KILLED
proof of input_bobfuNKmV1.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 11 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 243 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 50 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 1582 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 349 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (46) CdtProblem
    (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 46 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 44 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 11 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) 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:

   plus(0, x) -> x
   plus(s(x), y) -> s(plus(p(s(x)), y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   exp(x, 0) -> s(0)
   exp(x, s(y)) -> times(x, exp(x, y))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   tower(x, y) -> towerIter(x, y, s(0))
   towerIter(0, y, z) -> z
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

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:

   plus(0', x) -> x
   plus(s(x), y) -> s(plus(p(s(x)), y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   exp(x, 0') -> s(0')
   exp(x, s(y)) -> times(x, exp(x, y))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   tower(x, y) -> towerIter(x, y, s(0'))
   towerIter(0', y, z) -> z
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

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:

   plus(0, x) -> x
   plus(s(x), y) -> s(plus(p(s(x)), y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(p(s(x)), y))
   exp(x, 0) -> s(0)
   exp(x, s(y)) -> times(x, exp(x, y))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   tower(x, y) -> towerIter(x, y, s(0))
   towerIter(0, y, z) -> z
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z))

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:

   plus(0, x) -> x [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(p(s(x)), y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   tower(x, y) -> towerIter(x, y, s(0)) [1]
   towerIter(0, y, z) -> z [1]
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [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:

   plus(0, x) -> x [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(p(s(x)), y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   tower(x, y) -> towerIter(x, y, s(0)) [1]
   towerIter(0, y, z) -> z [1]
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s

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

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

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

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

   tower_2
   towerIter_3

(c) The following functions are completely defined:

   p_1
   times_2
   exp_2
   plus_2

Due to the following rules being added:

   p(v0) -> 0 [0]

And the following fresh constants: none

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

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

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

   plus(0, x) -> x [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(p(s(x)), y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   tower(x, y) -> towerIter(x, y, s(0)) [1]
   towerIter(0, y, z) -> z [1]
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s

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:

   plus(0, x) -> x [1]
   plus(s(0), y) -> s(plus(0, y)) [2]
   plus(s(s(x')), y) -> s(plus(s(p(s(x'))), y)) [2]
   plus(s(x), y) -> s(plus(0, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> plus(y, times(0, y)) [2]
   times(s(s(x'')), y) -> plus(y, times(s(p(s(x''))), y)) [2]
   times(s(x), y) -> plus(y, times(0, y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(0)) -> times(x, s(0)) [2]
   exp(x, s(s(y'))) -> times(x, times(x, exp(x, y'))) [2]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   tower(x, y) -> towerIter(x, y, s(0)) [1]
   towerIter(0, y, z) -> z [1]
   towerIter(s(0), y, 0) -> towerIter(0, y, s(0)) [3]
   towerIter(s(0), y, s(y'')) -> towerIter(0, y, times(y, exp(y, y''))) [3]
   towerIter(s(s(x1)), y, 0) -> towerIter(s(p(s(x1))), y, s(0)) [3]
   towerIter(s(s(x1)), y, s(y1)) -> towerIter(s(p(s(x1))), y, times(y, exp(y, y1))) [3]
   towerIter(s(x), y, 0) -> towerIter(0, y, s(0)) [2]
   towerIter(s(x), y, s(y2)) -> towerIter(0, y, times(y, exp(y, y2))) [2]
   p(v0) -> 0 [0]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s

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

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

   0 => 0

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

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

   exp(z', z'') -{ 2 }-> times(x, times(x, exp(x, y'))) :|: z' = x, x >= 0, z'' = 1 + (1 + y'), y' >= 0
   exp(z', z'') -{ 2 }-> times(x, 1 + 0) :|: z' = x, x >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' = x, x >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, y) :|: z'' = y, y >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + x'), y) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, times(0, y)) :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(y, times(0, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, times(1 + p(1 + x''), y)) :|: z' = 1 + (1 + x''), z'' = y, y >= 0, x'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, y, times(y, exp(y, y''))) :|: z'' = y, y >= 0, z1 = 1 + y'', y'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, y, times(y, exp(y, y2))) :|: z' = 1 + x, z1 = 1 + y2, z'' = y, x >= 0, y >= 0, y2 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, y, 1 + 0) :|: z1 = 0, z'' = y, y >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, y, 1 + 0) :|: z' = 1 + x, z1 = 0, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + x1), y, times(y, exp(y, y1))) :|: y1 >= 0, x1 >= 0, z' = 1 + (1 + x1), z1 = 1 + y1, z'' = y, y >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + x1), y, 1 + 0) :|: z1 = 0, x1 >= 0, z' = 1 + (1 + x1), z'' = y, y >= 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:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0


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

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { p }
   { plus }
   { times }
   { exp }
   { towerIter }
   { tower }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower}

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

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

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

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {plus}, {times}, {exp}, {towerIter}, {tower}
Previous analysis results are:
  p: runtime: ?, size: O(n^1) [z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 }-> 1 + plus(1 + p(1 + (z' - 2)), z'') :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(1 + p(1 + (z' - 2)), z'')) :|: z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', times(z'', exp(z'', z1 - 1))) :|: z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(1 + p(1 + (z' - 2)), z'', 1 + 0) :|: z1 = 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower}
Previous analysis results are:
  p: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower}
Previous analysis results are:
  p: runtime: O(n^1) [1 + z'], size: O(n^1) [z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower}
Previous analysis results are:
  p: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  plus: runtime: ?, size: INF

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 2 }-> times(z', 1 + 0) :|: z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= 1 + (z' - 2), z' - 2 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 2 }-> 1 + plus(0, z'') :|: z'' >= 0, z' = 1 + 0
   plus(z', z'') -{ 1 }-> 1 + plus(0, z'') :|: z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 2 + z' }-> 1 + plus(1 + s, z'') :|: s >= 0, s <= 1 + (z' - 2), z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', times(0, z'')) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', times(0, z'')) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + z' }-> plus(z'', times(1 + s', z'')) :|: s' >= 0, s' <= 1 + (z' - 2), z'' >= 0, z' - 2 >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z'' >= 0, z' = 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z'' >= 0, z1 - 1 >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', times(z'', exp(z'', z1 - 1))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   towerIter(z', z'', z1) -{ 3 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 1 + 0
   towerIter(z', z'', z1) -{ 2 }-> towerIter(0, z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s1, z'', 1 + 0) :|: s1 >= 0, s1 <= 1 + (z' - 2), z1 = 0, z' - 2 >= 0, z'' >= 0
   towerIter(z', z'', z1) -{ 3 + z' }-> towerIter(1 + s2, z'', times(z'', exp(z'', z1 - 1))) :|: s2 >= 0, s2 <= 1 + (z' - 2), z1 - 1 >= 0, z' - 2 >= 0, z'' >= 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {towerIter}, {tower}
Previous analysis results are:
  p: runtime: O(n^1) [1 + z'], size: O(n^1) [z']
  plus: runtime: INF, size: INF

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

(31) 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:

   p(v0) -> null_p [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   exp(v0, v1) -> null_exp [0]
   towerIter(v0, v1, v2) -> null_towerIter [0]

And the following fresh constants:    null_p, null_plus, null_times, null_exp, null_towerIter

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

(32)
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:

   plus(0, x) -> x [1]
   plus(s(x), y) -> s(plus(p(s(x)), y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(p(s(x)), y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   tower(x, y) -> towerIter(x, y, s(0)) [1]
   towerIter(0, y, z) -> z [1]
   towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(y, z)) [1]
   p(v0) -> null_p [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   exp(v0, v1) -> null_exp [0]
   towerIter(v0, v1, v2) -> null_towerIter [0]

The TRS has the following type information:
plus :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
0 :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
s :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
p :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
times :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
exp :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
tower :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
towerIter :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter -> 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
null_p :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
null_plus :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
null_times :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
null_exp :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter
null_towerIter :: 0:s:null_p:null_plus:null_times:null_exp:null_towerIter

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

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

   0 => 0
   null_p => 0
   null_plus => 0
   null_times => 0
   null_exp => 0
   null_towerIter => 0

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 1 }-> times(x, exp(x, y)) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   exp(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' = x, x >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(p(1 + x), y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> plus(y, times(p(1 + x), y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   tower(z', z'') -{ 1 }-> towerIter(x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1) -{ 1 }-> z :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   towerIter(z', z'', z1) -{ 1 }-> towerIter(p(1 + x), y, exp(y, z)) :|: z' = 1 + x, z1 = z, z >= 0, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   tower(z0, z1) -> towerIter(z0, z1, s(0))
   towerIter(0, z0, z1) -> z1
   towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2))
Tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, 0) -> c4
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(0)) -> c6
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0)))
   TOWERITER(0, z0, z1) -> c9
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
S tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, 0) -> c4
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(0)) -> c6
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0)))
   TOWERITER(0, z0, z1) -> c9
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, exp_2, p_1, tower_2, towerIter_3

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, P_1, TOWER_2, TOWERITER_3

Compound Symbols:   c, c1_2, c2, c3_3, c4, c5_2, c6, c7_1, c8_1, c9, c10_2, c11_2


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

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TOWER(z0, z1) -> c8(TOWERITER(z0, z1, s(0)))
Removed 5 trailing nodes:
   TIMES(0, z0) -> c2
   P(s(0)) -> c6
   EXP(z0, 0) -> c4
   TOWERITER(0, z0, z1) -> c9
   PLUS(0, z0) -> c

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   tower(z0, z1) -> towerIter(z0, z1, s(0))
   towerIter(0, z0, z1) -> z1
   towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, exp_2, p_1, tower_2, towerIter_3

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, P_1, TOWERITER_3

Compound Symbols:   c1_2, c3_3, c5_2, c7_1, c10_2, c11_2


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

(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   tower(z0, z1) -> towerIter(z0, z1, s(0))
   towerIter(0, z0, z1) -> z1
   towerIter(s(z0), z1, z2) -> towerIter(p(s(z0)), z1, exp(z1, z2))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0)))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, P_1, TOWERITER_3

Compound Symbols:   c1_2, c3_3, c5_2, c7_1, c10_2, c11_2


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

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PLUS(s(z0), z1) -> c1(PLUS(p(s(z0)), z1), P(s(z0))) by 
   PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   TIMES_2, EXP_2, P_1, TOWERITER_3, PLUS_2

Compound Symbols:   c3_3, c5_2, c7_1, c10_2, c11_2, c1_2


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PLUS(s(0), x1) -> c1(PLUS(0, x1), P(s(0)))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0)))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   TIMES_2, EXP_2, P_1, TOWERITER_3, PLUS_2

Compound Symbols:   c3_3, c5_2, c7_1, c10_2, c11_2, c1_2


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(s(z0), z1) -> c3(PLUS(z1, times(p(s(z0)), z1)), TIMES(p(s(z0)), z1), P(s(z0))) by 
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)), TIMES(0, x1), P(s(0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2

Compound Symbols:   c5_2, c7_1, c10_2, c11_2, c1_2, c3_3, c3_1


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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2

Compound Symbols:   c5_2, c7_1, c10_2, c11_2, c1_2, c3_3, c3_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(z0), z1, z2) -> c10(TOWERITER(p(s(z0)), z1, exp(z1, z2)), P(s(z0))) by 
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2

Compound Symbols:   c5_2, c7_1, c11_2, c1_2, c3_3, c3_1, c10_2


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TOWERITER(s(0), x1, x2) -> c10(TOWERITER(0, x1, exp(x1, x2)), P(s(0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TOWERITER_3, PLUS_2, TIMES_2

Compound Symbols:   c5_2, c7_1, c11_2, c1_2, c3_3, c3_1, c10_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(z0), z1, z2) -> c11(TOWERITER(p(s(z0)), z1, exp(z1, z2)), EXP(z1, z2)) by 
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(0), x1, x2) -> c11(TOWERITER(0, x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3

Compound Symbols:   c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2


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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
K tuples:none
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3

Compound Symbols:   c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2, c11_1


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
We considered the (Usable) Rules:none
And the Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EXP(x_1, x_2)) = 0
   POL(P(x_1)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3)) = [1]
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(exp(x_1, x_2)) = x_1 + x_2
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = x_2

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, PLUS_2, TIMES_2, TOWERITER_3

Compound Symbols:   c5_2, c7_1, c1_2, c3_3, c3_1, c10_2, c11_2, c11_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PLUS(s(s(z0)), x1) -> c1(PLUS(s(p(s(z0))), x1), P(s(s(z0)))) by 
   PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0))))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0))))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(0)), x1) -> c1(PLUS(s(0), x1), P(s(s(0))))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_3, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_3, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(p(s(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by 
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(s(0), x1) -> c3(PLUS(x1, times(p(s(0)), x1))) by 
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(x0), z0, 0) -> c10(TOWERITER(p(s(x0)), z0, s(0)), P(s(x0))) by 
   TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TOWERITER(s(0), x1, 0) -> c10(TOWERITER(0, x1, s(0)), P(s(0)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(x0), z0, s(z1)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), P(s(x0))) by 
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TOWERITER(s(0), x1, s(x2)) -> c10(TOWERITER(0, x1, times(x1, exp(x1, x2))), P(s(0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0))))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c10_2, c11_2, c11_1, c1_2, c1_1, c3_3


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(s(z0)), x1, x2) -> c10(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), P(s(s(z0)))) by 
   TOWERITER(s(s(x0)), z0, 0) -> c10(TOWERITER(s(p(s(x0))), z0, s(0)), P(s(s(x0))))
   TOWERITER(s(s(x0)), z0, s(z1)) -> c10(TOWERITER(s(p(s(x0))), z0, times(z0, exp(z0, z1))), P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

(77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
We considered the (Usable) Rules:none
And the Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EXP(x_1, x_2)) = 0
   POL(P(x_1)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3)) = [1]
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(exp(x_1, x_2)) = x_1 + x_2
   POL(p(x_1)) = [1]
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = 0

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
We considered the (Usable) Rules:
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
And the Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EXP(x_1, x_2)) = 0
   POL(P(x_1)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(exp(x_1, x_2)) = x_1 + x_2
   POL(p(x_1)) = x_1
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = 0

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(x0), z0, s(z1)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1))) by 
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(0), x1, s(x2)) -> c11(TOWERITER(0, x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
We considered the (Usable) Rules:none
And the Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EXP(x_1, x_2)) = 0
   POL(P(x_1)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3)) = [1]
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(exp(x_1, x_2)) = x_1 + x_2
   POL(p(x_1)) = [1]
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = x_1

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_2, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(s(z0)), x1, x2) -> c11(TOWERITER(s(p(s(z0))), x1, exp(x1, x2)), EXP(x1, x2)) by 
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(s(x0)), z0, s(z1)) -> c11(TOWERITER(s(p(s(x0))), z0, times(z0, exp(z0, z1))), EXP(z0, s(z1)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)), EXP(z0, 0))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2


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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2


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

(91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
We considered the (Usable) Rules:
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
And the Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(EXP(x_1, x_2)) = 0
   POL(P(x_1)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3)) = x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(exp(x_1, x_2)) = x_1 + x_2
   POL(p(x_1)) = x_1
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = x_1

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(x0), z0, 0) -> c11(TOWERITER(p(s(x0)), z0, s(0))) by 
   TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0)))
   TOWERITER(s(s(z0)), x1, 0) -> c11(TOWERITER(s(p(s(z0))), x1, s(0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2


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

(95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TOWERITER(s(0), x1, 0) -> c11(TOWERITER(0, x1, s(0)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c3_3, c10_2, c10_1, c11_2


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace TIMES(s(s(z0)), x1) -> c3(PLUS(x1, times(s(p(s(z0))), x1)), TIMES(s(p(s(z0))), x1), P(s(s(z0)))) by TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0)))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace TIMES(s(0), x0) -> c3(PLUS(x0, times(0, x0))) by TIMES(s(0), z0) -> c3(PLUS(z0, 0))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   P(s(s(z0))) -> c7(P(s(z0)))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, P_1, TIMES_2, TOWERITER_3, PLUS_2

Compound Symbols:   c5_2, c7_1, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace P(s(s(z0))) -> c7(P(s(z0))) by 
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1

Compound Symbols:   c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1


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

(103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PLUS(s(s(0)), x1) -> c1(P(s(s(0))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)), P(s(s(0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1

Compound Symbols:   c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
K tuples:
   TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, TOWERITER_3, PLUS_2, P_1

Compound Symbols:   c5_2, c3_1, c11_1, c1_2, c1_1, c10_2, c10_1, c11_2, c3_3, c7_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(0), x1, x2) -> c11(EXP(x1, x2)) by 
   TOWERITER(s(0), z0, s(y1)) -> c11(EXP(z0, s(y1)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   PLUS(s(s(x0)), x1) -> c1(P(s(s(x0))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
K tuples:
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c1_1, c10_2, c10_1, c11_2, c11_1, c3_3, c7_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(x0)), x1) -> c1(P(s(s(x0)))) by 
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
K tuples:
   TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0))))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c10_1, c11_2, c11_1, c3_3, c7_1, c1_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(s(x0)), x1, x2) -> c10(P(s(s(x0)))) by 
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), z0) -> c3(PLUS(z0, 0))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
K tuples:
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c7_1, c10_1, c1_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TIMES(s(0), z0) -> c3(PLUS(z0, 0)) by 
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   P(s(s(s(y0)))) -> c7(P(s(s(y0))))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
K tuples:
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c7_1, c10_1, c1_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace P(s(s(s(y0)))) -> c7(P(s(s(y0)))) by 
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0)))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
K tuples:
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c1_1, c7_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(s(y0))), z1) -> c1(P(s(s(s(y0))))) by 
   PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0))))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
   PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0))))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
   PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0))))))
K tuples:
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0)))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c7_1, c1_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(s(s(y0))), z1, z2) -> c10(P(s(s(s(y0))))) by 
   TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0))))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(p(s(z0)), z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(p(s(z0)), z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
   PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0))))))
   TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0))))))
S tuples:
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   TIMES(s(x0), x1) -> c3(PLUS(x1, times(p(s(x0)), x1)))
   PLUS(s(s(s(z0))), x1) -> c1(PLUS(s(s(p(s(z0)))), x1), P(s(s(s(z0)))))
   TIMES(s(s(x0)), x1) -> c3(TIMES(s(p(s(x0))), x1))
   TOWERITER(s(s(z0)), x1, 0) -> c10(TOWERITER(s(p(s(z0))), x1, s(0)), P(s(s(z0))))
   TOWERITER(s(x0), 0, s(x2)) -> c10(TOWERITER(p(s(x0)), 0, 0), P(s(x0)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c10(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), P(s(x0)))
   TOWERITER(s(x0), z0, s(0)) -> c10(TOWERITER(p(s(x0)), z0, times(z0, s(0))), P(s(x0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c10(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), P(s(x0)))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c10(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), P(s(s(z0))))
   TOWERITER(s(s(s(z0))), x1, x2) -> c10(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), P(s(s(s(z0)))))
   TOWERITER(s(x0), 0, s(x2)) -> c11(TOWERITER(p(s(x0)), 0, 0), EXP(0, s(x2)))
   TOWERITER(s(x0), s(z0), s(x2)) -> c11(TOWERITER(p(s(x0)), s(z0), plus(exp(s(z0), x2), times(p(s(z0)), exp(s(z0), x2)))), EXP(s(z0), s(x2)))
   TOWERITER(s(x0), z0, s(0)) -> c11(TOWERITER(p(s(x0)), z0, times(z0, s(0))), EXP(z0, s(0)))
   TOWERITER(s(x0), z0, s(s(z1))) -> c11(TOWERITER(p(s(x0)), z0, times(z0, times(z0, exp(z0, z1)))), EXP(z0, s(s(z1))))
   TOWERITER(s(s(z0)), x1, s(x2)) -> c11(TOWERITER(s(p(s(z0))), x1, times(x1, exp(x1, x2))), EXP(x1, s(x2)))
   TOWERITER(s(s(s(z0))), x1, x2) -> c11(TOWERITER(s(s(p(s(z0)))), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(x0)), z0, 0) -> c11(TOWERITER(s(p(s(x0))), z0, s(0)))
   TIMES(s(s(z0)), z1) -> c3(PLUS(z1, plus(z1, times(p(s(p(s(z0)))), z1))), TIMES(s(p(s(z0))), z1), P(s(s(z0))))
   TIMES(s(0), s(s(s(y0)))) -> c3(PLUS(s(s(s(y0))), 0))
   P(s(s(s(s(y0))))) -> c7(P(s(s(s(y0)))))
   PLUS(s(s(s(s(y0)))), z1) -> c1(P(s(s(s(s(y0))))))
K tuples:
   TOWERITER(s(x0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(0), x1, s(x2)) -> c11(EXP(x1, s(x2)))
   TOWERITER(s(s(0)), x1, x2) -> c11(TOWERITER(s(0), x1, exp(x1, x2)), EXP(x1, x2))
   TOWERITER(s(s(0)), x1, x2) -> c10(TOWERITER(s(0), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(y0)))), z1, z2) -> c10(P(s(s(s(s(y0))))))
Defined Rule Symbols:   p_1, times_2, plus_2, exp_2

Defined Pair Symbols:   EXP_2, TIMES_2, PLUS_2, TOWERITER_3, P_1

Compound Symbols:   c5_2, c3_1, c1_2, c10_2, c11_2, c11_1, c3_3, c10_1, c7_1, c1_1