KILLED
proof of input_jh1LYSIfo1.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 880 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
        (24) CpxRNTS
    (25) CompletionProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (32) CdtProblem
    (33) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (36) CdtProblem
    (37) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 109 ms]
    (42) CdtProblem
    (43) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 83 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) 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:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, 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:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0', Z) -> rnil
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0', Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0'))
   plus(0', Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0', Y) -> 0'
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, X)

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

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

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X)))
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, X)

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

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

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


The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X))) [1]
   2ndspos(0, Z) -> rnil [1]
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z)) [1]
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z)) [1]
   2ndsneg(0, Z) -> rnil [1]
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z)) [1]
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z)) [1]
   pi(X) -> 2ndspos(X, from(0)) [1]
   plus(0, Y) -> Y [1]
   plus(s(X), Y) -> s(plus(X, Y)) [1]
   times(0, Y) -> 0 [1]
   times(s(X), Y) -> plus(Y, times(X, Y)) [1]
   square(X) -> times(X, X) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   from(X) -> cons(X, from(s(X))) [1]
   2ndspos(0, Z) -> rnil [1]
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z)) [1]
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z)) [1]
   2ndsneg(0, Z) -> rnil [1]
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z)) [1]
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z)) [1]
   pi(X) -> 2ndspos(X, from(0)) [1]
   plus(0, Y) -> Y [1]
   plus(s(X), Y) -> s(plus(X, Y)) [1]
   times(0, Y) -> 0 [1]
   times(s(X), Y) -> plus(Y, times(X, Y)) [1]
   square(X) -> times(X, X) [1]

The TRS has the following type information:
from :: s:0 -> cons:cons2
cons :: s:0 -> cons:cons2 -> cons:cons2
s :: s:0 -> s:0
2ndspos :: s:0 -> cons:cons2 -> rnil:rcons
0 :: s:0
rnil :: rnil:rcons
cons2 :: s:0 -> cons:cons2 -> cons:cons2
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:0 -> posrecip:negrecip
2ndsneg :: s:0 -> cons:cons2 -> rnil:rcons
negrecip :: s:0 -> posrecip:negrecip
pi :: s:0 -> rnil:rcons
plus :: s:0 -> s:0 -> s:0
times :: s:0 -> s:0 -> s:0
square :: s:0 -> s:0

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:

   2ndspos_2
   2ndsneg_2
   pi_1
   square_1

(c) The following functions are completely defined:

   times_2
   from_1
   plus_2

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   from(X) -> cons(X, from(s(X))) [1]
   2ndspos(0, Z) -> rnil [1]
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z)) [1]
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z)) [1]
   2ndsneg(0, Z) -> rnil [1]
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z)) [1]
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z)) [1]
   pi(X) -> 2ndspos(X, from(0)) [1]
   plus(0, Y) -> Y [1]
   plus(s(X), Y) -> s(plus(X, Y)) [1]
   times(0, Y) -> 0 [1]
   times(s(X), Y) -> plus(Y, times(X, Y)) [1]
   square(X) -> times(X, X) [1]

The TRS has the following type information:
from :: s:0 -> cons:cons2
cons :: s:0 -> cons:cons2 -> cons:cons2
s :: s:0 -> s:0
2ndspos :: s:0 -> cons:cons2 -> rnil:rcons
0 :: s:0
rnil :: rnil:rcons
cons2 :: s:0 -> cons:cons2 -> cons:cons2
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:0 -> posrecip:negrecip
2ndsneg :: s:0 -> cons:cons2 -> rnil:rcons
negrecip :: s:0 -> posrecip:negrecip
pi :: s:0 -> rnil:rcons
plus :: s:0 -> s:0 -> s:0
times :: s:0 -> s:0 -> s:0
square :: s:0 -> s:0
const :: cons:cons2
const1 :: posrecip:negrecip

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:

   from(X) -> cons(X, from(s(X))) [1]
   2ndspos(0, Z) -> rnil [1]
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z)) [1]
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z)) [1]
   2ndsneg(0, Z) -> rnil [1]
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z)) [1]
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z)) [1]
   pi(X) -> 2ndspos(X, cons(0, from(s(0)))) [2]
   plus(0, Y) -> Y [1]
   plus(s(X), Y) -> s(plus(X, Y)) [1]
   times(0, Y) -> 0 [1]
   times(s(0), Y) -> plus(Y, 0) [2]
   times(s(s(X')), Y) -> plus(Y, plus(Y, times(X', Y))) [2]
   square(X) -> times(X, X) [1]

The TRS has the following type information:
from :: s:0 -> cons:cons2
cons :: s:0 -> cons:cons2 -> cons:cons2
s :: s:0 -> s:0
2ndspos :: s:0 -> cons:cons2 -> rnil:rcons
0 :: s:0
rnil :: rnil:rcons
cons2 :: s:0 -> cons:cons2 -> cons:cons2
rcons :: posrecip:negrecip -> rnil:rcons -> rnil:rcons
posrecip :: s:0 -> posrecip:negrecip
2ndsneg :: s:0 -> cons:cons2 -> rnil:rcons
negrecip :: s:0 -> posrecip:negrecip
pi :: s:0 -> rnil:rcons
plus :: s:0 -> s:0 -> s:0
times :: s:0 -> s:0 -> s:0
square :: s:0 -> s:0
const :: cons:cons2
const1 :: posrecip:negrecip

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
   rnil => 0
   const => 0
   const1 => 0

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

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

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + N, 1 + X + Z) :|: Z >= 0, z = 1 + N, X >= 0, z' = 1 + X + Z, N >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(N, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, z = 1 + N, X >= 0, N >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + N, 1 + X + Z) :|: Z >= 0, z = 1 + N, X >= 0, z' = 1 + X + Z, N >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(N, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, z = 1 + N, X >= 0, N >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   pi(z) -{ 2 }-> 2ndspos(X, 1 + 0 + from(1 + 0)) :|: X >= 0, z = X
   plus(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   square(z) -{ 1 }-> times(X, X) :|: X >= 0, z = X
   times(z, z') -{ 2 }-> plus(Y, plus(Y, times(X', Y))) :|: z' = Y, Y >= 0, X' >= 0, z = 1 + (1 + X')
   times(z, z') -{ 2 }-> plus(Y, 0) :|: z' = Y, Y >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0


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

(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:

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   pi(z) -{ 2 }-> 2ndspos(z, 1 + 0 + from(1 + 0)) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   square(z) -{ 1 }-> times(z, z) :|: z >= 0
   times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(z', 0) :|: z' >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


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

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

   { from }
   { 2ndspos, 2ndsneg }
   { plus }
   { pi }
   { times }
   { square }

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

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

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   pi(z) -{ 2 }-> 2ndspos(z, 1 + 0 + from(1 + 0)) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   square(z) -{ 1 }-> times(z, z) :|: z >= 0
   times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(z', 0) :|: z' >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndspos,2ndsneg}, {plus}, {pi}, {times}, {square}

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

(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:

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   pi(z) -{ 2 }-> 2ndspos(z, 1 + 0 + from(1 + 0)) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   square(z) -{ 1 }-> times(z, z) :|: z >= 0
   times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(z', 0) :|: z' >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndspos,2ndsneg}, {plus}, {pi}, {times}, {square}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   pi(z) -{ 2 }-> 2ndspos(z, 1 + 0 + from(1 + 0)) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   square(z) -{ 1 }-> times(z, z) :|: z >= 0
   times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(z', 0) :|: z' >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndspos,2ndsneg}, {plus}, {pi}, {times}, {square}
Previous analysis results are:
  from: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + (z - 1), 1 + X + Z) :|: Z >= 0, X >= 0, z' = 1 + X + Z, z - 1 >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(z - 1, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, X >= 0, z - 1 >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   pi(z) -{ 2 }-> 2ndspos(z, 1 + 0 + from(1 + 0)) :|: z >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z' >= 0, z - 1 >= 0
   square(z) -{ 1 }-> times(z, z) :|: z >= 0
   times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(z', 0) :|: z' >= 0, z = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndspos,2ndsneg}, {plus}, {pi}, {times}, {square}
Previous analysis results are:
  from: runtime: INF, size: O(1) [0]

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

(25) 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:

   2ndspos(v0, v1) -> null_2ndspos [0]
   2ndsneg(v0, v1) -> null_2ndsneg [0]

And the following fresh constants:    null_2ndspos, null_2ndsneg, const, const1

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

(26)
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:

   from(X) -> cons(X, from(s(X))) [1]
   2ndspos(0, Z) -> rnil [1]
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, Z)) [1]
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z)) [1]
   2ndsneg(0, Z) -> rnil [1]
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, Z)) [1]
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z)) [1]
   pi(X) -> 2ndspos(X, from(0)) [1]
   plus(0, Y) -> Y [1]
   plus(s(X), Y) -> s(plus(X, Y)) [1]
   times(0, Y) -> 0 [1]
   times(s(X), Y) -> plus(Y, times(X, Y)) [1]
   square(X) -> times(X, X) [1]
   2ndspos(v0, v1) -> null_2ndspos [0]
   2ndsneg(v0, v1) -> null_2ndsneg [0]

The TRS has the following type information:
from :: s:0 -> cons:cons2
cons :: s:0 -> cons:cons2 -> cons:cons2
s :: s:0 -> s:0
2ndspos :: s:0 -> cons:cons2 -> rnil:rcons:null_2ndspos:null_2ndsneg
0 :: s:0
rnil :: rnil:rcons:null_2ndspos:null_2ndsneg
cons2 :: s:0 -> cons:cons2 -> cons:cons2
rcons :: posrecip:negrecip -> rnil:rcons:null_2ndspos:null_2ndsneg -> rnil:rcons:null_2ndspos:null_2ndsneg
posrecip :: s:0 -> posrecip:negrecip
2ndsneg :: s:0 -> cons:cons2 -> rnil:rcons:null_2ndspos:null_2ndsneg
negrecip :: s:0 -> posrecip:negrecip
pi :: s:0 -> rnil:rcons:null_2ndspos:null_2ndsneg
plus :: s:0 -> s:0 -> s:0
times :: s:0 -> s:0 -> s:0
square :: s:0 -> s:0
null_2ndspos :: rnil:rcons:null_2ndspos:null_2ndsneg
null_2ndsneg :: rnil:rcons:null_2ndspos:null_2ndsneg
const :: cons:cons2
const1 :: posrecip:negrecip

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

(27) 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
   rnil => 0
   null_2ndspos => 0
   null_2ndsneg => 0
   const => 0
   const1 => 0

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

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

   2ndsneg(z, z') -{ 1 }-> 2ndsneg(1 + N, 1 + X + Z) :|: Z >= 0, z = 1 + N, X >= 0, z' = 1 + X + Z, N >= 0
   2ndsneg(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0
   2ndsneg(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   2ndsneg(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndspos(N, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, z = 1 + N, X >= 0, N >= 0
   2ndspos(z, z') -{ 1 }-> 2ndspos(1 + N, 1 + X + Z) :|: Z >= 0, z = 1 + N, X >= 0, z' = 1 + X + Z, N >= 0
   2ndspos(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0
   2ndspos(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   2ndspos(z, z') -{ 1 }-> 1 + (1 + Y) + 2ndsneg(N, Z) :|: Z >= 0, z' = 1 + X + (1 + Y + Z), Y >= 0, z = 1 + N, X >= 0, N >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   pi(z) -{ 1 }-> 2ndspos(X, from(0)) :|: X >= 0, z = X
   plus(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   square(z) -{ 1 }-> times(X, X) :|: X >= 0, z = X
   times(z, z') -{ 1 }-> plus(Y, times(X, Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   times(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   square(z0) -> times(z0, z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(0, z0) -> c1
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(0, z0) -> c4
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PI(z0) -> c7(2NDSPOS(z0, from(0)), FROM(0))
   PLUS(0, z0) -> c8
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(0, z0) -> c10
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   SQUARE(z0) -> c12(TIMES(z0, z0))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(0, z0) -> c1
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(0, z0) -> c4
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PI(z0) -> c7(2NDSPOS(z0, from(0)), FROM(0))
   PLUS(0, z0) -> c8
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(0, z0) -> c10
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   SQUARE(z0) -> c12(TIMES(z0, z0))
K tuples:none
Defined Rule Symbols:   from_1, 2ndspos_2, 2ndsneg_2, pi_1, plus_2, times_2, square_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PI_1, PLUS_2, TIMES_2, SQUARE_1

Compound Symbols:   c_1, c1, c2_1, c3_1, c4, c5_1, c6_1, c7_2, c8, c9_1, c10, c11_2, c12_1


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

(31) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SQUARE(z0) -> c12(TIMES(z0, z0))
Removed 4 trailing nodes:
   PLUS(0, z0) -> c8
   2NDSPOS(0, z0) -> c1
   2NDSNEG(0, z0) -> c4
   TIMES(0, z0) -> c10

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   square(z0) -> times(z0, z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PI(z0) -> c7(2NDSPOS(z0, from(0)), FROM(0))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PI(z0) -> c7(2NDSPOS(z0, from(0)), FROM(0))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
K tuples:none
Defined Rule Symbols:   from_1, 2ndspos_2, 2ndsneg_2, pi_1, plus_2, times_2, square_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PI_1, PLUS_2, TIMES_2

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c7_2, c9_1, c11_2


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

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   square(z0) -> times(z0, z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   PI(z0) -> c1(FROM(0))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   PI(z0) -> c1(FROM(0))
K tuples:none
Defined Rule Symbols:   from_1, 2ndspos_2, 2ndsneg_2, pi_1, plus_2, times_2, square_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(35) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   PI(z0) -> c1(FROM(0))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   square(z0) -> times(z0, z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
K tuples:none
Defined Rule Symbols:   from_1, 2ndspos_2, 2ndsneg_2, pi_1, plus_2, times_2, square_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(37) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, from(s(z0)))
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   square(z0) -> times(z0, z0)
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
Defined Rule Symbols:   from_1, 2ndspos_2, 2ndsneg_2, pi_1, plus_2, times_2, square_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   2ndspos(0, z0) -> rnil
   2ndspos(s(z0), cons(z1, z2)) -> 2ndspos(s(z0), cons2(z1, z2))
   2ndspos(s(z0), cons2(z1, cons(z2, z3))) -> rcons(posrecip(z2), 2ndsneg(z0, z3))
   2ndsneg(0, z0) -> rnil
   2ndsneg(s(z0), cons(z1, z2)) -> 2ndsneg(s(z0), cons2(z1, z2))
   2ndsneg(s(z0), cons2(z1, cons(z2, z3))) -> rcons(negrecip(z2), 2ndspos(z0, z3))
   pi(z0) -> 2ndspos(z0, from(0))
   square(z0) -> times(z0, z0)

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
We considered the (Usable) Rules:
   from(z0) -> cons(z0, from(s(z0)))
And the Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(2NDSNEG(x_1, x_2)) = [1] + x_1 + x_2
   POL(2NDSPOS(x_1, x_2)) = [1] + x_1 + x_2
   POL(FROM(x_1)) = 0
   POL(PI(x_1)) = [1] + x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = x_1
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(cons2(x_1, x_2)) = x_2
   POL(from(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [1] + x_1 + x_2

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(43) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(2NDSNEG(x_1, x_2)) = 0
   POL(2NDSPOS(x_1, x_2)) = 0
   POL(FROM(x_1)) = 0
   POL(PI(x_1)) = [1] + [2]x_1^2
   POL(PLUS(x_1, x_2)) = [2]x_1
   POL(TIMES(x_1, x_2)) = x_1*x_2
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = 0
   POL(cons2(x_1, x_2)) = 0
   POL(from(x_1)) = 0
   POL(plus(x_1, x_2)) = [2] + x_1^2
   POL(s(x_1)) = [2] + x_1
   POL(times(x_1, x_2)) = [2]x_2^2

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PI(z0) -> c1(2NDSPOS(z0, from(0))) by 
   PI(x0) -> c1(2NDSPOS(x0, cons(0, from(s(0)))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, from(s(0)))))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PI(x0) -> c1(2NDSPOS(x0, cons(0, from(s(0))))) by 
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c(FROM(s(z0)))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
S tuples:
   FROM(z0) -> c(FROM(s(z0)))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1

Compound Symbols:   c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1


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

(51) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c(FROM(s(z0))) by 
   FROM(s(x0)) -> c(FROM(s(s(x0))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
   FROM(s(x0)) -> c(FROM(s(s(x0))))
S tuples:
   FROM(s(x0)) -> c(FROM(s(s(x0))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1, FROM_1

Compound Symbols:   c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1, c_1


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

(53) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1, FROM_1

Compound Symbols:   c2_1, c3_1, c5_1, c6_1, c9_1, c11_2, c1_1, c_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSPOS(s(z0), cons(z1, z2)) -> c2(2NDSPOS(s(z0), cons2(z1, z2))) by 
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSPOS_2, 2NDSNEG_2, PLUS_2, TIMES_2, PI_1, FROM_1

Compound Symbols:   c3_1, c5_1, c6_1, c9_1, c11_2, c1_1, c_1, c2_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) -> c3(2NDSNEG(z0, z3)) by 
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2)))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSNEG_2, PLUS_2, TIMES_2, PI_1, FROM_1, 2NDSPOS_2

Compound Symbols:   c5_1, c6_1, c9_1, c11_2, c1_1, c_1, c2_1, c3_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(z0), cons(z1, z2)) -> c5(2NDSNEG(s(z0), cons2(z1, z2))) by 
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0)))))))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSNEG_2, PLUS_2, TIMES_2, PI_1, FROM_1, 2NDSPOS_2

Compound Symbols:   c6_1, c9_1, c11_2, c1_1, c_1, c2_1, c3_1, c5_1


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

(61) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PI(x0) -> c1(2NDSPOS(x0, cons(0, cons(s(0), from(s(s(0))))))) by PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), from(s(s(s(0)))))))))
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), from(s(s(s(0)))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   2NDSNEG_2, PLUS_2, TIMES_2, FROM_1, 2NDSPOS_2, PI_1

Compound Symbols:   c6_1, c9_1, c11_2, c_1, c2_1, c3_1, c5_1, c1_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) -> c6(2NDSPOS(z0, z3)) by 
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), from(s(s(s(0)))))))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c9(PLUS(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   PLUS_2, TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PI_1

Compound Symbols:   c9_1, c11_2, c_1, c2_1, c3_1, c5_1, c1_1, c6_1


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

(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c9(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), from(s(s(s(0)))))))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PI_1, PLUS_2

Compound Symbols:   c11_2, c_1, c2_1, c3_1, c5_1, c1_1, c6_1, c9_1


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), from(s(s(s(0))))))))) by PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, PI_1

Compound Symbols:   c11_2, c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c1_1


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

(69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSPOS(s(z0), cons(z1, cons(y2, y3))) -> c2(2NDSPOS(s(z0), cons2(z1, cons(y2, y3)))) by 
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2)))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, PI_1

Compound Symbols:   c11_2, c_1, c3_1, c5_1, c6_1, c9_1, c1_1, c2_1


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y1, y2)))) -> c3(2NDSNEG(s(y0), cons(y1, y2))) by 
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, PI_1

Compound Symbols:   c11_2, c_1, c3_1, c5_1, c6_1, c9_1, c1_1, c2_1


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

(73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y1, cons(y2, y3))))) -> c3(2NDSNEG(s(y0), cons2(y1, cons(y2, y3)))) by 
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSNEG_2, PLUS_2, PI_1, 2NDSPOS_2

Compound Symbols:   c11_2, c_1, c5_1, c6_1, c9_1, c1_1, c2_1, c3_1


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

(75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(z0), cons(z1, cons(y2, y3))) -> c5(2NDSNEG(s(z0), cons2(z1, cons(y2, y3)))) by 
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSNEG_2, PLUS_2, PI_1, 2NDSPOS_2

Compound Symbols:   c11_2, c_1, c6_1, c9_1, c1_1, c2_1, c3_1, c5_1


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

(77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y1, cons(y2, y3))))) -> c6(2NDSPOS(s(y0), cons(y1, cons(y2, y3)))) by 
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSNEG_2, PLUS_2, PI_1, 2NDSPOS_2

Compound Symbols:   c11_2, c_1, c6_1, c9_1, c1_1, c2_1, c3_1, c5_1


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

(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons(y3, y4))))) by 
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSNEG_2, PLUS_2, PI_1, 2NDSPOS_2

Compound Symbols:   c11_2, c_1, c6_1, c9_1, c1_1, c2_1, c3_1, c5_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace 2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y1, cons(y2, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons2(y1, cons(y2, cons2(y3, cons(y4, y5)))))) by 
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, PLUS_2, PI_1, 2NDSPOS_2, 2NDSNEG_2

Compound Symbols:   c11_2, c_1, c9_1, c1_1, c2_1, c3_1, c5_1, c6_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c9(PLUS(s(y0), z1)) by 
   PLUS(s(s(s(y0))), z1) -> c9(PLUS(s(s(y0)), z1))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
   PLUS(s(s(s(y0))), z1) -> c9(PLUS(s(s(y0)), z1))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
   PLUS(s(s(s(y0))), z1) -> c9(PLUS(s(s(y0)), z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, PI_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2

Compound Symbols:   c11_2, c_1, c1_1, c2_1, c3_1, c5_1, c6_1, c9_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PI(z0) -> c1(2NDSPOS(z0, cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0))))))))))) by 
   PI(s(s(y0))) -> c1(2NDSPOS(s(s(y0)), cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
   PLUS(s(s(s(y0))), z1) -> c9(PLUS(s(s(y0)), z1))
   PI(s(s(y0))) -> c1(2NDSPOS(s(s(y0)), cons(0, cons(s(0), cons(s(s(0)), cons(s(s(s(0))), from(s(s(s(s(0)))))))))))
S tuples:
   FROM(s(s(x0))) -> c(FROM(s(s(s(x0)))))
K tuples:
   PI(z0) -> c1(2NDSPOS(z0, from(0)))
   TIMES(s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons(y3, y4)))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons(y3, y4)))))
   2NDSPOS(s(s(y0)), cons(z1, cons(z2, cons2(y3, cons(y4, y5))))) -> c2(2NDSPOS(s(s(y0)), cons2(z1, cons(z2, cons2(y3, cons(y4, y5))))))
   2NDSPOS(s(s(z0)), cons2(z1, cons(z2, cons(z3, cons(y2, y3))))) -> c3(2NDSNEG(s(z0), cons(z3, cons(y2, y3))))
   2NDSPOS(s(s(s(y0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))) -> c3(2NDSNEG(s(s(y0)), cons2(z3, cons(z4, cons(y3, cons(y4, y5))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSPOS(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))) -> c3(2NDSNEG(s(s(s(y0))), cons2(z3, cons(z4, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(y0)), cons(z1, cons(z2, cons(y3, cons(y4, y5))))) -> c5(2NDSNEG(s(s(y0)), cons2(z1, cons(z2, cons(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons(y5, y6)))))))
   2NDSNEG(s(s(s(y0))), cons(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))) -> c5(2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons2(y3, cons(y4, cons2(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons(y3, y4)))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons(y3, y4)))))
   2NDSNEG(s(s(s(y0))), cons2(z1, cons(z2, cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(y0)), cons(z3, cons(z4, cons2(y3, cons(y4, y5))))))
   2NDSNEG(s(s(s(z0))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))) -> c6(2NDSPOS(s(s(z0)), cons2(z3, cons(z4, cons(z5, cons(y4, y5))))))
   2NDSNEG(s(s(s(s(y0)))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))) -> c6(2NDSPOS(s(s(s(y0))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons(y5, cons(y6, y7))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons(y7, y8)))))))))
   2NDSNEG(s(s(s(s(s(y0))))), cons2(z1, cons(z2, cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))) -> c6(2NDSPOS(s(s(s(s(y0)))), cons2(z3, cons(z4, cons2(z5, cons(z6, cons2(y5, cons(y6, cons2(y7, cons(y8, y9))))))))))
   PLUS(s(s(s(y0))), z1) -> c9(PLUS(s(s(y0)), z1))
Defined Rule Symbols:   times_2, plus_2, from_1

Defined Pair Symbols:   TIMES_2, FROM_1, 2NDSPOS_2, 2NDSNEG_2, PLUS_2, PI_1

Compound Symbols:   c11_2, c_1, c2_1, c3_1, c5_1, c6_1, c9_1, c1_1