KILLED
proof of input_2wRS7j11WU.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
        (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) typed CpxTrs
        (13) OrderProof [LOWER BOUND(ID), 16 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 753 ms]
        (16) typed CpxTrs
    (17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (18) TRS for Loop Detection
(19) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(20) CdtProblem
    (21) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (22) CdtProblem
    (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
        (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxTRS
        (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxWeightedTrs
            (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedTrs
            (33) CompletionProof [UPPER BOUND(ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
                (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (36) CpxTypedWeightedCompleteTrs
                (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (38) CpxRNTS
                (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxRNTS
            (41) CompletionProof [UPPER BOUND(ID), 0 ms]
            (42) CpxTypedWeightedCompleteTrs
                (43) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (44) CpxRNTS
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
        (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (48) CdtProblem
        (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (50) CdtProblem
        (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (52) CdtProblem
        (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
        (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
        (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CdtProblem
        (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CdtProblem
(75) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(76) CpxWeightedTrs
    (77) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CpxTypedWeightedTrs
    (79) CompletionProof [UPPER BOUND(ID), 0 ms]
    (80) CpxTypedWeightedCompleteTrs
        (81) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (82) CpxTypedWeightedCompleteTrs
        (83) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (84) CpxRNTS
        (85) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (86) CpxRNTS
    (87) CompletionProof [UPPER BOUND(ID), 0 ms]
    (88) CpxTypedWeightedCompleteTrs
        (89) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (90) CpxRNTS


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

(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
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


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

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

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


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
ACK :: Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2 -> c:c1:c2
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
c2 :: c:c1:c2
GOAL :: Cons:Nil -> Cons:Nil -> c3
c3 :: c:c1:c2 -> c3
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_4 :: c:c1:c2
hole_Cons:Nil2_4 :: Cons:Nil
hole_c33_4 :: c3
gen_c:c1:c24_4 :: Nat -> c:c1:c2
gen_Cons:Nil5_4 :: Nat -> Cons:Nil

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ACK, ack

They will be analysed ascendingly in the following order:
ack < ACK

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

(14)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
ACK :: Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2 -> c:c1:c2
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
c2 :: c:c1:c2
GOAL :: Cons:Nil -> Cons:Nil -> c3
c3 :: c:c1:c2 -> c3
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_4 :: c:c1:c2
hole_Cons:Nil2_4 :: Cons:Nil
hole_c33_4 :: c3
gen_c:c1:c24_4 :: Nat -> c:c1:c2
gen_Cons:Nil5_4 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c1:c24_4(0) <=> c2
gen_c:c1:c24_4(+(x, 1)) <=> c(gen_c:c1:c24_4(x))
gen_Cons:Nil5_4(0) <=> Nil
gen_Cons:Nil5_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil5_4(x))


The following defined symbols remain to be analysed:
ack, ACK

They will be analysed ascendingly in the following order:
ack < ACK

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ack(gen_Cons:Nil5_4(1), gen_Cons:Nil5_4(+(1, n7_4))) -> *6_4, rt in Omega(0)

Induction Base:
ack(gen_Cons:Nil5_4(1), gen_Cons:Nil5_4(+(1, 0)))

Induction Step:
ack(gen_Cons:Nil5_4(1), gen_Cons:Nil5_4(+(1, +(n7_4, 1)))) ->_R^Omega(0)
ack(gen_Cons:Nil5_4(0), ack(Cons(Nil, gen_Cons:Nil5_4(0)), gen_Cons:Nil5_4(+(1, n7_4)))) ->_IH
ack(gen_Cons:Nil5_4(0), *6_4)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
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))
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)

Types:
ACK :: Cons:Nil -> Cons:Nil -> c:c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1:c2 -> c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2 -> c:c1:c2
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
c2 :: c:c1:c2
GOAL :: Cons:Nil -> Cons:Nil -> c3
c3 :: c:c1:c2 -> c3
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_c:c1:c21_4 :: c:c1:c2
hole_Cons:Nil2_4 :: Cons:Nil
hole_c33_4 :: c3
gen_c:c1:c24_4 :: Nat -> c:c1:c2
gen_Cons:Nil5_4 :: Nat -> Cons:Nil


Lemmas:
ack(gen_Cons:Nil5_4(1), gen_Cons:Nil5_4(+(1, n7_4))) -> *6_4, rt in Omega(0)


Generator Equations:
gen_c:c1:c24_4(0) <=> c2
gen_c:c1:c24_4(+(x, 1)) <=> c(gen_c:c1:c24_4(x))
gen_Cons:Nil5_4(0) <=> Nil
gen_Cons:Nil5_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil5_4(x))


The following defined symbols remain to be analysed:
ACK
----------------------------------------

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

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

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

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

(20)
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


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

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

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

(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))
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


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

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

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

(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)
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


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

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

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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)

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

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

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   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(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)

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

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

(30)
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(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) [1]
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) [1]
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) [0]
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) [0]
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) [0]

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

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

(32)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   ACK(Cons(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) [1]
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) [1]
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) [0]
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) [0]
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) [0]

The TRS has the following type information:
ACK :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1 -> c:c1
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil

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

(33) 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:

   ACK_2

(c) The following functions are completely defined:

   ack_2

Due to the following rules being added:

   ack(v0, v1) -> Nil [0]

And the following fresh constants:    const

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

(34)
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(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) [1]
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) [1]
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) [0]
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) [0]
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) [0]
   ack(v0, v1) -> Nil [0]

The TRS has the following type information:
ACK :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1 -> c:c1
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
const :: c:c1

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

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

(36)
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(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) [1]
   ACK(Cons(z0, z1), Cons(z2, Nil)) -> c1(ACK(z1, ack(z1, Cons(Nil, Nil))), ACK(Cons(z0, z1), Nil)) [1]
   ACK(Cons(z0, z1), Cons(z2, Cons(z2', z3'))) -> c1(ACK(z1, ack(z1, ack(Cons(z0, z1), z3'))), ACK(Cons(z0, z1), Cons(z2', z3'))) [1]
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, Nil), ACK(Cons(z0, z1), z3)) [1]
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) [0]
   ack(Cons(z0, z1), Cons(z2, Nil)) -> ack(z1, ack(z1, Cons(Nil, Nil))) [0]
   ack(Cons(z0, z1), Cons(z2, Cons(z2'', z3''))) -> ack(z1, ack(z1, ack(Cons(z0, z1), z3''))) [0]
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, Nil) [0]
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) [0]
   ack(v0, v1) -> Nil [0]

The TRS has the following type information:
ACK :: Cons:Nil -> Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1 -> c:c1
ack :: Cons:Nil -> Cons:Nil -> Cons:Nil
const :: c:c1

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

(37) 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
   const => 0

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   ACK(z, z') -{ 1 }-> 1 + ACK(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, ack(z1, ack(1 + z0 + z1, z3'))) + ACK(1 + z0 + z1, 1 + z2' + z3') :|: z1 >= 0, z3' >= 0, z2' >= 0, z0 >= 0, z' = 1 + z2 + (1 + z2' + z3'), z = 1 + z0 + z1, z2 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, ack(z1, 1 + 0 + 0)) + ACK(1 + z0 + z1, 0) :|: z1 >= 0, z' = 1 + z2 + 0, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, 0) + ACK(1 + z0 + z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, ack(z1, ack(1 + z0 + z1, z3''))) :|: z1 >= 0, z' = 1 + z2 + (1 + z2'' + z3''), z0 >= 0, z3'' >= 0, z2'' >= 0, z = 1 + z0 + z1, z2 >= 0
   ack(z, z') -{ 0 }-> ack(z1, ack(z1, 1 + 0 + 0)) :|: z1 >= 0, z' = 1 + z2 + 0, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   ack(z, z') -{ 0 }-> ack(z1, 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ack(z, z') -{ 0 }-> 1 + (1 + 0 + 0) + z0 :|: z0 >= 0, z = 0, z' = z0


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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   ACK(z, z') -{ 1 }-> 1 + ACK(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, ack(z1, ack(1 + z0 + z1, z3'))) + ACK(1 + z0 + z1, 1 + z2' + z3') :|: z1 >= 0, z3' >= 0, z2' >= 0, z0 >= 0, z' = 1 + z2 + (1 + z2' + z3'), z = 1 + z0 + z1, z2 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, ack(z1, 1 + 0 + 0)) + ACK(1 + z0 + z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' - 1 >= 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, 0) + ACK(1 + z0 + z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, ack(z1, ack(1 + z0 + z1, z3''))) :|: z1 >= 0, z' = 1 + z2 + (1 + z2'' + z3''), z0 >= 0, z3'' >= 0, z2'' >= 0, z = 1 + z0 + z1, z2 >= 0
   ack(z, z') -{ 0 }-> ack(z1, ack(z1, 1 + 0 + 0)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' - 1 >= 0
   ack(z, z') -{ 0 }-> ack(z1, 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ack(z, z') -{ 0 }-> 1 + (1 + 0 + 0) + z' :|: z' >= 0, z = 0


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

(41) 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:

   ack(v0, v1) -> null_ack [0]
   ACK(v0, v1) -> null_ACK [0]

And the following fresh constants:    null_ack, null_ACK

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

(42)
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(z0, z1), Nil) -> c(ACK(z1, Cons(Nil, Nil))) [1]
   ACK(Cons(z0, z1), Cons(z2, z3)) -> c1(ACK(z1, ack(Cons(z0, z1), z3)), ACK(Cons(z0, z1), z3)) [1]
   ack(Cons(z0, z1), Nil) -> ack(z1, Cons(Nil, Nil)) [0]
   ack(Cons(z0, z1), Cons(z2, z3)) -> ack(z1, ack(Cons(z0, z1), z3)) [0]
   ack(Nil, z0) -> Cons(Cons(Nil, Nil), z0) [0]
   ack(v0, v1) -> null_ack [0]
   ACK(v0, v1) -> null_ACK [0]

The TRS has the following type information:
ACK :: Cons:Nil:null_ack -> Cons:Nil:null_ack -> c:c1:null_ACK
Cons :: Cons:Nil:null_ack -> Cons:Nil:null_ack -> Cons:Nil:null_ack
Nil :: Cons:Nil:null_ack
c :: c:c1:null_ACK -> c:c1:null_ACK
c1 :: c:c1:null_ACK -> c:c1:null_ACK -> c:c1:null_ACK
ack :: Cons:Nil:null_ack -> Cons:Nil:null_ack -> Cons:Nil:null_ack
null_ack :: Cons:Nil:null_ack
null_ACK :: c:c1:null_ACK

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

(43) 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
   null_ack => 0
   null_ACK => 0

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   ACK(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ACK(z, z') -{ 1 }-> 1 + ACK(z1, ack(1 + z0 + z1, z3)) + ACK(1 + z0 + z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, ack(1 + z0 + z1, z3)) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ack(z, z') -{ 0 }-> ack(z1, 1 + 0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' = 0
   ack(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ack(z, z') -{ 0 }-> 1 + (1 + 0 + 0) + z0 :|: z0 >= 0, z = 0, z' = z0

Only complete derivations are relevant for the runtime complexity.

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

(45) 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)))

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

(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), 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


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

(47) 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))

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

(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), 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


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

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

(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), 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


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

(51) 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))))

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

(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), 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


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

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

(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), 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


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

(55) 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))
----------------------------------------

(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), 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


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

(57) 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)))

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

(58)
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


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

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

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

(60)
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


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

(61) 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))

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

(62)
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


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

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

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

(64)
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


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

(65) 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))))

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

(66)
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


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

(67) 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)))

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

(68)
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


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

(69) 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))

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

(70)
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


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

(71) 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))

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

(72)
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


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

(73) 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))))))

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

(74)
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


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

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

(76)
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
----------------------------------------

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

(78)
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
----------------------------------------

(79) 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

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

(80)
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
----------------------------------------

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

(82)
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
----------------------------------------

(83) 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

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

(84)
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


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

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

(86)
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


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

(87) 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

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

(88)
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
----------------------------------------

(89) 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

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

(90)
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.