KILLED
proof of input_8nvDsNY9Kj.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtLeafRemovalProof [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:

   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   rev1(0, nil) -> 0
   rev1(s(x), nil) -> s(x)
   rev1(x, cons(y, l)) -> rev1(y, l)
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

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:

   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   rev1(0', nil) -> 0'
   rev1(s(x), nil) -> s(x)
   rev1(x, cons(y, l)) -> rev1(y, l)
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

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:

   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   rev1(0, nil) -> 0
   rev1(s(x), nil) -> s(x)
   rev1(x, cons(y, l)) -> rev1(y, l)
   rev2(x, nil) -> nil
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l)))

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:

   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   rev1(0, nil) -> 0 [1]
   rev1(s(x), nil) -> s(x) [1]
   rev1(x, cons(y, l)) -> rev1(y, l) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [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:

   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   rev1(0, nil) -> 0 [1]
   rev1(s(x), nil) -> s(x) [1]
   rev1(x, cons(y, l)) -> rev1(y, l) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1]

The TRS has the following type information:
rev :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
rev1 :: 0:s -> nil:cons -> 0:s
rev2 :: 0:s -> nil:cons -> nil:cons
0 :: 0:s
s :: a -> 0:s

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:
none

(c) The following functions are completely defined:

   rev2_2
   rev_1
   rev1_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   rev1(0, nil) -> 0 [1]
   rev1(s(x), nil) -> s(x) [1]
   rev1(x, cons(y, l)) -> rev1(y, l) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1]

The TRS has the following type information:
rev :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
rev1 :: 0:s -> nil:cons -> 0:s
rev2 :: 0:s -> nil:cons -> nil:cons
0 :: 0:s
s :: a -> 0:s
const :: a

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:

   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   rev1(0, nil) -> 0 [1]
   rev1(s(x), nil) -> s(x) [1]
   rev1(x, cons(y, l)) -> rev1(y, l) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, nil)) -> rev(cons(x, nil)) [2]
   rev2(x, cons(y, cons(y', l'))) -> rev(cons(x, rev(cons(y, rev2(y', l'))))) [2]

The TRS has the following type information:
rev :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
rev1 :: 0:s -> nil:cons -> 0:s
rev2 :: 0:s -> nil:cons -> nil:cons
0 :: 0:s
s :: a -> 0:s
const :: a

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
   0 => 0
   const => 0

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

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

   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev1(x, l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   rev1(z, z') -{ 1 }-> rev1(y, l) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x
   rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   rev1(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
   rev2(z, z') -{ 2 }-> rev(1 + x + rev(1 + y + rev2(y', l'))) :|: x >= 0, y >= 0, l' >= 0, y' >= 0, z = x, z' = 1 + y + (1 + y' + l')
   rev2(z, z') -{ 2 }-> rev(1 + x + 0) :|: x >= 0, y >= 0, z' = 1 + y + 0, z = x
   rev2(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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

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

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

   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev1(x, l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   rev1(z, z') -{ 1 }-> rev1(y, l) :|: z >= 0, y >= 0, z' = 1 + y + l, l >= 0
   rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   rev1(z, z') -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0, z' = 0
   rev2(z, z') -{ 2 }-> rev(1 + z + rev(1 + y + rev2(y', l'))) :|: z >= 0, y >= 0, l' >= 0, y' >= 0, z' = 1 + y + (1 + y' + l')
   rev2(z, z') -{ 2 }-> rev(1 + z + 0) :|: z >= 0, z' - 1 >= 0
   rev2(z, z') -{ 1 }-> 0 :|: 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:    const

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

(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:

   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   rev1(0, nil) -> 0 [1]
   rev1(s(x), nil) -> s(x) [1]
   rev1(x, cons(y, l)) -> rev1(y, l) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1]

The TRS has the following type information:
rev :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
rev1 :: 0:s -> nil:cons -> 0:s
rev2 :: 0:s -> nil:cons -> nil:cons
0 :: 0:s
s :: a -> 0:s
const :: a

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
   0 => 0
   const => 0

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

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

   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + rev1(x, l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   rev1(z, z') -{ 1 }-> rev1(y, l) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x
   rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   rev1(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0
   rev2(z, z') -{ 1 }-> rev(1 + x + rev2(y, l)) :|: x >= 0, y >= 0, z' = 1 + y + l, l >= 0, z = x
   rev2(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 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:
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0, nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   REV(nil) -> c
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(0, nil) -> c3
   REV1(s(z0), nil) -> c4
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev_1, rev1_2, rev2_2

Defined Pair Symbols:   REV_1, REV1_2, REV2_2

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_1, c6, c7_2


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   REV2(z0, nil) -> c6
   REV1(0, nil) -> c3
   REV(nil) -> c
   REV1(s(z0), nil) -> c4

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev_1, rev1_2, rev2_2

Defined Pair Symbols:   REV_1, REV1_2, REV2_2

Compound Symbols:   c1_1, c2_1, c5_1, c7_2


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

(25) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   rev(nil) -> nil

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   REV(cons(z0, z1)) -> c1(REV1(z0, z1))
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV_1, REV1_2, REV2_2

Compound Symbols:   c1_1, c2_1, c5_1, c7_2


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

(27) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, z1)) -> c1(REV1(z0, z1)) by 
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
S tuples:
   REV(cons(z0, z1)) -> c2(REV2(z0, z1))
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV_1, REV1_2, REV2_2

Compound Symbols:   c2_1, c5_1, c7_2, c1_1


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

(29) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, z1)) -> c2(REV2(z0, z1)) by 
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
S tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c5_1, c7_2, c1_1, c2_1


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

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) by 
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
S tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c5_1, c1_1, c2_1, c7_2


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, nil)), REV2(z0, nil))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
S tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c5_1, c1_1, c2_1, c7_2


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) by 
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c5_1, c1_1, c2_1, c7_2


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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c5_1, c1_1, c2_1, c7_2, c7_1


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV1(z0, cons(z1, z2)) -> c5(REV1(z1, z2)) by 
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c1_1, c2_1, c7_2, c7_1, c5_1


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

(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c1(REV1(z0, cons(y1, y2))) by 
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c2_1, c7_2, c7_1, c5_1, c1_1


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

(43) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c7(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2)))) by REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c2_1, c7_2, c7_1, c5_1, c1_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c2(REV2(z0, cons(y1, y2))) by 
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_2, c7_1, c5_1, c1_1, c2_1


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

(47) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(x1, cons(z0, nil))) -> c7(REV(cons(x0, rev(cons(x1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_2, c5_1, c1_1, c2_1, c7_1


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

(49) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, rev(cons(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4)))) by REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_2, c5_1, c1_1, c2_1, c7_1


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

(51) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil))))
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_2, c5_1, c1_1, c2_1, c7_1


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

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REV2(z0, cons(z1, cons(z2, nil))) -> c7(REV(cons(z0, cons(rev1(z1, nil), nil))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_2, c5_1, c1_1, c2_1


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

(55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV1(z0, cons(z1, cons(y1, y2))) -> c5(REV1(z1, cons(y1, y2))) by 
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c7(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c1(REV1(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c2(REV2(z0, cons(z1, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(y2, nil)))) -> c2(REV2(z0, cons(z1, cons(y2, nil))))
   REV(cons(z0, cons(z1, cons(y2, cons(y3, y4))))) -> c2(REV2(z0, cons(z1, cons(y2, cons(y3, y4)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c7(REV(cons(z0, cons(rev1(z1, rev(cons(z2, rev2(z3, z4)))), rev2(z1, cons(rev1(z2, rev2(z3, z4)), rev2(z2, rev2(z3, z4))))))), REV2(z1, cons(z2, cons(z3, z4))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c5(REV1(z1, cons(z2, cons(y2, y3))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, rev1_2

Defined Pair Symbols:   REV2_2, REV_1, REV1_2

Compound Symbols:   c7_2, c1_1, c2_1, c5_1