KILLED
proof of input_zs55Dg0807.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (42) CdtProblem
    (43) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) 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:

   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(X, Y)) -> cons(s(X), incr(Y))
   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
   hd(cons(X, Y)) -> X
   tl(cons(X, Y)) -> Y

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:

   nats -> adx(zeros)
   zeros -> cons(0', zeros)
   incr(cons(X, Y)) -> cons(s(X), incr(Y))
   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
   hd(cons(X, Y)) -> X
   tl(cons(X, Y)) -> Y

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:

   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(X, Y)) -> cons(s(X), incr(Y))
   adx(cons(X, Y)) -> incr(cons(X, adx(Y)))
   hd(cons(X, Y)) -> X
   tl(cons(X, Y)) -> Y

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:

   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   incr(cons(X, Y)) -> cons(s(X), incr(Y)) [1]
   adx(cons(X, Y)) -> incr(cons(X, adx(Y))) [1]
   hd(cons(X, Y)) -> X [1]
   tl(cons(X, Y)) -> Y [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:

   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   incr(cons(X, Y)) -> cons(s(X), incr(Y)) [1]
   adx(cons(X, Y)) -> incr(cons(X, adx(Y))) [1]
   hd(cons(X, Y)) -> X [1]
   tl(cons(X, Y)) -> Y [1]

The TRS has the following type information:
nats :: cons
adx :: cons -> cons
zeros :: cons
cons :: 0:s -> cons -> cons
0 :: 0:s
incr :: cons -> cons
s :: 0:s -> 0:s
hd :: cons -> 0:s
tl :: cons -> 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
   hd_1
   tl_1

(c) The following functions are completely defined:

   zeros
   adx_1
   incr_1

Due to the following rules being added:

   adx(v0) -> const [0]
   incr(v0) -> const [0]

And the following fresh constants:    const

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   incr(cons(X, Y)) -> cons(s(X), incr(Y)) [1]
   adx(cons(X, Y)) -> incr(cons(X, adx(Y))) [1]
   hd(cons(X, Y)) -> X [1]
   tl(cons(X, Y)) -> Y [1]
   adx(v0) -> const [0]
   incr(v0) -> const [0]

The TRS has the following type information:
nats :: cons:const
adx :: cons:const -> cons:const
zeros :: cons:const
cons :: 0:s -> cons:const -> cons:const
0 :: 0:s
incr :: cons:const -> cons:const
s :: 0:s -> 0:s
hd :: cons:const -> 0:s
tl :: cons:const -> cons:const
const :: cons:const

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   nats -> adx(cons(0, zeros)) [2]
   zeros -> cons(0, zeros) [1]
   incr(cons(X, Y)) -> cons(s(X), incr(Y)) [1]
   adx(cons(X, cons(X', Y'))) -> incr(cons(X, incr(cons(X', adx(Y'))))) [2]
   adx(cons(X, Y)) -> incr(cons(X, const)) [1]
   hd(cons(X, Y)) -> X [1]
   tl(cons(X, Y)) -> Y [1]
   adx(v0) -> const [0]
   incr(v0) -> const [0]

The TRS has the following type information:
nats :: cons:const
adx :: cons:const -> cons:const
zeros :: cons:const
cons :: 0:s -> cons:const -> cons:const
0 :: 0:s
incr :: cons:const -> cons:const
s :: 0:s -> 0:s
hd :: cons:const -> 0:s
tl :: cons:const -> cons:const
const :: cons:const

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

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

   0 => 0
   const => 0

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 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(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 


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

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

   { incr }
   { zeros }
   { tl }
   { hd }
   { adx }
   { nats }

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {zeros}, {tl}, {hd}, {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(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {zeros}, {tl}, {hd}, {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(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {incr}, {zeros}, {tl}, {hd}, {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: z

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

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

   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 1 }-> incr(1 + X + 0) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Function symbols to be analyzed: {zeros}, {tl}, {hd}, {adx}, {nats}
Previous analysis results are:
  incr: runtime: O(n^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) -{ 2 + X }-> s' :|: s' >= 0, s' <= 2 * (1 + X + 0), Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 + Y }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * Y, Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

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

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

(27) 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

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

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

   adx(z) -{ 2 + X }-> s' :|: s' >= 0, s' <= 2 * (1 + X + 0), Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 + Y }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * Y, Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

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

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

(29) 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: ?

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

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

   adx(z) -{ 2 + X }-> s' :|: s' >= 0, s' <= 2 * (1 + X + 0), Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 2 }-> incr(1 + X + incr(1 + X' + adx(Y'))) :|: Y' >= 0, X >= 0, z = 1 + X + (1 + X' + Y'), X' >= 0
   adx(z) -{ 0 }-> 0 :|: z >= 0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 1 + Y }-> 1 + (1 + X) + s :|: s >= 0, s <= 2 * Y, Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 2 }-> adx(1 + 0 + zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

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

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

(31) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   incr(v0) -> null_incr [0]
   adx(v0) -> null_adx [0]
   hd(v0) -> null_hd [0]
   tl(v0) -> null_tl [0]

And the following fresh constants:    null_incr, null_adx, null_hd, null_tl

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

(32)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   nats -> adx(zeros) [1]
   zeros -> cons(0, zeros) [1]
   incr(cons(X, Y)) -> cons(s(X), incr(Y)) [1]
   adx(cons(X, Y)) -> incr(cons(X, adx(Y))) [1]
   hd(cons(X, Y)) -> X [1]
   tl(cons(X, Y)) -> Y [1]
   incr(v0) -> null_incr [0]
   adx(v0) -> null_adx [0]
   hd(v0) -> null_hd [0]
   tl(v0) -> null_tl [0]

The TRS has the following type information:
nats :: cons:null_incr:null_adx:null_tl
adx :: cons:null_incr:null_adx:null_tl -> cons:null_incr:null_adx:null_tl
zeros :: cons:null_incr:null_adx:null_tl
cons :: 0:s:null_hd -> cons:null_incr:null_adx:null_tl -> cons:null_incr:null_adx:null_tl
0 :: 0:s:null_hd
incr :: cons:null_incr:null_adx:null_tl -> cons:null_incr:null_adx:null_tl
s :: 0:s:null_hd -> 0:s:null_hd
hd :: cons:null_incr:null_adx:null_tl -> 0:s:null_hd
tl :: cons:null_incr:null_adx:null_tl -> cons:null_incr:null_adx:null_tl
null_incr :: cons:null_incr:null_adx:null_tl
null_adx :: cons:null_incr:null_adx:null_tl
null_hd :: 0:s:null_hd
null_tl :: cons:null_incr:null_adx:null_tl

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

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

   0 => 0
   null_incr => 0
   null_adx => 0
   null_hd => 0
   null_tl => 0

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

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

   adx(z) -{ 1 }-> incr(1 + X + adx(Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0
   adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   hd(z) -{ 1 }-> X :|: Y >= 0, z = 1 + X + Y, X >= 0
   hd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 1 }-> 1 + (1 + X) + incr(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0
   nats -{ 1 }-> adx(zeros) :|: 
   tl(z) -{ 1 }-> Y :|: Y >= 0, z = 1 + X + Y, X >= 0
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   zeros -{ 1 }-> 1 + 0 + zeros :|: 

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5
S tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   HD(cons(z0, z1)) -> c4
   TL(cons(z0, z1)) -> c5
K tuples:none
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   NATS, ZEROS, INCR_1, ADX_1, HD_1, TL_1

Compound Symbols:   c_2, c1_1, c2_1, c3_2, c4, c5


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   TL(cons(z0, z1)) -> c5
   HD(cons(z0, z1)) -> c4

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
S tuples:
   NATS -> c(ADX(zeros), ZEROS)
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:none
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   NATS, ZEROS, INCR_1, ADX_1

Compound Symbols:   c_2, c1_1, c2_1, c3_2


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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
   NATS -> c4(ZEROS)
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
   NATS -> c4(ZEROS)
K tuples:none
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(41) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   NATS -> c4(ZEROS)

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
K tuples:none
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(43) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   NATS -> c4(ADX(zeros))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> adx(zeros)
   zeros -> cons(0, zeros)
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   nats, zeros, incr_1, adx_1, hd_1, tl_1

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(zeros))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NATS -> c4(ADX(zeros)) by 
   NATS -> c4(ADX(cons(0, zeros)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(cons(0, zeros)))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace NATS -> c4(ADX(cons(0, zeros))) by 
   NATS -> c4(ADX(cons(0, cons(0, zeros))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(cons(0, cons(0, zeros))))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(51) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NATS -> c4(ADX(cons(0, cons(0, zeros)))) by NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros)))))
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros)))))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, z1)) -> c2(INCR(z1))
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, z1)) -> c2(INCR(z1)) by 
   INCR(cons(z0, cons(y0, y1))) -> c2(INCR(cons(y0, y1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c2(INCR(cons(y0, y1)))
S tuples:
   ZEROS -> c1(ZEROS)
   ADX(cons(z0, z1)) -> c3(INCR(cons(z0, adx(z1))), ADX(z1))
   INCR(cons(z0, cons(y0, y1))) -> c2(INCR(cons(y0, y1)))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c1_1, c3_2, c4_1, c2_1


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

(55) 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, cons(z0, z1))) -> c3(INCR(cons(x0, incr(cons(z0, adx(z1))))), ADX(cons(z0, z1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c2(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 -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(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:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c1_1, c4_1, c2_1, c3_2


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

(57) 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, cons(z0, z1)))) -> c3(INCR(cons(x0, incr(cons(x1, incr(cons(z0, adx(z1))))))), ADX(cons(x1, cons(z0, z1))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros)))))
   INCR(cons(z0, cons(y0, y1))) -> c2(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))))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(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:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, NATS, INCR_1, ADX_1

Compound Symbols:   c1_1, c4_1, c2_1, c3_2


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

(59) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace NATS -> c4(ADX(cons(0, cons(0, cons(0, zeros))))) by NATS -> c4(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(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))))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
S tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(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:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(61) 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))))
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(INCR(cons(y0, y1)))
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   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 -> c1(ZEROS)
   INCR(cons(z0, cons(y0, y1))) -> c2(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(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:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, INCR_1, ADX_1, NATS

Compound Symbols:   c1_1, c2_1, c3_2, c4_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, cons(y0, y1))) -> c2(INCR(cons(y0, y1))) by 
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c2(INCR(cons(z1, cons(y1, y2))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   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))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c2(INCR(cons(z1, cons(y1, y2))))
S tuples:
   ZEROS -> c1(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   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))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c2(INCR(cons(z1, cons(y1, y2))))
K tuples:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c1_1, c3_2, c4_1, c2_1


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(65) 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))))
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(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   adx(cons(z0, z1)) -> incr(cons(z0, adx(z1)))
   incr(cons(z0, z1)) -> cons(s(z0), incr(z1))
   zeros -> cons(0, zeros)
Tuples:
   ZEROS -> c1(ZEROS)
   ADX(cons(x0, cons(z0, x2))) -> c3(INCR(cons(x0, cons(s(z0), incr(adx(x2))))), ADX(cons(z0, x2)))
   NATS -> c4(ADX(cons(0, cons(0, cons(0, cons(0, zeros))))))
   INCR(cons(z0, cons(z1, cons(y1, y2)))) -> c2(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 -> c1(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)))) -> c2(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:
   NATS -> c4(ADX(zeros))
Defined Rule Symbols:   adx_1, incr_1, zeros

Defined Pair Symbols:   ZEROS, ADX_1, NATS, INCR_1

Compound Symbols:   c1_1, c3_2, c4_1, c2_1