr_1, adx_1, nats, zeros, head_1, tail_1

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(49) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   NATS -> c(ADX(zeros))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   incr(nil) -> nil
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   incr_1, adx_1, nats, zeros, head_1, tail_1

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   nats -> adx(zeros)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(zeros))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NATS -> c(ADX(zeros)) by 
   NATS -> c(ADX(cons(0, zeros)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, zeros)))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NATS -> c(ADX(cons(0, zeros))) by 
   NATS -> c(ADX(cons(0, cons(0, zeros))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, zeros))))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(57) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NATS -> c(ADX(cons(0, cons(0, zeros)))) by NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
S tuples:
   INCR(cons(z0, z1)) -> c1(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   INCR_1, ADX_1, ZEROS, NATS

Compound Symbols:   c1_1, c3_2, c5_1, c_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, z1)) -> c1(INCR(z1)) by 
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
S tuples:
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ADX_1, ZEROS, NATS, INCR_1

Compound Symbols:   c3_2, c5_1, c_1, c1_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1)) by 
   ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil))
   ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil))
   ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil))
   ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))
K tuples:
   NATS -> c(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c5_1, c_1, c1_1, c3_2


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

(63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ADX(cons(x0, nil)) -> c3(INCR(cons(x0, nil)), ADX(nil))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c5_1, c_1, c1_1, c3_2


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ADX(cons(x0, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1))) by 
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))), ADX(cons(x1, nil)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
K tuples:none
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c5_1, c_1, c1_1, c3_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
K tuples:none
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c5_1, c_1, c1_1, c3_2, c3_1


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
We considered the (Usable) Rules:none
And the Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ADX(x_1)) = x_1
   POL(INCR(x_1)) = 0
   POL(NATS) = [1]
   POL(ZEROS) = 0
   POL(adx(x_1)) = [1] + x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1]
   POL(incr(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(zeros) = [1]

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   NATS -> c(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c5_1, c_1, c1_1, c3_2, c3_1


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

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NATS -> c(ADX(cons(0, cons(0, cons(0, zeros))))) by NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
S tuples:
   ZEROS -> c5(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c5_1, c1_1, c3_2, c3_1, c_1


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

(73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, cons(y0, y1))) -> c1(INCR(cons(y0, y1))) by 
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c3_1, c_1, c1_1


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ADX(cons(x0, cons(x1, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1)))) by ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c3_1, c_1, c1_1


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil))))) by ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil)))))
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
   ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil)))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c_1, c1_1, c3_1


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(incr(cons(z2, adx(z3))))))), ADX(cons(z1, cons(z2, z3)))) by ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil)))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c_1, c1_1, c3_1


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), incr(nil))))) by ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil))))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
   ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
K tuples:
   ADX(cons(x0, cons(x1, nil))) -> c3(INCR(cons(x0, incr(cons(x1, nil)))))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c_1, c1_1, c3_1


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ADX(cons(z0, cons(z1, nil))) -> c3(INCR(cons(z0, cons(s(z1), nil))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(nil) -> nil
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   incr(nil) -> nil
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
S tuples:
   ZEROS -> c5(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c1(INCR(cons(z1, cons(y1, y2))))
   ADX(cons(z0, cons(z1, cons(z2, z3)))) -> c3(INCR(cons(z0, cons(s(z1), incr(cons(s(z2), incr(adx(z3))))))), ADX(cons(z1, cons(z2, z3))))
K tuples:none
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c5_1, c3_2, c_1, c1_1