KILLED
proof of input_vkXr3JcEb3.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (40) CdtProblem
    (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 4 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 11 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 8 ms]
    (64) CdtProblem
    (65) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) 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:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(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:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(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:

   empty(nil) -> true
   empty(cons(x, l)) -> false
   head(cons(x, l)) -> x
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   rev(nil) -> nil
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l))
   last(x, l) -> if(empty(l), x, l)
   if(true, x, l) -> x
   if(false, x, l) -> last(head(l), tail(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:

   empty(nil) -> true [1]
   empty(cons(x, l)) -> false [1]
   head(cons(x, l)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   last(x, l) -> if(empty(l), x, l) [1]
   if(true, x, l) -> x [1]
   if(false, x, l) -> last(head(l), tail(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:

   empty(nil) -> true [1]
   empty(cons(x, l)) -> false [1]
   head(cons(x, l)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   last(x, l) -> if(empty(l), x, l) [1]
   if(true, x, l) -> x [1]
   if(false, x, l) -> last(head(l), tail(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:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1 -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1 -> nil:cons -> rev1
rev2 :: rev1 -> nil:cons -> nil:cons
last :: rev1 -> nil:cons -> rev1
if :: true:false -> rev1 -> nil:cons -> rev1

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:

   last_2
   if_3

(c) The following functions are completely defined:

   rev2_2
   head_1
   tail_1
   empty_1
   rev_1

Due to the following rules being added:

   head(v0) -> const [0]

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:

   empty(nil) -> true [1]
   empty(cons(x, l)) -> false [1]
   head(cons(x, l)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   last(x, l) -> if(empty(l), x, l) [1]
   if(true, x, l) -> x [1]
   if(false, x, l) -> last(head(l), tail(l)) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1]
   head(v0) -> const [0]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1:const -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1:const
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1:const -> nil:cons -> rev1:const
rev2 :: rev1:const -> nil:cons -> nil:cons
last :: rev1:const -> nil:cons -> rev1:const
if :: true:false -> rev1:const -> nil:cons -> rev1:const
const :: rev1:const

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:

   empty(nil) -> true [1]
   empty(cons(x, l)) -> false [1]
   head(cons(x, l)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   last(x, nil) -> if(true, x, nil) [2]
   last(x, cons(x', l')) -> if(false, x, cons(x', l')) [2]
   if(true, x, l) -> x [1]
   if(false, x, cons(x'', l'')) -> last(x'', l'') [3]
   if(false, x, nil) -> last(const, nil) [2]
   if(false, x, cons(x1, l1)) -> last(const, l1) [2]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, nil)) -> rev(cons(x, nil)) [2]
   rev2(x, cons(y, cons(y', l2))) -> rev(cons(x, rev(cons(y, rev2(y', l2))))) [2]
   head(v0) -> const [0]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1:const -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1:const
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1:const -> nil:cons -> rev1:const
rev2 :: rev1:const -> nil:cons -> nil:cons
last :: rev1:const -> nil:cons -> rev1:const
if :: true:false -> rev1:const -> nil:cons -> rev1:const
const :: rev1:const

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
   true => 1
   false => 0
   const => 0

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

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

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if(z, z', z'') -{ 1 }-> x :|: z' = x, z = 1, x >= 0, l >= 0, z'' = l
   if(z, z', z'') -{ 3 }-> last(x'', l'') :|: z' = x, l'' >= 0, z'' = 1 + x'' + l'', x >= 0, x'' >= 0, z = 0
   if(z, z', z'') -{ 2 }-> last(0, l1) :|: x1 >= 0, z' = x, x >= 0, l1 >= 0, z = 0, z'' = 1 + x1 + l1
   if(z, z', z'') -{ 2 }-> last(0, 0) :|: z'' = 0, z' = x, x >= 0, z = 0
   last(z, z') -{ 2 }-> if(1, x, 0) :|: x >= 0, z = x, z' = 0
   last(z, z') -{ 2 }-> if(0, x, 1 + x' + l') :|: x >= 0, x' >= 0, l' >= 0, z' = 1 + x' + l', z = x
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + (1 + x + l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   rev2(z, z') -{ 2 }-> rev(1 + x + rev(1 + y + rev2(y', l2))) :|: x >= 0, y >= 0, y' >= 0, z = x, l2 >= 0, z' = 1 + y + (1 + y' + l2)
   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
   tail(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tail(z) -{ 1 }-> 0 :|: 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:

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 3 }-> last(x'', l'') :|: l'' >= 0, z'' = 1 + x'' + l'', z' >= 0, x'' >= 0, z = 0
   if(z, z', z'') -{ 2 }-> last(0, l1) :|: x1 >= 0, z' >= 0, l1 >= 0, z = 0, z'' = 1 + x1 + l1
   if(z, z', z'') -{ 2 }-> last(0, 0) :|: z'' = 0, z' >= 0, z = 0
   last(z, z') -{ 2 }-> if(1, z, 0) :|: z >= 0, z' = 0
   last(z, z') -{ 2 }-> if(0, z, 1 + x' + l') :|: z >= 0, x' >= 0, l' >= 0, z' = 1 + x' + l'
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + (1 + x + l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   rev2(z, z') -{ 2 }-> rev(1 + z + rev(1 + y + rev2(y', l2))) :|: z >= 0, y >= 0, y' >= 0, l2 >= 0, z' = 1 + y + (1 + y' + l2)
   rev2(z, z') -{ 2 }-> rev(1 + z + 0) :|: z >= 0, z' - 1 >= 0
   rev2(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   tail(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tail(z) -{ 1 }-> 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:

   head(v0) -> null_head [0]

And the following fresh constants:    null_head

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

(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:

   empty(nil) -> true [1]
   empty(cons(x, l)) -> false [1]
   head(cons(x, l)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, l)) -> l [1]
   rev(nil) -> nil [1]
   rev(cons(x, l)) -> cons(rev1(x, l), rev2(x, l)) [1]
   last(x, l) -> if(empty(l), x, l) [1]
   if(true, x, l) -> x [1]
   if(false, x, l) -> last(head(l), tail(l)) [1]
   rev2(x, nil) -> nil [1]
   rev2(x, cons(y, l)) -> rev(cons(x, rev2(y, l))) [1]
   head(v0) -> null_head [0]

The TRS has the following type information:
empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: rev1:null_head -> nil:cons -> nil:cons
false :: true:false
head :: nil:cons -> rev1:null_head
tail :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons
rev1 :: rev1:null_head -> nil:cons -> rev1:null_head
rev2 :: rev1:null_head -> nil:cons -> nil:cons
last :: rev1:null_head -> nil:cons -> rev1:null_head
if :: true:false -> rev1:null_head -> nil:cons -> rev1:null_head
null_head :: rev1:null_head

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
   true => 1
   false => 0
   null_head => 0

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

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

   empty(z) -{ 1 }-> 1 :|: z = 0
   empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if(z, z', z'') -{ 1 }-> x :|: z' = x, z = 1, x >= 0, l >= 0, z'' = l
   if(z, z', z'') -{ 1 }-> last(head(l), tail(l)) :|: z' = x, x >= 0, l >= 0, z = 0, z'' = l
   last(z, z') -{ 1 }-> if(empty(l), x, l) :|: z' = l, x >= 0, l >= 0, z = x
   rev(z) -{ 1 }-> 0 :|: z = 0
   rev(z) -{ 1 }-> 1 + (1 + x + l) + rev2(x, l) :|: x >= 0, l >= 0, z = 1 + x + l
   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
   tail(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l
   tail(z) -{ 1 }-> 0 :|: 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:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   last(z0, z1) -> if(empty(z1), z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> last(head(z1), tail(z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   HEAD(cons(z0, z1)) -> c2
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   REV(nil) -> c5
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1), EMPTY(z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)), HEAD(z1))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)), TAIL(z1))
   REV2(z0, nil) -> c11
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   EMPTY(nil) -> c
   EMPTY(cons(z0, z1)) -> c1
   HEAD(cons(z0, z1)) -> c2
   TAIL(nil) -> c3
   TAIL(cons(z0, z1)) -> c4
   REV(nil) -> c5
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1), EMPTY(z1))
   IF(true, z0, z1) -> c8
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)), HEAD(z1))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)), TAIL(z1))
   REV2(z0, nil) -> c11
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   empty_1, head_1, tail_1, rev_1, last_2, if_3, rev2_2

Defined Pair Symbols:   EMPTY_1, HEAD_1, TAIL_1, REV_1, LAST_2, IF_3, REV2_2

Compound Symbols:   c, c1, c2, c3, c4, c5, c6_1, c7_2, c8, c9_2, c10_2, c11, c12_2


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing nodes:
   EMPTY(cons(z0, z1)) -> c1
   REV2(z0, nil) -> c11
   HEAD(cons(z0, z1)) -> c2
   IF(true, z0, z1) -> c8
   TAIL(cons(z0, z1)) -> c4
   REV(nil) -> c5
   EMPTY(nil) -> c
   TAIL(nil) -> c3

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   last(z0, z1) -> if(empty(z1), z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> last(head(z1), tail(z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1), EMPTY(z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)), HEAD(z1))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)), TAIL(z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1), EMPTY(z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)), HEAD(z1))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)), TAIL(z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   empty_1, head_1, tail_1, rev_1, last_2, if_3, rev2_2

Defined Pair Symbols:   REV_1, LAST_2, IF_3, REV2_2

Compound Symbols:   c6_1, c7_2, c9_2, c10_2, c12_2


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

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   last(z0, z1) -> if(empty(z1), z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> last(head(z1), tail(z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev2(z1, z2)))
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
K tuples:none
Defined Rule Symbols:   empty_1, head_1, tail_1, rev_1, last_2, if_3, rev2_2

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(27) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   rev(nil) -> nil
   last(z0, z1) -> if(empty(z1), z0, z1)
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> last(head(z1), tail(z1))

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

(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))
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(z0, z1) -> c7(IF(empty(z1), z0, z1))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, empty_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LAST(z0, z1) -> c7(IF(empty(z1), z0, z1)) by 
   LAST(x0, nil) -> c7(IF(true, x0, nil))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))

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

(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))
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, nil) -> c7(IF(true, x0, nil))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, nil) -> c7(IF(true, x0, nil))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, empty_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c9_1, c10_1, c7_1


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LAST(x0, nil) -> c7(IF(true, x0, nil))

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

(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))
   empty(nil) -> true
   empty(cons(z0, z1)) -> false
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, empty_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c9_1, c10_1, c7_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   empty(nil) -> true
   empty(cons(z0, z1)) -> false

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1)))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c9_1, c10_1, c7_1


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, z1) -> c9(LAST(head(z1), tail(z1))) by 
   IF(false, x0, nil) -> c9(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, nil) -> c9(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, nil) -> c9(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c10_1, c7_1, c9_1


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF(false, x0, nil) -> c9(LAST(head(nil), nil))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
K tuples:none
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c10_1, c7_1, c9_1


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

(39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(IF(x_1, x_2, x_3)) = x_3
   POL(LAST(x_1, x_2)) = x_2
   POL(REV(x_1)) = 0
   POL(REV2(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(rev(x_1)) = [1]
   POL(rev1(x_1, x_2)) = [1] + x_1 + x_2
   POL(rev2(x_1, x_2)) = [1] + x_1 + x_2
   POL(tail(x_1)) = x_1

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1)))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, IF_3, LAST_2

Compound Symbols:   c6_1, c12_2, c10_1, c7_1, c9_1


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

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, z1) -> c10(LAST(head(z1), tail(z1))) by 
   IF(false, x0, nil) -> c10(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, nil) -> c10(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, nil) -> c10(LAST(head(nil), nil))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF(false, x0, nil) -> c10(LAST(head(nil), nil))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

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

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
And the Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(IF(x_1, x_2, x_3)) = x_3
   POL(LAST(x_1, x_2)) = x_2
   POL(REV(x_1)) = 0
   POL(REV2(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(rev(x_1)) = [1]
   POL(rev1(x_1, x_2)) = [1] + x_1 + x_2
   POL(rev2(x_1, x_2)) = [1] + x_1 + x_2
   POL(tail(x_1)) = x_1

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1)) by 
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, tail(cons(z0, z1)))) by 
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c10_1, c9_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
And the Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(IF(x_1, x_2, x_3)) = x_3
   POL(LAST(x_1, x_2)) = x_2
   POL(REV(x_1)) = 0
   POL(REV2(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(head(x_1)) = [1] + x_1
   POL(nil) = [1]
   POL(rev(x_1)) = [1] + x_1
   POL(rev1(x_1, x_2)) = [1] + x_1 + x_2
   POL(rev2(x_1, x_2)) = [1] + x_1 + x_2
   POL(tail(x_1)) = x_1

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c10_1, c9_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1)) by 
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))

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

(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))
   head(cons(z0, z1)) -> z0
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1, head_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c10_1, c9_1


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

(57) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   head(cons(z0, z1)) -> z0

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

(58)
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))
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c10_1, c9_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, tail(cons(z0, z1)))) by 
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))

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

(60)
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))
   tail(cons(z0, z1)) -> z1
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1, tail_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   tail(cons(z0, z1)) -> z1

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

(62)
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))
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(IF(x_1, x_2, x_3)) = x_2 + x_3
   POL(LAST(x_1, x_2)) = [1] + x_1 + x_2
   POL(REV(x_1)) = 0
   POL(REV2(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = [1]
   POL(nil) = [1]
   POL(rev(x_1)) = [1] + x_1
   POL(rev1(x_1, x_2)) = [1] + x_1 + x_2
   POL(rev2(x_1, x_2)) = [1] + x_1 + x_2

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

(64)
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))
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(65) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))

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

(66)
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))
Tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
S tuples:
   REV(cons(z0, z1)) -> c6(REV2(z0, z1))
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV_1, REV2_2, LAST_2, IF_3

Compound Symbols:   c6_1, c12_2, c7_1, c9_1, c10_1


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

(67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(z0, z1)) -> c6(REV2(z0, z1)) by 
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))

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

(68)
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))
Tuples:
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
S tuples:
   REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, LAST_2, IF_3, REV_1

Compound Symbols:   c12_2, c7_1, c9_1, c10_1, c6_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REV2(z0, cons(z1, z2)) -> c12(REV(cons(z0, rev2(z1, z2))), REV2(z1, z2)) by 
   REV2(x0, cons(z0, nil)) -> c12(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))

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

(70)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, nil)) -> c12(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, nil)) -> c12(REV(cons(x0, nil)), REV2(z0, nil))
   REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(head(cons(z0, z1)), z1))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REV2(x0, cons(z0, nil)) -> c12(REV(cons(x0, nil)), REV2(z0, nil))

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

(72)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2)))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace REV2(x0, cons(z0, cons(z1, z2))) -> c12(REV(cons(x0, rev(cons(z0, rev2(z1, z2))))), REV2(z0, cons(z1, z2))) by 
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c12(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))

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

(74)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c12(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))), REV2(x1, cons(z0, nil)))
   REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c12(REV(cons(x0, rev(cons(x1, rev(cons(z0, rev2(z1, z2))))))), REV2(x1, cons(z0, cons(z1, z2))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2


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

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

(76)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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)))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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)))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2, c12_1


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(x1, cons(z0, cons(z1, z2)))) -> c12(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)))) -> c12(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))))
----------------------------------------

(78)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(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))) -> c12(REV(cons(x0, rev(cons(x1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2, c12_1


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(x0, cons(x1, cons(z0, nil))) -> c12(REV(cons(x0, rev(cons(x1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
----------------------------------------

(80)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2, c12_1


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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)))) -> c12(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))))
----------------------------------------

(82)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil)))))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2, c12_1


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), rev2(z1, nil))))) by REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
----------------------------------------

(84)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
K tuples:
   IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1))
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c9_1, c10_1, c6_1, c12_2, c12_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(false, x0, cons(z0, z1)) -> c9(LAST(z0, z1)) by 
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))

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

(86)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
S tuples:
   REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
K tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV_1, REV2_2

Compound Symbols:   c7_1, c10_1, c6_1, c12_2, c12_1, c9_1


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

(87) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace REV(cons(z0, cons(y1, y2))) -> c6(REV2(z0, cons(y1, y2))) by 
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   REV(cons(x0, cons(rev1(x1, nil), nil))) -> c6(REV2(x0, cons(rev1(x1, nil), nil)))

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

(88)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   REV(cons(x0, cons(rev1(x1, nil), nil))) -> c6(REV2(x0, cons(rev1(x1, nil), nil)))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   REV(cons(x0, cons(rev1(x1, nil), nil))) -> c6(REV2(x0, cons(rev1(x1, nil), nil)))
K tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV2_2, REV_1

Compound Symbols:   c7_1, c10_1, c12_2, c12_1, c9_1, c6_1


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REV(cons(x0, cons(rev1(x1, nil), nil))) -> c6(REV2(x0, cons(rev1(x1, nil), nil)))

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

(90)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
K tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, IF_3, REV2_2, REV_1

Compound Symbols:   c7_1, c10_1, c12_2, c12_1, c9_1, c6_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(false, x0, cons(z0, z1)) -> c10(LAST(z0, z1)) by 
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))

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

(92)
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))
Tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
K tuples:
   LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   LAST_2, REV2_2, IF_3, REV_1

Compound Symbols:   c7_1, c12_2, c12_1, c9_1, c6_1, c10_1


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

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LAST(x0, cons(z0, z1)) -> c7(IF(false, x0, cons(z0, z1))) by 
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))

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

(94)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
K tuples:
   IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, IF_3, REV_1, LAST_2

Compound Symbols:   c12_2, c12_1, c9_1, c6_1, c10_1, c7_1


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

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(false, z0, cons(z1, cons(y1, y2))) -> c9(LAST(z1, cons(y1, y2))) by 
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))

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

(96)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2)))
K tuples:
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, REV_1, IF_3, LAST_2

Compound Symbols:   c12_2, c12_1, c6_1, c10_1, c7_1, c9_1


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

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace REV(cons(x0, cons(rev1(x1, y0), y2))) -> c6(REV2(x0, cons(rev1(x1, y0), y2))) by 
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))

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

(98)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
K tuples:
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, IF_3, LAST_2, REV_1

Compound Symbols:   c12_2, c12_1, c10_1, c7_1, c9_1, c6_1


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

(99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   REV2(z0, cons(z1, cons(z2, nil))) -> c12(REV(cons(z0, cons(rev1(z1, nil), nil))))

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

(100)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
K tuples:
   IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2)))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, IF_3, LAST_2, REV_1

Compound Symbols:   c12_2, c10_1, c7_1, c9_1, c6_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(false, z0, cons(z1, cons(y1, y2))) -> c10(LAST(z1, cons(y1, y2))) by 
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c10(LAST(z1, cons(z2, cons(y2, y3))))

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

(102)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c10(LAST(z1, cons(z2, cons(y2, y3))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
K tuples:
   LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c10(LAST(z1, cons(z2, cons(y2, y3))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, LAST_2, IF_3, REV_1

Compound Symbols:   c12_2, c7_1, c9_1, c6_1, c10_1


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

(103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LAST(z0, cons(z1, cons(y2, y3))) -> c7(IF(false, z0, cons(z1, cons(y2, y3)))) by 
   LAST(z0, cons(z1, cons(z2, cons(y3, y4)))) -> c7(IF(false, z0, cons(z1, cons(z2, cons(y3, y4)))))

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

(104)
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))
Tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c10(LAST(z1, cons(z2, cons(y2, y3))))
   LAST(z0, cons(z1, cons(z2, cons(y3, y4)))) -> c7(IF(false, z0, cons(z1, cons(z2, cons(y3, y4)))))
S tuples:
   REV2(x0, cons(z0, cons(x2, x3))) -> c12(REV(cons(x0, cons(rev1(z0, rev2(x2, x3)), rev2(z0, rev2(x2, x3))))), REV2(z0, cons(x2, x3)))
   REV2(z0, cons(z1, cons(z2, cons(z3, z4)))) -> c12(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))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, y3)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, y3))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4))))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, cons(y3, y4)))))
   REV(cons(z0, cons(rev1(z1, z2), cons(y2, nil)))) -> c6(REV2(z0, cons(rev1(z1, z2), cons(y2, nil))))
K tuples:
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c9(LAST(z1, cons(z2, cons(y2, y3))))
   IF(false, z0, cons(z1, cons(z2, cons(y2, y3)))) -> c10(LAST(z1, cons(z2, cons(y2, y3))))
   LAST(z0, cons(z1, cons(z2, cons(y3, y4)))) -> c7(IF(false, z0, cons(z1, cons(z2, cons(y3, y4)))))
Defined Rule Symbols:   rev2_2, rev_1

Defined Pair Symbols:   REV2_2, IF_3, REV_1, LAST_2

Compound Symbols:   c12_2, c9_1, c6_1, c10_1, c7_1