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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (24) CdtProblem
    (25) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRhsSimplificationProcessorProof [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) CdtLeafRemovalProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) 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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
   goal(m, n) -> ack(m, 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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
   goal(m, n) -> ack(m, 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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil))
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs))
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n)
   goal(m, n) -> ack(m, 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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1]
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1]
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1]
   goal(m, n) -> ack(m, 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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1]
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1]
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1]
   goal(m, n) -> ack(m, n) [1]

The TRS has the following type information:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   goal_2

(c) The following functions are completely defined:

   ack_2

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1]
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1]
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1]
   goal(m, n) -> ack(m, n) [1]

The TRS has the following type information:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

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

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

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

   ack(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1]
   ack(Cons(x', xs'), Cons(x, Nil)) -> ack(xs', ack(xs', Cons(Nil, Nil))) [2]
   ack(Cons(x', xs'), Cons(x, Cons(x1, xs1))) -> ack(xs', ack(xs', ack(Cons(x', xs'), xs1))) [2]
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1]
   goal(m, n) -> ack(m, n) [1]

The TRS has the following type information:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

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

   Nil => 0

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

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

   ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   ack(z, z') -{ 2 }-> ack(xs', ack(xs', ack(1 + x' + xs', xs1))) :|: x1 >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + (1 + x1 + xs1), xs1 >= 0, z = 1 + x' + xs'
   ack(z, z') -{ 2 }-> ack(xs', ack(xs', 1 + 0 + 0)) :|: x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + 0, z = 1 + x' + xs'
   ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + n :|: n >= 0, z' = n, z = 0
   goal(z, z') -{ 1 }-> ack(m, n) :|: z = m, n >= 0, z' = n, m >= 0


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   ack(z, z') -{ 2 }-> ack(xs', ack(xs', ack(1 + x' + xs', xs1))) :|: x1 >= 0, x' >= 0, xs' >= 0, x >= 0, z' = 1 + x + (1 + x1 + xs1), xs1 >= 0, z = 1 + x' + xs'
   ack(z, z') -{ 2 }-> ack(xs', ack(xs', 1 + 0 + 0)) :|: x' >= 0, xs' >= 0, z' - 1 >= 0, z = 1 + x' + xs'
   ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> ack(z, z') :|: z' >= 0, z >= 0


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

(17) 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:
none

And the following fresh constants: none

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

(18)
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(Cons(x, xs), Nil) -> ack(xs, Cons(Nil, Nil)) [1]
   ack(Cons(x', xs'), Cons(x, xs)) -> ack(xs', ack(Cons(x', xs'), xs)) [1]
   ack(Nil, n) -> Cons(Cons(Nil, Nil), n) [1]
   goal(m, n) -> ack(m, n) [1]

The TRS has the following type information:
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

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

   Nil => 0

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

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

   ack(z, z') -{ 1 }-> ack(xs, 1 + 0 + 0) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 0
   ack(z, z') -{ 1 }-> ack(xs', ack(1 + x' + xs', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, xs' >= 0, x >= 0, z = 1 + x' + xs'
   ack(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + n :|: n >= 0, z' = n, z = 0
   goal(z, z') -{ 1 }-> ack(m, n) :|: z = m, n >= 0, z' = n, m >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
   goal(z0, z1) -> ack(z0, z1)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
   ACK(Nil, z0) -> c2
   GOAL(z0, z1) -> c3(ACK(z0, z1))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
   ACK(Nil, z0) -> c2
   GOAL(z0, z1) -> c3(ACK(z0, z1))
K tuples:none
Defined Rule Symbols:   ack_2, goal_2

Defined Pair Symbols:   ACK_2, GOAL_2

Compound Symbols:   c_1, c1_2, c2, c3_1


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

(23) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c3(ACK(z0, z1))
Removed 1 trailing nodes:
   ACK(Nil, z0) -> c2

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
   goal(z0, z1) -> ack(z0, z1)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
K tuples:none
Defined Rule Symbols:   ack_2, goal_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2


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

(25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> ack(z0, z1)

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2


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

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) by 
   ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(Cons(z0, z1), Cons(x2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) by 
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), Cons(Nil, Nil))), ACK(Cons(x0, Nil), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2, c1_1


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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2, c1_1


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ACK(Cons(z0, z1), Cons(x2, Cons(z2, z3))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3))), ACK(Cons(z0, z1), Cons(z2, z3))) by 
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Nil, Cons(Cons(Nil, Nil), ack(Cons(x0, Nil), x4))), ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2, c1_1


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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_2, c1_1


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

(37) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace ACK(Cons(x0, Cons(z0, z1)), Cons(x2, Nil)) -> c1(ACK(Cons(z0, z1), ack(z1, ack(Cons(z0, z1), Nil))), ACK(Cons(x0, Cons(z0, z1)), Nil)) by ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
S tuples:
   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil)))
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c_1, c1_1, c1_2


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) by 
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
S tuples:
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_1, c1_2, c_1


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

(41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ACK(Cons(x0, Nil), Cons(x2, Nil)) -> c1(ACK(Cons(x0, Nil), Nil))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
S tuples:
   ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_1, c1_2, c_1


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(x0, x1), Cons(x2, Nil)) -> c1(ACK(Cons(x0, x1), Nil)) by 
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c1_1, c_1


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

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ACK(Cons(z0, Cons(y0, Nil)), Nil) -> c(ACK(Cons(y0, Nil), Cons(Nil, Nil)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4)))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c1_1, c_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(x0, Nil), Cons(x2, Cons(x3, x4))) -> c1(ACK(Cons(x0, Nil), Cons(x3, x4))) by 
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c_1, c1_1


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

(49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(z0, Cons(y0, y1)), Nil) -> c(ACK(Cons(y0, y1), Cons(Nil, Nil))) by 
   ACK(Cons(z0, Cons(z1, Cons(y1, y2))), Nil) -> c(ACK(Cons(z1, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c_1, c1_1


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   ACK(Cons(z0, Cons(y1, Nil)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Nil)), Nil))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c_1, c1_1


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

(53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(z0, Cons(y1, y2)), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, y2)), Nil)) by 
   ACK(Cons(z0, Cons(z1, Cons(y2, y3))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c_1, c1_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y2, y3)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y2, y3)))) by 
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y4, y5)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5))))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil))
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3))
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0)
Tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5))))))
S tuples:
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Nil))) -> c1(ACK(z1, ack(z1, ack(z1, Cons(Nil, Nil)))), ACK(Cons(z0, z1), Cons(x3, Nil)))
   ACK(Cons(z0, z1), Cons(x2, Cons(x3, Cons(z2, z3)))) -> c1(ACK(z1, ack(z1, ack(z1, ack(Cons(z0, z1), z3)))), ACK(Cons(z0, z1), Cons(x3, Cons(z2, z3))))
   ACK(Cons(z0, Cons(z1, z2)), Cons(z3, Nil)) -> c1(ACK(Cons(z1, z2), ack(z2, ack(z2, Cons(Nil, Nil)))), ACK(Cons(z0, Cons(z1, z2)), Nil))
   ACK(Cons(z0, Cons(y0, Cons(y1, y2))), Nil) -> c(ACK(Cons(y0, Cons(y1, y2)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Cons(z2, Nil)) -> c1(ACK(Cons(z0, Cons(y1, Cons(y2, y3))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Nil)))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Nil))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(y3, Cons(y4, y5))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(y3, Cons(y4, y5)))))
   ACK(Cons(z0, Cons(z1, Cons(y1, Nil))), Nil) -> c(ACK(Cons(z1, Cons(y1, Nil)), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y1, Cons(y2, y3)))), Nil) -> c(ACK(Cons(z1, Cons(y1, Cons(y2, y3))), Cons(Nil, Nil)))
   ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Nil))), Nil))
   ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Cons(z3, Nil)) -> c1(ACK(Cons(z0, Cons(z1, Cons(y2, Cons(y3, y4)))), Nil))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Nil))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Nil)))))
   ACK(Cons(z0, Nil), Cons(z1, Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5)))))) -> c1(ACK(Cons(z0, Nil), Cons(z2, Cons(z3, Cons(y3, Cons(y4, y5))))))
K tuples:none
Defined Rule Symbols:   ack_2

Defined Pair Symbols:   ACK_2

Compound Symbols:   c1_2, c_1, c1_1