KILLED
proof of input_JgtXAVHV5a.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (52) CdtProblem
    (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) 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:

   rev1(0, nil) -> 0
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(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:

   rev1(0', nil) -> 0'
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(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:

   rev1(0, nil) -> 0
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(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:

   rev1(0, nil) -> 0 [1]
   rev1(s(X), nil) -> s(X) [1]
   rev1(X, cons(Y, L)) -> rev1(Y, L) [1]
   rev(nil) -> nil [1]
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1]
   rev2(X, nil) -> nil [1]
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(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:

   rev1(0, nil) -> 0 [1]
   rev1(s(X), nil) -> s(X) [1]
   rev1(X, cons(Y, L)) -> rev1(Y, L) [1]
   rev(nil) -> nil [1]
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1]
   rev2(X, nil) -> nil [1]
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1]

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

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:

   rev_1
   rev2_2
   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:

   rev1(0, nil) -> 0 [1]
   rev1(s(X), nil) -> s(X) [1]
   rev1(X, cons(Y, L)) -> rev1(Y, L) [1]
   rev(nil) -> nil [1]
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1]
   rev2(X, nil) -> nil [1]
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1]

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

   rev1(0, nil) -> 0 [1]
   rev1(s(X), nil) -> s(X) [1]
   rev1(X, cons(Y, L)) -> rev1(Y, L) [1]
   rev(nil) -> nil [1]
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1]
   rev2(X, nil) -> nil [1]
   rev2(X, cons(Y, nil)) -> rev(cons(X, rev(nil))) [2]
   rev2(X, cons(Y, cons(Y', L'))) -> rev(cons(X, rev(rev(cons(Y, rev(rev2(Y', L'))))))) [2]

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

   0 => 0
   nil => 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) :|: z = 1 + X + L, X >= 0, L >= 0
   rev1(z, z') -{ 1 }-> rev1(Y, L) :|: Y >= 0, X >= 0, z = X, L >= 0, z' = 1 + Y + L
   rev1(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   rev1(z, z') -{ 1 }-> 1 + X :|: z = 1 + X, X >= 0, z' = 0
   rev2(z, z') -{ 2 }-> rev(1 + X + rev(rev(1 + Y + rev(rev2(Y', L'))))) :|: L' >= 0, Y >= 0, Y' >= 0, X >= 0, z' = 1 + Y + (1 + Y' + L'), z = X
   rev2(z, z') -{ 2 }-> rev(1 + X + rev(0)) :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 0
   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) :|: z = 1 + X + L, X >= 0, L >= 0
   rev1(z, z') -{ 1 }-> rev1(Y, L) :|: Y >= 0, z >= 0, L >= 0, z' = 1 + Y + L
   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(rev(1 + Y + rev(rev2(Y', L'))))) :|: L' >= 0, Y >= 0, Y' >= 0, z >= 0, z' = 1 + Y + (1 + Y' + L')
   rev2(z, z') -{ 2 }-> rev(1 + z + rev(0)) :|: z' - 1 >= 0, z >= 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:

   rev1(0, nil) -> 0 [1]
   rev1(s(X), nil) -> s(X) [1]
   rev1(X, cons(Y, L)) -> rev1(Y, L) [1]
   rev(nil) -> nil [1]
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L)) [1]
   rev2(X, nil) -> nil [1]
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L)))) [1]

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

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

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5_1, c6, c7_3


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   REV1(s(z0), nil) -> c1
   REV2(z0, nil) -> c6
   REV1(0, nil) -> c
   REV(nil) -> c3

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
S tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c2_1, c4_1, c5_1, c7_3


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

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

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))), REV(nil), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c2_1, c4_1, c5_1, c7_3, c7_1


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

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev2(z0, cons(z1, z2))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c2_1, c4_1, c5_1, c7_3, c7_1


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

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(rev2(z0, nil)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c2_1, c4_1, c5_1, c7_1, c7_3


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

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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
S tuples:
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c2_1, c4_1, c5_1, c7_1, c7_3


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

(33) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2)) by 
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
S tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))), REV(rev(cons(z0, rev(rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c4_1, c5_1, c7_1, c7_3, c2_1


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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
S tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3))))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c4_1, c5_1, c7_1, c2_1, c7_3


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

(37) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(x1, cons(x2, x3))) -> c7(REV(rev(cons(x1, rev(rev2(x2, x3)))))) by REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
S tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c4_1, c5_1, c7_1, c2_1, c7_3


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

(39) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) by REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
----------------------------------------

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
S tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(rev(cons(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c4_1, c5_1, c7_1, c2_1, c7_3


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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
S tuples:
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c4_1, c5_1, c7_1, c2_1, c7_3


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

(43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, z1)) -> c4(REV1(z0, z1)) by 
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
S tuples:
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, REV1_2

Compound Symbols:   c5_1, c7_1, c2_1, c7_3, c4_1


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

(45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, z1)) -> c5(REV2(z0, z1)) by 
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
S tuples:
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, nil)) -> c7(REV(cons(z0, nil)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_1, c2_1, c7_3, c4_1, c5_1


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

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
S tuples:
   REV2(x0, cons(x1, x2)) -> c7(REV(cons(x0, rev(rev2(x1, x2)))))
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV2_2, REV1_2, REV_1

Compound Symbols:   c7_1, c2_1, c7_3, c4_1, c5_1


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

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
S tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
   REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c2_1, c7_1, c7_3, c4_1, c5_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REV2(x0, cons(z0, nil)) -> c7(REV(cons(x0, rev(nil)))) by 
   REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
   REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil)))
S tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
   REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil)))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c2_1, c7_1, c7_3, c4_1, c5_1


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

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   REV(cons(z0, cons(y1, nil))) -> c5(REV2(z0, cons(y1, nil)))
   REV2(x0, cons(x1, nil)) -> c7(REV(cons(x0, nil)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
S tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(x0, cons(z0, cons(z1, z2))) -> c7(REV(cons(x0, rev(rev(cons(z0, rev(rev2(z1, z2))))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c2_1, c7_1, c7_3, c4_1, c5_1


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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
S tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c2_1, c7_1, c7_3, c4_1, c5_1


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
S tuples:
   REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2)))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV2_2, REV_1

Compound Symbols:   c2_1, c7_1, c7_3, c4_1, c5_1


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

(59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV1(z0, cons(z1, cons(y1, y2))) -> c2(REV1(z1, cons(y1, y2))) by 
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
S tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3))))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV2_2, REV_1, REV1_2

Compound Symbols:   c7_1, c7_3, c4_1, c5_1, c2_1


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

(61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, cons(y1, cons(y2, y3)))) -> c4(REV1(z0, cons(y1, cons(y2, y3)))) by 
   REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4)))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4)))))
S tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4)))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV2_2, REV_1, REV1_2

Compound Symbols:   c7_1, c7_3, c5_1, c2_1, c4_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, cons(y1, y2))) -> c5(REV2(z0, cons(y1, y2))) by 
   REV(cons(z0, cons(z1, cons(y2, y3)))) -> c5(REV2(z0, cons(z1, cons(y2, y3))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4)))))
S tuples:
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, rev(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))), REV(cons(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))), REV2(z1, cons(z2, z3)))
   REV(cons(z0, cons(y1, cons(y2, y3)))) -> c5(REV2(z0, cons(y1, cons(y2, y3))))
   REV2(z0, cons(z1, cons(z2, z3))) -> c7(REV(cons(z0, cons(rev1(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3)))), rev2(rev1(z1, rev(rev2(z2, z3))), rev2(z1, rev(rev2(z2, z3))))))))
   REV1(z0, cons(z1, cons(z2, cons(y2, y3)))) -> c2(REV1(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c4(REV1(z0, cons(z1, cons(z2, cons(y3, y4)))))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV2_2, REV_1, REV1_2

Compound Symbols:   c7_1, c7_3, c5_1, c2_1, c4_1