KILLED
proof of input_OZpwPJ9H5i.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (10) TRS for Loop Detection
    (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRelTRS
        (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) typed CpxTrs
        (15) OrderProof [LOWER BOUND(ID), 1 ms]
        (16) typed CpxTrs
(17) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(18) CpxWeightedTrs
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxTypedWeightedTrs
    (21) CompletionProof [UPPER BOUND(ID), 0 ms]
    (22) CpxTypedWeightedCompleteTrs
        (23) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxTypedWeightedCompleteTrs
        (25) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) InliningProof [UPPER BOUND(ID), 118 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CpxRelTRS
        (45) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxTRS
        (47) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CpxWeightedTrs
            (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (50) CpxTypedWeightedTrs
            (51) CompletionProof [UPPER BOUND(ID), 0 ms]
            (52) CpxTypedWeightedCompleteTrs
                (53) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (54) CpxTypedWeightedCompleteTrs
                (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (56) CpxRNTS
                (57) InliningProof [UPPER BOUND(ID), 75 ms]
                (58) CpxRNTS
                (59) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (60) CpxRNTS
            (61) CompletionProof [UPPER BOUND(ID), 0 ms]
            (62) CpxTypedWeightedCompleteTrs
                (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (64) CpxRNTS
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
        (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
        (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (76) CdtProblem
        (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CdtProblem
        (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (80) CdtProblem
        (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CdtProblem
        (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CdtProblem
        (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CdtProblem
        (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (88) CdtProblem
        (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (90) CdtProblem
        (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (92) CdtProblem
        (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (94) CdtProblem
        (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (96) CdtProblem
        (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 8 ms]
        (98) CdtProblem
        (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   nats -> cons(0, n__incr(n__nats))
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   nats -> n__nats
   odds -> n__odds
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(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:

   nats -> cons(0', n__incr(n__nats))
   pairs -> cons(0', n__incr(n__odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   nats -> n__nats
   odds -> n__odds
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(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:

   nats -> cons(0, n__incr(n__nats))
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS)))
   head(cons(X, XS)) -> X
   tail(cons(X, XS)) -> activate(XS)
   incr(X) -> n__incr(X)
   nats -> n__nats
   odds -> n__odds
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(X) -> X

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
Tuples:
   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12
S tuples:
   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12
K tuples:none
Defined Rule Symbols:   nats, pairs, odds, incr_1, head_1, tail_1, activate_1

Defined Pair Symbols:   NATS, PAIRS, ODDS, INCR_1, HEAD_1, TAIL_1, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4, c5_1, c6, c7, c8_1, c9_2, c10_1, c11_1, c12


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

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

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


The TRS R consists of the following rules:

   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12

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

   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0

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

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

(10)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12

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

   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12

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

   nats -> cons(0', n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0', n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0

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

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

(14)
Obligation:
Innermost TRS:
Rules:
NATS -> c
NATS -> c1
PAIRS -> c2
ODDS -> c3(INCR(pairs), PAIRS)
ODDS -> c4
INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
INCR(z0) -> c6
HEAD(cons(z0, z1)) -> c7
TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__nats) -> c10(NATS)
ACTIVATE(n__odds) -> c11(ODDS)
ACTIVATE(z0) -> c12
nats -> cons(0', n__incr(n__nats))
nats -> n__nats
pairs -> cons(0', n__incr(n__odds))
odds -> incr(pairs)
odds -> n__odds
incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
incr(z0) -> n__incr(z0)
head(cons(z0, z1)) -> z0
tail(cons(z0, z1)) -> activate(z1)
activate(n__incr(z0)) -> incr(activate(z0))
activate(n__nats) -> nats
activate(n__odds) -> odds
activate(z0) -> z0

Types:
NATS :: c:c1
c :: c:c1
c1 :: c:c1
PAIRS :: c2
c2 :: c2
ODDS :: c3:c4
c3 :: c5:c6 -> c2 -> c3:c4
INCR :: cons:n__incr:n__nats:n__odds -> c5:c6
pairs :: cons:n__incr:n__nats:n__odds
c4 :: c3:c4
cons :: 0':s -> cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
c5 :: c9:c10:c11:c12 -> c5:c6
ACTIVATE :: cons:n__incr:n__nats:n__odds -> c9:c10:c11:c12
c6 :: c5:c6
HEAD :: cons:n__incr:n__nats:n__odds -> c7
c7 :: c7
TAIL :: cons:n__incr:n__nats:n__odds -> c8
c8 :: c9:c10:c11:c12 -> c8
n__incr :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
c9 :: c5:c6 -> c9:c10:c11:c12 -> c9:c10:c11:c12
activate :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
n__nats :: cons:n__incr:n__nats:n__odds
c10 :: c:c1 -> c9:c10:c11:c12
n__odds :: cons:n__incr:n__nats:n__odds
c11 :: c3:c4 -> c9:c10:c11:c12
c12 :: c9:c10:c11:c12
nats :: cons:n__incr:n__nats:n__odds
0' :: 0':s
odds :: cons:n__incr:n__nats:n__odds
incr :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
s :: 0':s -> 0':s
head :: cons:n__incr:n__nats:n__odds -> 0':s
tail :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
hole_c:c11_13 :: c:c1
hole_c22_13 :: c2
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_cons:n__incr:n__nats:n__odds5_13 :: cons:n__incr:n__nats:n__odds
hole_0':s6_13 :: 0':s
hole_c9:c10:c11:c127_13 :: c9:c10:c11:c12
hole_c78_13 :: c7
hole_c89_13 :: c8
gen_cons:n__incr:n__nats:n__odds10_13 :: Nat -> cons:n__incr:n__nats:n__odds
gen_0':s11_13 :: Nat -> 0':s
gen_c9:c10:c11:c1212_13 :: Nat -> c9:c10:c11:c12

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ODDS, INCR, ACTIVATE, activate, odds, incr

They will be analysed ascendingly in the following order:
ODDS = INCR
ODDS = ACTIVATE
INCR = ACTIVATE
activate < ACTIVATE
activate = odds
activate = incr
odds = incr

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

(16)
Obligation:
Innermost TRS:
Rules:
NATS -> c
NATS -> c1
PAIRS -> c2
ODDS -> c3(INCR(pairs), PAIRS)
ODDS -> c4
INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
INCR(z0) -> c6
HEAD(cons(z0, z1)) -> c7
TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__nats) -> c10(NATS)
ACTIVATE(n__odds) -> c11(ODDS)
ACTIVATE(z0) -> c12
nats -> cons(0', n__incr(n__nats))
nats -> n__nats
pairs -> cons(0', n__incr(n__odds))
odds -> incr(pairs)
odds -> n__odds
incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
incr(z0) -> n__incr(z0)
head(cons(z0, z1)) -> z0
tail(cons(z0, z1)) -> activate(z1)
activate(n__incr(z0)) -> incr(activate(z0))
activate(n__nats) -> nats
activate(n__odds) -> odds
activate(z0) -> z0

Types:
NATS :: c:c1
c :: c:c1
c1 :: c:c1
PAIRS :: c2
c2 :: c2
ODDS :: c3:c4
c3 :: c5:c6 -> c2 -> c3:c4
INCR :: cons:n__incr:n__nats:n__odds -> c5:c6
pairs :: cons:n__incr:n__nats:n__odds
c4 :: c3:c4
cons :: 0':s -> cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
c5 :: c9:c10:c11:c12 -> c5:c6
ACTIVATE :: cons:n__incr:n__nats:n__odds -> c9:c10:c11:c12
c6 :: c5:c6
HEAD :: cons:n__incr:n__nats:n__odds -> c7
c7 :: c7
TAIL :: cons:n__incr:n__nats:n__odds -> c8
c8 :: c9:c10:c11:c12 -> c8
n__incr :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
c9 :: c5:c6 -> c9:c10:c11:c12 -> c9:c10:c11:c12
activate :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
n__nats :: cons:n__incr:n__nats:n__odds
c10 :: c:c1 -> c9:c10:c11:c12
n__odds :: cons:n__incr:n__nats:n__odds
c11 :: c3:c4 -> c9:c10:c11:c12
c12 :: c9:c10:c11:c12
nats :: cons:n__incr:n__nats:n__odds
0' :: 0':s
odds :: cons:n__incr:n__nats:n__odds
incr :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
s :: 0':s -> 0':s
head :: cons:n__incr:n__nats:n__odds -> 0':s
tail :: cons:n__incr:n__nats:n__odds -> cons:n__incr:n__nats:n__odds
hole_c:c11_13 :: c:c1
hole_c22_13 :: c2
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_cons:n__incr:n__nats:n__odds5_13 :: cons:n__incr:n__nats:n__odds
hole_0':s6_13 :: 0':s
hole_c9:c10:c11:c127_13 :: c9:c10:c11:c12
hole_c78_13 :: c7
hole_c89_13 :: c8
gen_cons:n__incr:n__nats:n__odds10_13 :: Nat -> cons:n__incr:n__nats:n__odds
gen_0':s11_13 :: Nat -> 0':s
gen_c9:c10:c11:c1212_13 :: Nat -> c9:c10:c11:c12


Generator Equations:
gen_cons:n__incr:n__nats:n__odds10_13(0) <=> n__nats
gen_cons:n__incr:n__nats:n__odds10_13(+(x, 1)) <=> cons(0', gen_cons:n__incr:n__nats:n__odds10_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c9:c10:c11:c1212_13(0) <=> c10(c)
gen_c9:c10:c11:c1212_13(+(x, 1)) <=> c9(c5(c10(c)), gen_c9:c10:c11:c1212_13(x))


The following defined symbols remain to be analysed:
incr, ODDS, INCR, ACTIVATE, activate, odds

They will be analysed ascendingly in the following order:
ODDS = INCR
ODDS = ACTIVATE
INCR = ACTIVATE
activate < ACTIVATE
activate = odds
activate = incr
odds = incr

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   nats -> cons(0, n__incr(n__nats)) [1]
   pairs -> cons(0, n__incr(n__odds)) [1]
   odds -> incr(pairs) [1]
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) [1]
   head(cons(X, XS)) -> X [1]
   tail(cons(X, XS)) -> activate(XS) [1]
   incr(X) -> n__incr(X) [1]
   nats -> n__nats [1]
   odds -> n__odds [1]
   activate(n__incr(X)) -> incr(activate(X)) [1]
   activate(n__nats) -> nats [1]
   activate(n__odds) -> odds [1]
   activate(X) -> X [1]

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

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

(20)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   nats -> cons(0, n__incr(n__nats)) [1]
   pairs -> cons(0, n__incr(n__odds)) [1]
   odds -> incr(pairs) [1]
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) [1]
   head(cons(X, XS)) -> X [1]
   tail(cons(X, XS)) -> activate(XS) [1]
   incr(X) -> n__incr(X) [1]
   nats -> n__nats [1]
   odds -> n__odds [1]
   activate(n__incr(X)) -> incr(activate(X)) [1]
   activate(n__nats) -> nats [1]
   activate(n__odds) -> odds [1]
   activate(X) -> X [1]

The TRS has the following type information:
nats :: n__nats:n__incr:cons:n__odds
cons :: 0:s -> n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
0 :: 0:s
n__incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
n__nats :: n__nats:n__incr:cons:n__odds
pairs :: n__nats:n__incr:cons:n__odds
n__odds :: n__nats:n__incr:cons:n__odds
odds :: n__nats:n__incr:cons:n__odds
incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
s :: 0:s -> 0:s
activate :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
head :: n__nats:n__incr:cons:n__odds -> 0:s
tail :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds

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

(21) 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:

   head_1
   tail_1

(c) The following functions are completely defined:

   pairs
   activate_1
   incr_1
   odds
   nats

Due to the following rules being added:
none

And the following fresh constants: none

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

(22)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   nats -> cons(0, n__incr(n__nats)) [1]
   pairs -> cons(0, n__incr(n__odds)) [1]
   odds -> incr(pairs) [1]
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) [1]
   head(cons(X, XS)) -> X [1]
   tail(cons(X, XS)) -> activate(XS) [1]
   incr(X) -> n__incr(X) [1]
   nats -> n__nats [1]
   odds -> n__odds [1]
   activate(n__incr(X)) -> incr(activate(X)) [1]
   activate(n__nats) -> nats [1]
   activate(n__odds) -> odds [1]
   activate(X) -> X [1]

The TRS has the following type information:
nats :: n__nats:n__incr:cons:n__odds
cons :: 0:s -> n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
0 :: 0:s
n__incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
n__nats :: n__nats:n__incr:cons:n__odds
pairs :: n__nats:n__incr:cons:n__odds
n__odds :: n__nats:n__incr:cons:n__odds
odds :: n__nats:n__incr:cons:n__odds
incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
s :: 0:s -> 0:s
activate :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
head :: n__nats:n__incr:cons:n__odds -> 0:s
tail :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds

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

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

(24)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   nats -> cons(0, n__incr(n__nats)) [1]
   pairs -> cons(0, n__incr(n__odds)) [1]
   odds -> incr(cons(0, n__incr(n__odds))) [2]
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) [1]
   head(cons(X, XS)) -> X [1]
   tail(cons(X, XS)) -> activate(XS) [1]
   incr(X) -> n__incr(X) [1]
   nats -> n__nats [1]
   odds -> n__odds [1]
   activate(n__incr(n__incr(X'))) -> incr(incr(activate(X'))) [2]
   activate(n__incr(n__nats)) -> incr(nats) [2]
   activate(n__incr(n__odds)) -> incr(odds) [2]
   activate(n__incr(X)) -> incr(X) [2]
   activate(n__nats) -> nats [1]
   activate(n__odds) -> odds [1]
   activate(X) -> X [1]

The TRS has the following type information:
nats :: n__nats:n__incr:cons:n__odds
cons :: 0:s -> n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
0 :: 0:s
n__incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
n__nats :: n__nats:n__incr:cons:n__odds
pairs :: n__nats:n__incr:cons:n__odds
n__odds :: n__nats:n__incr:cons:n__odds
odds :: n__nats:n__incr:cons:n__odds
incr :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
s :: 0:s -> 0:s
activate :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds
head :: n__nats:n__incr:cons:n__odds -> 0:s
tail :: n__nats:n__incr:cons:n__odds -> n__nats:n__incr:cons:n__odds

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

(25) 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
   n__nats => 0
   n__odds => 1

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> odds :|: z = 1
   activate(z) -{ 1 }-> nats :|: z = 0
   activate(z) -{ 2 }-> incr(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 2 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 2 }-> incr(nats) :|: z = 1 + 0
   activate(z) -{ 2 }-> incr(incr(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
   head(z) -{ 1 }-> X :|: z = 1 + X + XS, X >= 0, XS >= 0
   incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   incr(z) -{ 1 }-> 1 + (1 + X) + (1 + activate(XS)) :|: z = 1 + X + XS, X >= 0, XS >= 0
   nats -{ 1 }-> 0 :|: 
   nats -{ 1 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 2 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 1 }-> 1 :|: 
   pairs -{ 1 }-> 1 + 0 + (1 + 1) :|: 
   tail(z) -{ 1 }-> activate(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0


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

(27) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   nats -{ 1 }-> 0 :|: 
   nats -{ 1 }-> 1 + 0 + (1 + 0) :|: 

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> odds :|: z = 1
   activate(z) -{ 2 }-> incr(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 2 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 2 }-> incr(incr(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
   activate(z) -{ 3 }-> incr(0) :|: z = 1 + 0
   activate(z) -{ 3 }-> incr(1 + 0 + (1 + 0)) :|: z = 1 + 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + (1 + 0) :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + XS, X >= 0, XS >= 0
   incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   incr(z) -{ 1 }-> 1 + (1 + X) + (1 + activate(XS)) :|: z = 1 + X + XS, X >= 0, XS >= 0
   nats -{ 1 }-> 0 :|: 
   nats -{ 1 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 2 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 1 }-> 1 :|: 
   pairs -{ 1 }-> 1 + 0 + (1 + 1) :|: 
   tail(z) -{ 1 }-> activate(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> odds :|: z = 1
   activate(z) -{ 2 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 2 }-> incr(incr(activate(z - 2))) :|: z - 2 >= 0
   activate(z) -{ 3 }-> incr(0) :|: z = 1 + 0
   activate(z) -{ 2 }-> incr(z - 1) :|: z - 1 >= 0
   activate(z) -{ 3 }-> incr(1 + 0 + (1 + 0)) :|: z = 1 + 0
   activate(z) -{ 2 }-> 0 :|: z = 0
   activate(z) -{ 2 }-> 1 + 0 + (1 + 0) :|: z = 0
   head(z) -{ 1 }-> X :|: z = 1 + X + XS, X >= 0, XS >= 0
   incr(z) -{ 1 }-> 1 + z :|: z >= 0
   incr(z) -{ 1 }-> 1 + (1 + X) + (1 + activate(XS)) :|: z = 1 + X + XS, X >= 0, XS >= 0
   nats -{ 1 }-> 0 :|: 
   nats -{ 1 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 2 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 1 }-> 1 :|: 
   pairs -{ 1 }-> 1 + 0 + (1 + 1) :|: 
   tail(z) -{ 1 }-> activate(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0


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

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

   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]

And the following fresh constants:    null_head, null_tail

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

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

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

   nats -> cons(0, n__incr(n__nats)) [1]
   pairs -> cons(0, n__incr(n__odds)) [1]
   odds -> incr(pairs) [1]
   incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) [1]
   head(cons(X, XS)) -> X [1]
   tail(cons(X, XS)) -> activate(XS) [1]
   incr(X) -> n__incr(X) [1]
   nats -> n__nats [1]
   odds -> n__odds [1]
   activate(n__incr(X)) -> incr(activate(X)) [1]
   activate(n__nats) -> nats [1]
   activate(n__odds) -> odds [1]
   activate(X) -> X [1]
   head(v0) -> null_head [0]
   tail(v0) -> null_tail [0]

The TRS has the following type information:
nats :: n__nats:n__incr:cons:n__odds:null_tail
cons :: 0:s:null_head -> n__nats:n__incr:cons:n__odds:null_tail -> n__nats:n__incr:cons:n__odds:null_tail
0 :: 0:s:null_head
n__incr :: n__nats:n__incr:cons:n__odds:null_tail -> n__nats:n__incr:cons:n__odds:null_tail
n__nats :: n__nats:n__incr:cons:n__odds:null_tail
pairs :: n__nats:n__incr:cons:n__odds:null_tail
n__odds :: n__nats:n__incr:cons:n__odds:null_tail
odds :: n__nats:n__incr:cons:n__odds:null_tail
incr :: n__nats:n__incr:cons:n__odds:null_tail -> n__nats:n__incr:cons:n__odds:null_tail
s :: 0:s:null_head -> 0:s:null_head
activate :: n__nats:n__incr:cons:n__odds:null_tail -> n__nats:n__incr:cons:n__odds:null_tail
head :: n__nats:n__incr:cons:n__odds:null_tail -> 0:s:null_head
tail :: n__nats:n__incr:cons:n__odds:null_tail -> n__nats:n__incr:cons:n__odds:null_tail
null_head :: 0:s:null_head
null_tail :: n__nats:n__incr:cons:n__odds:null_tail

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

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

   0 => 0
   n__nats => 0
   n__odds => 1
   null_head => 0
   null_tail => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> odds :|: z = 1
   activate(z) -{ 1 }-> nats :|: z = 0
   activate(z) -{ 1 }-> incr(activate(X)) :|: z = 1 + X, X >= 0
   head(z) -{ 1 }-> X :|: z = 1 + X + XS, X >= 0, XS >= 0
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   incr(z) -{ 1 }-> 1 + (1 + X) + (1 + activate(XS)) :|: z = 1 + X + XS, X >= 0, XS >= 0
   nats -{ 1 }-> 0 :|: 
   nats -{ 1 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 1 }-> incr(pairs) :|: 
   odds -{ 1 }-> 1 :|: 
   pairs -{ 1 }-> 1 + 0 + (1 + 1) :|: 
   tail(z) -{ 1 }-> activate(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
Tuples:
   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12
S tuples:
   NATS -> c
   NATS -> c1
   PAIRS -> c2
   ODDS -> c3(INCR(pairs), PAIRS)
   ODDS -> c4
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   INCR(z0) -> c6
   HEAD(cons(z0, z1)) -> c7
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__nats) -> c10(NATS)
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(z0) -> c12
K tuples:none
Defined Rule Symbols:   nats, pairs, odds, incr_1, head_1, tail_1, activate_1

Defined Pair Symbols:   NATS, PAIRS, ODDS, INCR_1, HEAD_1, TAIL_1, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4, c5_1, c6, c7, c8_1, c9_2, c10_1, c11_1, c12


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

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TAIL(cons(z0, z1)) -> c8(ACTIVATE(z1))
Removed 8 trailing nodes:
   INCR(z0) -> c6
   NATS -> c1
   ACTIVATE(z0) -> c12
   HEAD(cons(z0, z1)) -> c7
   NATS -> c
   PAIRS -> c2
   ACTIVATE(n__nats) -> c10(NATS)
   ODDS -> c4

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
Tuples:
   ODDS -> c3(INCR(pairs), PAIRS)
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
S tuples:
   ODDS -> c3(INCR(pairs), PAIRS)
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
K tuples:none
Defined Rule Symbols:   nats, pairs, odds, incr_1, head_1, tail_1, activate_1

Defined Pair Symbols:   ODDS, INCR_1, ACTIVATE_1

Compound Symbols:   c3_2, c5_1, c9_2, c11_1


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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   pairs -> cons(0, n__incr(n__odds))
   odds -> incr(pairs)
   odds -> n__odds
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> activate(z1)
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
K tuples:none
Defined Rule Symbols:   nats, pairs, odds, incr_1, head_1, tail_1, activate_1

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c9_2, c11_1, c3_1


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

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c9_2, c11_1, c3_1


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

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

(44)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))

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

   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))

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

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

(46)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))

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

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0)) [1]
   ACTIVATE(n__odds) -> c11(ODDS) [1]
   ODDS -> c3(INCR(pairs)) [1]
   activate(n__incr(z0)) -> incr(activate(z0)) [0]
   activate(n__nats) -> nats [0]
   activate(n__odds) -> odds [0]
   activate(z0) -> z0 [0]
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1))) [0]
   incr(z0) -> n__incr(z0) [0]
   nats -> cons(0, n__incr(n__nats)) [0]
   nats -> n__nats [0]
   odds -> incr(pairs) [0]
   odds -> n__odds [0]
   pairs -> cons(0, n__incr(n__odds)) [0]

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

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

(50)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0)) [1]
   ACTIVATE(n__odds) -> c11(ODDS) [1]
   ODDS -> c3(INCR(pairs)) [1]
   activate(n__incr(z0)) -> incr(activate(z0)) [0]
   activate(n__nats) -> nats [0]
   activate(n__odds) -> odds [0]
   activate(z0) -> z0 [0]
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1))) [0]
   incr(z0) -> n__incr(z0) [0]
   nats -> cons(0, n__incr(n__nats)) [0]
   nats -> n__nats [0]
   odds -> incr(pairs) [0]
   odds -> n__odds [0]
   pairs -> cons(0, n__incr(n__odds)) [0]

The TRS has the following type information:
INCR :: cons:n__incr:n__odds:n__nats -> c5
cons :: s:0 -> cons:n__incr:n__odds:n__nats -> cons:n__incr:n__odds:n__nats
c5 :: c9:c11 -> c5
ACTIVATE :: cons:n__incr:n__odds:n__nats -> c9:c11
n__incr :: cons:n__incr:n__odds:n__nats -> cons:n__incr:n__odds:n__nats
c9 :: c5 -> c9:c11 -> c9:c11
activate :: cons:n__incr:n__odds:n__nats -> cons:n__incr:n__odds:n__nats
n__odds :: cons:n__incr:n__odds:n__nats
c11 :: c3 -> c9:c11
ODDS :: c3
c3 :: c5 -> c3
pairs :: cons:n__incr:n__odds:n__nats
incr :: cons:n__incr:n__odds:n__nats -> cons:n__incr:n__odds:n__nats
n__nats :: cons:n__incr:n__odds:n__nats
nats :: cons:n__incr:n__odds:n__nats
odds :: cons:n__incr:n__odds:n__nats
s :: s:0 -> s:0
0 :: s:0

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

(51) 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:

   INCR_1
   ACTIVATE_1
   ODDS

(c) The following functions are completely defined:

   activate_1
   incr_1
   nats
   odds
   pairs

Due to the following rules being added:

   activate(v0) -> null_activate [0]
   incr(v0) -> null_incr [0]
   nats -> null_nats [0]
   odds -> null_odds [0]
   pairs -> null_pairs [0]

And the following fresh constants:    null_activate, null_incr, null_nats, null_odds, null_pairs, const, const1, const2

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

(52)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0)) [1]
   ACTIVATE(n__odds) -> c11(ODDS) [1]
   ODDS -> c3(INCR(pairs)) [1]
   activate(n__incr(z0)) -> incr(activate(z0)) [0]
   activate(n__nats) -> nats [0]
   activate(n__odds) -> odds [0]
   activate(z0) -> z0 [0]
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1))) [0]
   incr(z0) -> n__incr(z0) [0]
   nats -> cons(0, n__incr(n__nats)) [0]
   nats -> n__nats [0]
   odds -> incr(pairs) [0]
   odds -> n__odds [0]
   pairs -> cons(0, n__incr(n__odds)) [0]
   activate(v0) -> null_activate [0]
   incr(v0) -> null_incr [0]
   nats -> null_nats [0]
   odds -> null_odds [0]
   pairs -> null_pairs [0]

The TRS has the following type information:
INCR :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c5
cons :: s:0 -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c5 :: c9:c11 -> c5
ACTIVATE :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c9:c11
n__incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c9 :: c5 -> c9:c11 -> c9:c11
activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c11 :: c3 -> c9:c11
ODDS :: c3
c3 :: c5 -> c3
pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
s :: s:0 -> s:0
0 :: s:0
null_activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
const :: c5
const1 :: c9:c11
const2 :: c3

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

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

(54)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) [1]
   ACTIVATE(n__incr(n__incr(z0'))) -> c9(INCR(incr(activate(z0'))), ACTIVATE(n__incr(z0'))) [1]
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats), ACTIVATE(n__nats)) [1]
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(null_activate), ACTIVATE(z0)) [1]
   ACTIVATE(n__odds) -> c11(ODDS) [1]
   ODDS -> c3(INCR(cons(0, n__incr(n__odds)))) [1]
   ODDS -> c3(INCR(null_pairs)) [1]
   activate(n__incr(n__incr(z0''))) -> incr(incr(activate(z0''))) [0]
   activate(n__incr(n__nats)) -> incr(nats) [0]
   activate(n__incr(n__odds)) -> incr(odds) [0]
   activate(n__incr(z0)) -> incr(z0) [0]
   activate(n__incr(z0)) -> incr(null_activate) [0]
   activate(n__nats) -> nats [0]
   activate(n__odds) -> odds [0]
   activate(z0) -> z0 [0]
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1))) [0]
   incr(z0) -> n__incr(z0) [0]
   nats -> cons(0, n__incr(n__nats)) [0]
   nats -> n__nats [0]
   odds -> incr(cons(0, n__incr(n__odds))) [0]
   odds -> incr(null_pairs) [0]
   odds -> n__odds [0]
   pairs -> cons(0, n__incr(n__odds)) [0]
   activate(v0) -> null_activate [0]
   incr(v0) -> null_incr [0]
   nats -> null_nats [0]
   odds -> null_odds [0]
   pairs -> null_pairs [0]

The TRS has the following type information:
INCR :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c5
cons :: s:0 -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c5 :: c9:c11 -> c5
ACTIVATE :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c9:c11
n__incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c9 :: c5 -> c9:c11 -> c9:c11
activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c11 :: c3 -> c9:c11
ODDS :: c3
c3 :: c5 -> c3
pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
s :: s:0 -> s:0
0 :: s:0
null_activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
const :: c5
const1 :: c9:c11
const2 :: c3

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

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

   n__odds => 1
   n__nats => 0
   0 => 0
   null_activate => 0
   null_incr => 0
   null_nats => 0
   null_odds => 0
   null_pairs => 0
   const => 0
   const1 => 0
   const2 => 0

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 1 + ODDS :|: z = 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(z0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(odds) + ACTIVATE(1) :|: z = 1 + 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(nats) + ACTIVATE(0) :|: z = 1 + 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(incr(activate(z0'))) + ACTIVATE(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0
   INCR(z) -{ 1 }-> 1 + ACTIVATE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ODDS -{ 1 }-> 1 + INCR(0) :|: 
   ODDS -{ 1 }-> 1 + INCR(1 + 0 + (1 + 1)) :|: 
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> odds :|: z = 1
   activate(z) -{ 0 }-> nats :|: z = 0
   activate(z) -{ 0 }-> incr(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 0 }-> incr(nats) :|: z = 1 + 0
   activate(z) -{ 0 }-> incr(incr(activate(z0''))) :|: z = 1 + (1 + z0''), z0'' >= 0
   activate(z) -{ 0 }-> incr(0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   incr(z) -{ 0 }-> 1 + (1 + z0) + (1 + activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   nats -{ 0 }-> 0 :|: 
   nats -{ 0 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 0 }-> incr(0) :|: 
   odds -{ 0 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 0 }-> 1 :|: 
   odds -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 1 + 0 + (1 + 1) :|: 


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

(57) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   nats -{ 0 }-> 1 + 0 + (1 + 0) :|: 
   nats -{ 0 }-> 0 :|: 

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 1 + ODDS :|: z = 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(z0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(odds) + ACTIVATE(1) :|: z = 1 + 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(incr(activate(z0'))) + ACTIVATE(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(0) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(0) + ACTIVATE(0) :|: z = 1 + 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(1 + 0 + (1 + 0)) + ACTIVATE(0) :|: z = 1 + 0
   INCR(z) -{ 1 }-> 1 + ACTIVATE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ODDS -{ 1 }-> 1 + INCR(0) :|: 
   ODDS -{ 1 }-> 1 + INCR(1 + 0 + (1 + 1)) :|: 
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> odds :|: z = 1
   activate(z) -{ 0 }-> incr(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 0 }-> incr(incr(activate(z0''))) :|: z = 1 + (1 + z0''), z0'' >= 0
   activate(z) -{ 0 }-> incr(0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> incr(0) :|: z = 1 + 0
   activate(z) -{ 0 }-> incr(1 + 0 + (1 + 0)) :|: z = 1 + 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + (1 + 0) :|: z = 0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   incr(z) -{ 0 }-> 1 + (1 + z0) + (1 + activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   nats -{ 0 }-> 0 :|: 
   nats -{ 0 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 0 }-> incr(0) :|: 
   odds -{ 0 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 0 }-> 1 :|: 
   odds -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 1 + 0 + (1 + 1) :|: 


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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 1 + ODDS :|: z = 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(odds) + ACTIVATE(1) :|: z = 1 + 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(incr(activate(z - 2))) + ACTIVATE(1 + (z - 2)) :|: z - 2 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(0) + ACTIVATE(0) :|: z = 1 + 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(0) + ACTIVATE(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(z - 1) + ACTIVATE(z - 1) :|: z - 1 >= 0
   ACTIVATE(z) -{ 1 }-> 1 + INCR(1 + 0 + (1 + 0)) + ACTIVATE(0) :|: z = 1 + 0
   INCR(z) -{ 1 }-> 1 + ACTIVATE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ODDS -{ 1 }-> 1 + INCR(0) :|: 
   ODDS -{ 1 }-> 1 + INCR(1 + 0 + (1 + 1)) :|: 
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> odds :|: z = 1
   activate(z) -{ 0 }-> incr(odds) :|: z = 1 + 1
   activate(z) -{ 0 }-> incr(incr(activate(z - 2))) :|: z - 2 >= 0
   activate(z) -{ 0 }-> incr(0) :|: z - 1 >= 0
   activate(z) -{ 0 }-> incr(0) :|: z = 1 + 0
   activate(z) -{ 0 }-> incr(z - 1) :|: z - 1 >= 0
   activate(z) -{ 0 }-> incr(1 + 0 + (1 + 0)) :|: z = 1 + 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z = 0
   activate(z) -{ 0 }-> 1 + 0 + (1 + 0) :|: z = 0
   incr(z) -{ 0 }-> 0 :|: z >= 0
   incr(z) -{ 0 }-> 1 + z :|: z >= 0
   incr(z) -{ 0 }-> 1 + (1 + z0) + (1 + activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   nats -{ 0 }-> 0 :|: 
   nats -{ 0 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 0 }-> incr(0) :|: 
   odds -{ 0 }-> incr(1 + 0 + (1 + 1)) :|: 
   odds -{ 0 }-> 1 :|: 
   odds -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 1 + 0 + (1 + 1) :|: 


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

(61) 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:

   activate(v0) -> null_activate [0]
   incr(v0) -> null_incr [0]
   nats -> null_nats [0]
   odds -> null_odds [0]
   pairs -> null_pairs [0]
   INCR(v0) -> null_INCR [0]
   ACTIVATE(v0) -> null_ACTIVATE [0]

And the following fresh constants:    null_activate, null_incr, null_nats, null_odds, null_pairs, null_INCR, null_ACTIVATE, const

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

(62)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) [1]
   ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0)) [1]
   ACTIVATE(n__odds) -> c11(ODDS) [1]
   ODDS -> c3(INCR(pairs)) [1]
   activate(n__incr(z0)) -> incr(activate(z0)) [0]
   activate(n__nats) -> nats [0]
   activate(n__odds) -> odds [0]
   activate(z0) -> z0 [0]
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1))) [0]
   incr(z0) -> n__incr(z0) [0]
   nats -> cons(0, n__incr(n__nats)) [0]
   nats -> n__nats [0]
   odds -> incr(pairs) [0]
   odds -> n__odds [0]
   pairs -> cons(0, n__incr(n__odds)) [0]
   activate(v0) -> null_activate [0]
   incr(v0) -> null_incr [0]
   nats -> null_nats [0]
   odds -> null_odds [0]
   pairs -> null_pairs [0]
   INCR(v0) -> null_INCR [0]
   ACTIVATE(v0) -> null_ACTIVATE [0]

The TRS has the following type information:
INCR :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c5:null_INCR
cons :: s:0 -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c5 :: c9:c11:null_ACTIVATE -> c5:null_INCR
ACTIVATE :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> c9:c11:null_ACTIVATE
n__incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c9 :: c5:null_INCR -> c9:c11:null_ACTIVATE -> c9:c11:null_ACTIVATE
activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
c11 :: c3 -> c9:c11:null_ACTIVATE
ODDS :: c3
c3 :: c5:null_INCR -> c3
pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs -> cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
n__nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
s :: s:0 -> s:0
0 :: s:0
null_activate :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_incr :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_nats :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_odds :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_pairs :: cons:n__incr:n__odds:n__nats:null_activate:null_incr:null_nats:null_odds:null_pairs
null_INCR :: c5:null_INCR
null_ACTIVATE :: c9:c11:null_ACTIVATE
const :: c3

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

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

   n__odds => 1
   n__nats => 0
   0 => 0
   null_activate => 0
   null_incr => 0
   null_nats => 0
   null_odds => 0
   null_pairs => 0
   null_INCR => 0
   null_ACTIVATE => 0
   const => 0

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   ACTIVATE(z) -{ 1 }-> 1 + ODDS :|: z = 1
   ACTIVATE(z) -{ 1 }-> 1 + INCR(activate(z0)) + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0
   INCR(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   INCR(z) -{ 1 }-> 1 + ACTIVATE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   ODDS -{ 1 }-> 1 + INCR(pairs) :|: 
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> odds :|: z = 1
   activate(z) -{ 0 }-> nats :|: z = 0
   activate(z) -{ 0 }-> incr(activate(z0)) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   incr(z) -{ 0 }-> 1 + (1 + z0) + (1 + activate(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   nats -{ 0 }-> 0 :|: 
   nats -{ 0 }-> 1 + 0 + (1 + 0) :|: 
   odds -{ 0 }-> incr(pairs) :|: 
   odds -{ 0 }-> 1 :|: 
   odds -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 0 :|: 
   pairs -{ 0 }-> 1 + 0 + (1 + 1) :|: 

Only complete derivations are relevant for the runtime complexity.

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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACTIVATE(n__incr(z0)) -> c9(INCR(activate(z0)), ACTIVATE(z0)) by 
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats), ACTIVATE(n__nats))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats), ACTIVATE(n__nats))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats), ACTIVATE(n__nats))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c3_1, c9_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(pairs))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c3_1, c9_2, c9_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ODDS -> c3(INCR(pairs)) by 
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c9_1, c3_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(activate(z0))), ACTIVATE(n__incr(z0))) by 
   ACTIVATE(n__incr(n__incr(x0))) -> c9(INCR(n__incr(activate(x0))), ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(INCR(n__incr(activate(x0))), ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(INCR(n__incr(activate(x0))), ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c9_1, c3_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c9_1, c3_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACTIVATE(n__incr(n__odds)) -> c9(INCR(odds), ACTIVATE(n__odds)) by 
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c9_1, c3_1


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c9_1, c3_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACTIVATE(n__incr(n__nats)) -> c9(INCR(nats)) by 
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(n__nats))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(n__nats))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(n__nats))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c3_1, c9_1


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(n__nats))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
S tuples:
   INCR(cons(z0, z1)) -> c5(ACTIVATE(z1))
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   INCR_1, ACTIVATE_1, ODDS

Compound Symbols:   c5_1, c11_1, c9_2, c3_1, c9_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, z1)) -> c5(ACTIVATE(z1)) by 
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c9_2, c3_1, c9_1, c5_1


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(pairs)), ACTIVATE(n__odds)) by ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0))
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c9_2, c3_1, c9_1, c5_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__incr(z0)) -> c9(INCR(z0), ACTIVATE(z0)) by 
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(y0))) -> c9(INCR(n__incr(y0)), ACTIVATE(n__incr(y0)))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(INCR(n__incr(n__incr(n__incr(y0)))), ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(INCR(n__incr(n__incr(n__nats))), ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(INCR(n__incr(n__incr(n__odds))), ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(INCR(n__incr(n__incr(y0))), ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(n__incr(n__odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(n__incr(n__nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)), ACTIVATE(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))), ACTIVATE(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))), ACTIVATE(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))), ACTIVATE(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))), ACTIVATE(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))), ACTIVATE(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))), ACTIVATE(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))), ACTIVATE(cons(y0, n__incr(n__nats))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(y0))) -> c9(INCR(n__incr(y0)), ACTIVATE(n__incr(y0)))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(INCR(n__incr(n__incr(n__incr(y0)))), ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(INCR(n__incr(n__incr(n__nats))), ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(INCR(n__incr(n__incr(n__odds))), ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(INCR(n__incr(n__incr(y0))), ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(n__incr(n__odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(n__incr(n__nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)), ACTIVATE(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))), ACTIVATE(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))), ACTIVATE(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))), ACTIVATE(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))), ACTIVATE(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))), ACTIVATE(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))), ACTIVATE(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))), ACTIVATE(cons(y0, n__incr(n__nats))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(n__odds), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(y0))) -> c9(INCR(n__incr(y0)), ACTIVATE(n__incr(y0)))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(INCR(n__incr(n__incr(n__incr(y0)))), ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(INCR(n__incr(n__incr(n__nats))), ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(INCR(n__incr(n__incr(n__odds))), ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(INCR(n__incr(n__incr(y0))), ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(n__incr(n__odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(n__incr(n__nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)), ACTIVATE(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))), ACTIVATE(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))), ACTIVATE(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))), ACTIVATE(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))), ACTIVATE(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))), ACTIVATE(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))), ACTIVATE(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))), ACTIVATE(cons(y0, n__incr(n__nats))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 16 trailing tuple parts
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0))) by 
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0)))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace INCR(cons(z0, n__incr(y0))) -> c5(ACTIVATE(n__incr(y0))) by 
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__incr(x0))) -> c9(ACTIVATE(n__incr(x0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__incr(n__incr(y0))) -> c9(ACTIVATE(n__incr(y0))) by 
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__incr(cons(y0, n__incr(y1)))) -> c9(INCR(cons(y0, n__incr(y1)))) by 
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(y1))))) -> c9(INCR(cons(z0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__odds)))) -> c9(INCR(cons(z0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__nats)))) -> c9(INCR(cons(z0, n__incr(n__nats))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(y2))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(y2))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(y2))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1))))) by 
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__odds)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__incr(z0)) -> incr(activate(z0))
   activate(n__nats) -> nats
   activate(n__odds) -> odds
   activate(z0) -> z0
   incr(cons(z0, z1)) -> cons(s(z0), n__incr(activate(z1)))
   incr(z0) -> n__incr(z0)
   nats -> cons(0, n__incr(n__nats))
   nats -> n__nats
   odds -> incr(pairs)
   odds -> n__odds
   pairs -> cons(0, n__incr(n__odds))
Tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(y2))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__odds)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))))
S tuples:
   ACTIVATE(n__odds) -> c11(ODDS)
   ODDS -> c3(INCR(cons(0, n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(z0)))) -> c9(INCR(incr(incr(activate(z0)))), ACTIVATE(n__incr(n__incr(z0))))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(INCR(incr(nats)), ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(INCR(incr(odds)), ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(z0))) -> c9(INCR(incr(z0)), ACTIVATE(n__incr(z0)))
   ACTIVATE(n__incr(n__odds)) -> c9(ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__nats)) -> c9(INCR(cons(0, n__incr(n__nats))))
   INCR(cons(z0, n__odds)) -> c5(ACTIVATE(n__odds))
   INCR(cons(z0, n__incr(n__incr(n__incr(y0))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   INCR(cons(z0, n__incr(n__incr(n__nats)))) -> c5(ACTIVATE(n__incr(n__incr(n__nats))))
   INCR(cons(z0, n__incr(n__incr(n__odds)))) -> c5(ACTIVATE(n__incr(n__incr(n__odds))))
   INCR(cons(z0, n__incr(n__incr(y0)))) -> c5(ACTIVATE(n__incr(n__incr(y0))))
   INCR(cons(z0, n__incr(n__odds))) -> c5(ACTIVATE(n__incr(n__odds)))
   INCR(cons(z0, n__incr(n__nats))) -> c5(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(n__odds)) -> c9(INCR(incr(cons(0, n__incr(n__odds)))), ACTIVATE(n__odds))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(y0)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__nats)))) -> c9(ACTIVATE(n__incr(n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__odds)))) -> c9(ACTIVATE(n__incr(n__incr(n__odds))))
   ACTIVATE(n__incr(n__incr(n__incr(y0)))) -> c9(ACTIVATE(n__incr(n__incr(y0))))
   ACTIVATE(n__incr(n__incr(n__odds))) -> c9(ACTIVATE(n__incr(n__odds)))
   ACTIVATE(n__incr(n__incr(n__nats))) -> c9(ACTIVATE(n__incr(n__nats)))
   ACTIVATE(n__incr(cons(y0, n__odds))) -> c9(INCR(cons(y0, n__odds)))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))) -> c9(INCR(cons(y0, n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))) -> c9(INCR(cons(y0, n__incr(n__incr(y1)))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))) -> c9(INCR(cons(y0, n__incr(n__odds))))
   ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))) -> c9(INCR(cons(y0, n__incr(n__nats))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))) -> c9(ACTIVATE(n__incr(cons(y0, n__odds))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))) -> c9(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y0)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(y0))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__odds)))) -> c5(ACTIVATE(n__incr(cons(y0, n__odds))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(y1))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(y1)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__incr(y1)))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__incr(y1))))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__odds))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__odds)))))
   INCR(cons(z0, n__incr(cons(y0, n__incr(n__nats))))) -> c5(ACTIVATE(n__incr(cons(y0, n__incr(n__nats)))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y0))))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__incr(y0)))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__nats))))))
   INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(n__incr(n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__odds))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__odds)))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(y1)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(y1))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1)))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__incr(y1))))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__nats)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(n__odds)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__incr(y1))))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__incr(y1)))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__odds)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__odds))))))
   INCR(cons(z0, n__incr(n__incr(cons(y0, n__incr(n__nats)))))) -> c5(ACTIVATE(n__incr(n__incr(cons(y0, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__odds)))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(y2))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))) -> c9(INCR(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))) -> c9(INCR(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(y1))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__odds)))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__odds))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(y2)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(cons(y1, n__incr(n__nats)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__incr(y1)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__nats))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(n__incr(n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__odds)))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(y2))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2)))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__incr(y2))))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__nats)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(n__odds)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2))))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__incr(y2)))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__odds))))))))
   ACTIVATE(n__incr(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats)))))))) -> c9(ACTIVATE(n__incr(cons(z0, n__incr(n__incr(cons(y1, n__incr(n__nats))))))))
K tuples:none
Defined Rule Symbols:   activate_1, incr_1, nats, odds, pairs

Defined Pair Symbols:   ACTIVATE_1, ODDS, INCR_1

Compound Symbols:   c11_1, c3_1, c9_2, c9_1, c5_1