KILLED
proof of input_xP9kURaklA.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) InliningProof [UPPER BOUND(ID), 145 ms]
        (20) CpxRNTS
        (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
(23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(24) CdtProblem
    (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms]
    (48) CdtProblem
    (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms]
    (76) CdtProblem
    (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 8 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 33 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 10 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 35 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) 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:

   ack_in(0, n) -> ack_out(s(n))
   ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
   u11(ack_out(n)) -> ack_out(n)
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

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:

   ack_in(0', n) -> ack_out(s(n))
   ack_in(s(m), 0') -> u11(ack_in(m, s(0')))
   u11(ack_out(n)) -> ack_out(n)
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

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:

   ack_in(0, n) -> ack_out(s(n))
   ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
   u11(ack_out(n)) -> ack_out(n)
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

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

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

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


The TRS R consists of the following rules:

   ack_in(0, n) -> ack_out(s(n)) [1]
   ack_in(s(m), 0) -> u11(ack_in(m, s(0))) [1]
   u11(ack_out(n)) -> ack_out(n) [1]
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m) [1]
   u21(ack_out(n), m) -> u22(ack_in(m, n)) [1]
   u22(ack_out(n)) -> ack_out(n) [1]

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

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

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

   ack_in(0, n) -> ack_out(s(n)) [1]
   ack_in(s(m), 0) -> u11(ack_in(m, s(0))) [1]
   u11(ack_out(n)) -> ack_out(n) [1]
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m) [1]
   u21(ack_out(n), m) -> u22(ack_in(m, n)) [1]
   u22(ack_out(n)) -> ack_out(n) [1]

The TRS has the following type information:
ack_in :: 0:s -> 0:s -> ack_out
0 :: 0:s
ack_out :: 0:s -> ack_out
s :: 0:s -> 0:s
u11 :: ack_out -> ack_out
u21 :: ack_out -> 0:s -> ack_out
u22 :: ack_out -> ack_out

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

(9) 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:

   u11(v0) -> null_u11 [0]
   u21(v0, v1) -> null_u21 [0]
   u22(v0) -> null_u22 [0]

And the following fresh constants:    null_u11, null_u21, null_u22

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

(10)
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:

   ack_in(0, n) -> ack_out(s(n)) [1]
   ack_in(s(m), 0) -> u11(ack_in(m, s(0))) [1]
   u11(ack_out(n)) -> ack_out(n) [1]
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m) [1]
   u21(ack_out(n), m) -> u22(ack_in(m, n)) [1]
   u22(ack_out(n)) -> ack_out(n) [1]
   u11(v0) -> null_u11 [0]
   u21(v0, v1) -> null_u21 [0]
   u22(v0) -> null_u22 [0]

The TRS has the following type information:
ack_in :: 0:s -> 0:s -> ack_out:null_u11:null_u21:null_u22
0 :: 0:s
ack_out :: 0:s -> ack_out:null_u11:null_u21:null_u22
s :: 0:s -> 0:s
u11 :: ack_out:null_u11:null_u21:null_u22 -> ack_out:null_u11:null_u21:null_u22
u21 :: ack_out:null_u11:null_u21:null_u22 -> 0:s -> ack_out:null_u11:null_u21:null_u22
u22 :: ack_out:null_u11:null_u21:null_u22 -> ack_out:null_u11:null_u21:null_u22
null_u11 :: ack_out:null_u11:null_u21:null_u22
null_u21 :: ack_out:null_u11:null_u21:null_u22
null_u22 :: ack_out:null_u11:null_u21:null_u22

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

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

   0 => 0
   null_u11 => 0
   null_u21 => 0
   null_u22 => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   ack_in(z, z') -{ 1 }-> u21(ack_in(1 + m, n), m) :|: n >= 0, z' = 1 + n, z = 1 + m, m >= 0
   ack_in(z, z') -{ 1 }-> u11(ack_in(m, 1 + 0)) :|: z = 1 + m, z' = 0, m >= 0
   ack_in(z, z') -{ 1 }-> 1 + (1 + n) :|: n >= 0, z' = n, z = 0
   u11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u11(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n
   u21(z, z') -{ 1 }-> u22(ack_in(m, n)) :|: n >= 0, z' = m, z = 1 + n, m >= 0
   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u22(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u22(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n

Only complete derivations are relevant for the runtime complexity.

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

(13) 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:
none

(c) The following functions are completely defined:

   ack_in_2
   u11_1
   u21_2
   u22_1

Due to the following rules being added:

   u11(v0) -> const [0]
   u21(v0, v1) -> const [0]
   u22(v0) -> const [0]

And the following fresh constants:    const

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

(14)
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:

   ack_in(0, n) -> ack_out(s(n)) [1]
   ack_in(s(m), 0) -> u11(ack_in(m, s(0))) [1]
   u11(ack_out(n)) -> ack_out(n) [1]
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m) [1]
   u21(ack_out(n), m) -> u22(ack_in(m, n)) [1]
   u22(ack_out(n)) -> ack_out(n) [1]
   u11(v0) -> const [0]
   u21(v0, v1) -> const [0]
   u22(v0) -> const [0]

The TRS has the following type information:
ack_in :: 0:s -> 0:s -> ack_out:const
0 :: 0:s
ack_out :: 0:s -> ack_out:const
s :: 0:s -> 0:s
u11 :: ack_out:const -> ack_out:const
u21 :: ack_out:const -> 0:s -> ack_out:const
u22 :: ack_out:const -> ack_out:const
const :: ack_out:const

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

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

(16)
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:

   ack_in(0, n) -> ack_out(s(n)) [1]
   ack_in(s(0), 0) -> u11(ack_out(s(s(0)))) [2]
   ack_in(s(s(m')), 0) -> u11(u21(ack_in(s(m'), 0), m')) [2]
   u11(ack_out(n)) -> ack_out(n) [1]
   ack_in(s(m), s(0)) -> u21(u11(ack_in(m, s(0))), m) [2]
   ack_in(s(m), s(s(n'))) -> u21(u21(ack_in(s(m), n'), m), m) [2]
   u21(ack_out(n), 0) -> u22(ack_out(s(n))) [2]
   u21(ack_out(0), s(m'')) -> u22(u11(ack_in(m'', s(0)))) [2]
   u21(ack_out(s(n'')), s(m1)) -> u22(u21(ack_in(s(m1), n''), m1)) [2]
   u22(ack_out(n)) -> ack_out(n) [1]
   u11(v0) -> const [0]
   u21(v0, v1) -> const [0]
   u22(v0) -> const [0]

The TRS has the following type information:
ack_in :: 0:s -> 0:s -> ack_out:const
0 :: 0:s
ack_out :: 0:s -> ack_out:const
s :: 0:s -> 0:s
u11 :: ack_out:const -> ack_out:const
u21 :: ack_out:const -> 0:s -> ack_out:const
u22 :: ack_out:const -> ack_out:const
const :: ack_out:const

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

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

   0 => 0
   const => 0

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

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

   ack_in(z, z') -{ 2 }-> u21(u21(ack_in(1 + m, n'), m), m) :|: z' = 1 + (1 + n'), n' >= 0, z = 1 + m, m >= 0
   ack_in(z, z') -{ 2 }-> u21(u11(ack_in(m, 1 + 0)), m) :|: z' = 1 + 0, z = 1 + m, m >= 0
   ack_in(z, z') -{ 2 }-> u11(u21(ack_in(1 + m', 0), m')) :|: z = 1 + (1 + m'), m' >= 0, z' = 0
   ack_in(z, z') -{ 2 }-> u11(1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 0
   ack_in(z, z') -{ 1 }-> 1 + (1 + n) :|: n >= 0, z' = n, z = 0
   u11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u11(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n
   u21(z, z') -{ 2 }-> u22(u21(ack_in(1 + m1, n''), m1)) :|: z' = 1 + m1, n'' >= 0, z = 1 + (1 + n''), m1 >= 0
   u21(z, z') -{ 2 }-> u22(u11(ack_in(m'', 1 + 0))) :|: m'' >= 0, z = 1 + 0, z' = 1 + m''
   u21(z, z') -{ 2 }-> u22(1 + (1 + n)) :|: n >= 0, z = 1 + n, z' = 0
   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u22(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u22(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n


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

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

   u11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u11(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n
   u22(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u22(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   ack_in(z, z') -{ 2 }-> u21(u21(ack_in(1 + m, n'), m), m) :|: z' = 1 + (1 + n'), n' >= 0, z = 1 + m, m >= 0
   ack_in(z, z') -{ 2 }-> u21(u11(ack_in(m, 1 + 0)), m) :|: z' = 1 + 0, z = 1 + m, m >= 0
   ack_in(z, z') -{ 2 }-> u11(u21(ack_in(1 + m', 0), m')) :|: z = 1 + (1 + m'), m' >= 0, z' = 0
   ack_in(z, z') -{ 2 }-> 0 :|: z = 1 + 0, z' = 0, v0 >= 0, 1 + (1 + (1 + 0)) = v0
   ack_in(z, z') -{ 3 }-> 1 + n :|: z = 1 + 0, z' = 0, n >= 0, 1 + (1 + (1 + 0)) = 1 + n
   ack_in(z, z') -{ 1 }-> 1 + (1 + n) :|: n >= 0, z' = n, z = 0
   u11(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u11(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n
   u21(z, z') -{ 2 }-> u22(u21(ack_in(1 + m1, n''), m1)) :|: z' = 1 + m1, n'' >= 0, z = 1 + (1 + n''), m1 >= 0
   u21(z, z') -{ 2 }-> u22(u11(ack_in(m'', 1 + 0))) :|: m'' >= 0, z = 1 + 0, z' = 1 + m''
   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u21(z, z') -{ 2 }-> 0 :|: n >= 0, z = 1 + n, z' = 0, v0 >= 0, 1 + (1 + n) = v0
   u21(z, z') -{ 3 }-> 1 + n' :|: n >= 0, z = 1 + n, z' = 0, n' >= 0, 1 + (1 + n) = 1 + n'
   u22(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   u22(z) -{ 1 }-> 1 + n :|: n >= 0, z = 1 + n


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

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

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

   ack_in(z, z') -{ 2 }-> u21(u21(ack_in(1 + (z - 1), z' - 2), z - 1), z - 1) :|: z' - 2 >= 0, z - 1 >= 0
   ack_in(z, z') -{ 2 }-> u21(u11(ack_in(z - 1, 1 + 0)), z - 1) :|: z' = 1 + 0, z - 1 >= 0
   ack_in(z, z') -{ 2 }-> u11(u21(ack_in(1 + (z - 2), 0), z - 2)) :|: z - 2 >= 0, z' = 0
   ack_in(z, z') -{ 2 }-> 0 :|: z = 1 + 0, z' = 0, v0 >= 0, 1 + (1 + (1 + 0)) = v0
   ack_in(z, z') -{ 3 }-> 1 + n :|: z = 1 + 0, z' = 0, n >= 0, 1 + (1 + (1 + 0)) = 1 + n
   ack_in(z, z') -{ 1 }-> 1 + (1 + z') :|: z' >= 0, z = 0
   u11(z) -{ 0 }-> 0 :|: z >= 0
   u11(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
   u21(z, z') -{ 2 }-> u22(u21(ack_in(1 + (z' - 1), z - 2), z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   u21(z, z') -{ 2 }-> u22(u11(ack_in(z' - 1, 1 + 0))) :|: z' - 1 >= 0, z = 1 + 0
   u21(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   u21(z, z') -{ 2 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (1 + (z - 1)) = v0
   u21(z, z') -{ 3 }-> 1 + n' :|: z - 1 >= 0, z' = 0, n' >= 0, 1 + (1 + (z - 1)) = 1 + n'
   u22(z) -{ 0 }-> 0 :|: z >= 0
   u22(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0


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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(0, z0) -> c
   ACK_IN(s(z0), 0) -> c1(U11(ack_in(z0, s(0))), ACK_IN(z0, s(0)))
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   U11(ack_out(z0)) -> c3
   U21(ack_out(z0), z1) -> c4(U22(ack_in(z1, z0)), ACK_IN(z1, z0))
   U22(ack_out(z0)) -> c5
S tuples:
   ACK_IN(0, z0) -> c
   ACK_IN(s(z0), 0) -> c1(U11(ack_in(z0, s(0))), ACK_IN(z0, s(0)))
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   U11(ack_out(z0)) -> c3
   U21(ack_out(z0), z1) -> c4(U22(ack_in(z1, z0)), ACK_IN(z1, z0))
   U22(ack_out(z0)) -> c5
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U11_1, U21_2, U22_1

Compound Symbols:   c, c1_2, c2_2, c3, c4_2, c5


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

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   U11(ack_out(z0)) -> c3
   ACK_IN(0, z0) -> c
   U22(ack_out(z0)) -> c5

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(U11(ack_in(z0, s(0))), ACK_IN(z0, s(0)))
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   U21(ack_out(z0), z1) -> c4(U22(ack_in(z1, z0)), ACK_IN(z1, z0))
S tuples:
   ACK_IN(s(z0), 0) -> c1(U11(ack_in(z0, s(0))), ACK_IN(z0, s(0)))
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   U21(ack_out(z0), z1) -> c4(U22(ack_in(z1, z0)), ACK_IN(z1, z0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_2, c2_2, c4_2


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

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
S tuples:
   ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1))
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK_IN(s(z0), s(z1)) -> c2(U21(ack_in(s(z0), z1), z0), ACK_IN(s(z0), z1)) by 
   ACK_IN(s(z0), s(0)) -> c2(U21(u11(ack_in(z0, s(0))), z0), ACK_IN(s(z0), 0))
   ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1)))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(z0), s(0)) -> c2(U21(u11(ack_in(z0, s(0))), z0), ACK_IN(s(z0), 0))
   ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1)))
S tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(z0), s(0)) -> c2(U21(u11(ack_in(z0, s(0))), z0), ACK_IN(s(z0), 0))
   ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_1, c4_1, c2_2


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

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK_IN(s(z0), s(0)) -> c2(U21(u11(ack_in(z0, s(0))), z0), ACK_IN(s(z0), 0)) by 
   ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1)))
   ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
S tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1)))
   ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_1, c4_1, c2_2, c2_1


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK_IN(s(z0), s(s(z1))) -> c2(U21(u21(ack_in(s(z0), z1), z0), z0), ACK_IN(s(z0), s(z1))) by 
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
S tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_1, c4_1, c2_2, c2_1


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

(35) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(0), s(0)) -> c2(U21(u11(ack_out(s(s(0)))), 0), ACK_IN(s(0), 0)) by ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
S tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_1, c4_1, c2_2, c2_1


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

(37) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), ACK_IN(s(s(z0)), 0)) by ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
S tuples:
   ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0)))
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c1_1, c4_1, c2_1, c2_2


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(z0), 0) -> c1(ACK_IN(z0, s(0))) by 
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
S tuples:
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0), ACK_IN(s(0), 0))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   U21_2, ACK_IN_2

Compound Symbols:   c4_1, c2_1, c2_2, c1_1


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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0))
S tuples:
   U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0))
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   U21_2, ACK_IN_2

Compound Symbols:   c4_1, c2_1, c2_2, c1_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U21(ack_out(z0), z1) -> c4(ACK_IN(z1, z0)) by 
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
S tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_1, c2_2, c1_1, c4_1


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

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ACK_IN(s(0), s(0)) -> c2(U21(ack_out(s(s(0))), 0))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
S tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_1, c2_2, c1_1, c4_1


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

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
We considered the (Usable) Rules:
   u11(ack_out(z0)) -> ack_out(z0)
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   ack_in(0, z0) -> ack_out(s(z0))
And the Tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(ACK_IN(x_1, x_2)) = [3]x_1
   POL(U21(x_1, x_2)) = x_1
   POL(ack_in(x_1, x_2)) = 0
   POL(ack_out(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(s(x_1)) = 0
   POL(u11(x_1)) = [2]x_1
   POL(u21(x_1, x_2)) = 0
   POL(u22(x_1)) = x_1

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
S tuples:
   ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0))
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_1, c2_2, c1_1, c4_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(x0), s(0)) -> c2(ACK_IN(s(x0), 0)) by 
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
S tuples:
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   U21(ack_out(s(0)), s(0)) -> c4(ACK_IN(s(0), s(0)))
   ACK_IN(s(s(0)), 0) -> c1(ACK_IN(s(0), s(0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
S tuples:
   ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0)))
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(z0), s(s(0))) -> c2(U21(u21(u11(ack_in(z0, s(0))), z0), z0), ACK_IN(s(z0), s(0))) by 
   ACK_IN(s(s(y0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(y0), s(0))), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(y0), s(0))), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
S tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(y0), s(0))), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(55) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(y0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(y0), s(0))), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(0))) by ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
S tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(57) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(y1))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(y1)), s(0))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(0))) by ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
S tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(59) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0))) by ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
S tuples:
   ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1))))
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(z0), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), ACK_IN(s(z0), s(s(z1)))) by 
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(63) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(s(y1)))), s(s(0))) -> c2(U21(u21(u11(ack_in(s(s(s(y1))), s(0))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(0))) by ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), ACK_IN(s(s(0)), s(0))) by ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0))) by ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0))) by ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0))) by ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1)))))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(73) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(z0), s(s(s(s(y1))))) -> c2(U21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), ACK_IN(s(z0), s(s(s(y1))))) by ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(s(y0)))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(0)))) by ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0)))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(s(y0)), 0) -> c1(ACK_IN(s(y0), s(0))) by 
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ACK_IN(s(s(0)), s(0)) -> c2(ACK_IN(s(s(0)), 0))
   U21(ack_out(0), s(s(0))) -> c4(ACK_IN(s(s(0)), 0))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U21(ack_out(s(0)), s(y0)) -> c4(ACK_IN(s(y0), s(0))) by 
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0))))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U21(ack_out(s(s(0))), s(y0)) -> c4(ACK_IN(s(y0), s(s(0)))) by 
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0)))) by ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(y0)), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(0)))) by ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(y0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(0)))) by ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0)))) by ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0))) by ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0))) by ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1)))))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U21(ack_out(s(s(s(y1)))), s(y0)) -> c4(ACK_IN(s(y0), s(s(s(y1))))) by 
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace U21(ack_out(0), s(s(y0))) -> c4(ACK_IN(s(s(y0)), 0)) by 
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(s(y0)), s(0)) -> c2(ACK_IN(s(s(y0)), 0)) by 
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0)))) by ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0)))) by ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0)))) by ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0)))) by ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0)))) by ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK_IN(s(z0), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(z1))))) by 
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(z0), s(s(s(s(s(y1)))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(y1))))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(z0), s(s(s(s(s(y1)))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(y1))))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(z0), s(s(s(s(s(y1)))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(y1))))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(z0), s(s(s(s(s(y1)))))) -> c2(U21(u21(u21(u21(ack_in(s(z0), s(y1)), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(y1)))))) by ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(s(y0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y0)))), 0), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), s(s(s(y0)))), ACK_IN(s(s(s(s(y0)))), s(s(s(0))))) by ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(119) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(0)), 0), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0))))) by ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(y0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), 0), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(0))))) by ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(y0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y0))), 0), s(s(y0))), s(s(y0))), s(s(y0))), s(s(y0))), ACK_IN(s(s(s(y0))), s(s(s(0))))) by ACK_IN(s(s(s(z0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(s(0)))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
   ACK_IN(s(s(s(z0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
   ACK_IN(s(s(s(z0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), 0), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0))))) by ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack_in(0, z0) -> ack_out(s(z0))
   ack_in(s(z0), 0) -> u11(ack_in(z0, s(0)))
   ack_in(s(z0), s(z1)) -> u21(ack_in(s(z0), z1), z0)
   u11(ack_out(z0)) -> ack_out(z0)
   u21(ack_out(z0), z1) -> u22(ack_in(z1, z0))
   u22(ack_out(z0)) -> ack_out(z0)
Tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
   ACK_IN(s(s(s(z0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
S tuples:
   ACK_IN(s(s(z0)), s(0)) -> c2(U21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), ACK_IN(s(s(z0)), 0))
   ACK_IN(s(s(s(y0))), 0) -> c1(ACK_IN(s(s(y0)), s(0)))
   U21(ack_out(s(0)), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(0)))
   ACK_IN(s(s(s(y0))), s(0)) -> c2(ACK_IN(s(s(s(y0))), 0))
   ACK_IN(s(s(y1)), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(y1)), z1), s(y1)), s(y1)), s(y1)), ACK_IN(s(s(y1)), s(s(z1))))
   ACK_IN(s(s(s(y1))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(z1))))
   ACK_IN(s(s(s(0))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(z1))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))) -> c2(U21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(z1))))
   ACK_IN(s(s(z0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(z0, s(0))), z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(0)))
   ACK_IN(s(s(s(z0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(z0), s(0))), s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(0)))
   ACK_IN(s(s(s(0))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(0), s(0))), s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(0)))
   ACK_IN(s(s(s(0))), 0) -> c1(ACK_IN(s(s(0)), s(0)))
   ACK_IN(s(s(s(s(y0)))), 0) -> c1(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(0)), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(0)))
   U21(ack_out(s(s(0))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(0))))
   U21(ack_out(s(s(0))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(s(z0)))), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(0)))
   ACK_IN(s(s(0)), s(s(0))) -> c2(U21(u21(u11(u21(u11(ack_in(0, s(0))), 0)), s(0)), s(0)), ACK_IN(s(s(0)), s(0)))
   U21(ack_out(s(s(s(z0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(z0)))))
   U21(ack_out(s(s(s(z0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(z0)))))
   U21(ack_out(s(s(s(s(y1))))), s(z1)) -> c4(ACK_IN(s(z1), s(s(s(s(y1))))))
   U21(ack_out(s(s(s(0)))), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(0))) -> c4(ACK_IN(s(s(0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(y0))) -> c4(ACK_IN(s(s(y0)), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), s(s(s(0)))))
   U21(ack_out(s(s(s(0)))), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(0)) -> c2(ACK_IN(s(s(s(0))), 0))
   ACK_IN(s(s(s(s(y0)))), s(0)) -> c2(ACK_IN(s(s(s(s(y0)))), 0))
   ACK_IN(s(s(s(s(z0)))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(s(z0))), 0), s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(0))))
   ACK_IN(s(s(0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(0), 0), 0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(0))))
   ACK_IN(s(s(z0)), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(z0), 0), z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(0))))
   ACK_IN(s(s(s(z0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(z0)), 0), s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(0))))
   ACK_IN(s(s(s(0))), s(s(s(0)))) -> c2(U21(u21(u21(u11(u21(ack_in(s(s(0)), 0), s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(0))))
   ACK_IN(s(s(s(y1))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(y1))), z1), s(s(y1))), s(s(y1))), s(s(y1))), s(s(y1))), ACK_IN(s(s(s(y1))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(y1)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(y1)))), z1), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), s(s(s(y1)))), ACK_IN(s(s(s(s(y1)))), s(s(s(z1)))))
   ACK_IN(s(s(y0)), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(y0)), z1), s(y0)), s(y0)), s(y0)), s(y0)), ACK_IN(s(s(y0)), s(s(s(z1)))))
   ACK_IN(s(s(s(0))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(0))), z1), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(s(y1))))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(s(y1))))), z1), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), s(s(s(s(y1))))), ACK_IN(s(s(s(s(s(y1))))), s(s(s(z1)))))
   ACK_IN(s(s(s(s(0)))), s(s(s(s(z1))))) -> c2(U21(u21(u21(u21(ack_in(s(s(s(s(0)))), z1), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), s(s(s(0)))), ACK_IN(s(s(s(s(0)))), s(s(s(z1)))))
   ACK_IN(s(z0), s(s(s(s(s(z1)))))) -> c2(U21(u21(u21(u21(u21(ack_in(s(z0), z1), z0), z0), z0), z0), z0), ACK_IN(s(z0), s(s(s(s(z1))))))
   ACK_IN(s(s(s(s(z0)))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(s(z0))), s(0))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), s(s(s(z0)))), ACK_IN(s(s(s(s(z0)))), s(s(s(0)))))
   ACK_IN(s(s(0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(0), s(0))), s(0)), s(0)), s(0)), s(0)), ACK_IN(s(s(0)), s(s(s(0)))))
   ACK_IN(s(s(z0)), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(z0), s(0))), s(z0)), s(z0)), s(z0)), s(z0)), ACK_IN(s(s(z0)), s(s(s(0)))))
   ACK_IN(s(s(s(z0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(z0)), s(0))), s(s(z0))), s(s(z0))), s(s(z0))), s(s(z0))), ACK_IN(s(s(s(z0))), s(s(s(0)))))
   ACK_IN(s(s(s(0))), s(s(s(s(0))))) -> c2(U21(u21(u21(u21(u11(ack_in(s(s(0)), s(0))), s(s(0))), s(s(0))), s(s(0))), s(s(0))), ACK_IN(s(s(s(0))), s(s(s(0)))))
K tuples:
   U21(ack_out(0), s(s(s(y0)))) -> c4(ACK_IN(s(s(s(y0))), 0))
   U21(ack_out(0), s(s(s(0)))) -> c4(ACK_IN(s(s(s(0))), 0))
   U21(ack_out(0), s(s(s(s(y0))))) -> c4(ACK_IN(s(s(s(s(y0)))), 0))
Defined Rule Symbols:   ack_in_2, u11_1, u21_2, u22_1

Defined Pair Symbols:   ACK_IN_2, U21_2

Compound Symbols:   c2_2, c1_1, c4_1, c2_1