KILLED
proof of input_jdbotDhWi1.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 21 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 217 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 267 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 357 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 879 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
        (42) CpxRNTS
    (43) CompletionProof [UPPER BOUND(ID), 0 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
(47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(48) CdtProblem
    (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (68) CdtProblem
    (69) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (72) CdtProblem
    (73) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [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


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

(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(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   exp(x, 0) -> s(0)
   exp(x, s(y)) -> times(x, exp(x, y))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   tower(x, y) -> towerIter(0, x, y, s(0))
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
   help(true, c, x, y, z) -> z
   help(false, c, x, y, z) -> towerIter(s(c), 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(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   exp(x, 0') -> s(0')
   exp(x, s(y)) -> times(x, exp(x, y))
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   tower(x, y) -> towerIter(0', x, y, s(0'))
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
   help(true, c, x, y, z) -> z
   help(false, c, x, y, z) -> towerIter(s(c), 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(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   exp(x, 0) -> s(0)
   exp(x, s(y)) -> times(x, exp(x, y))
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   tower(x, y) -> towerIter(0, x, y, s(0))
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z)
   help(true, c, x, y, z) -> z
   help(false, c, x, y, z) -> towerIter(s(c), 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(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   tower(x, y) -> towerIter(0, x, y, s(0)) [1]
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) [1]
   help(true, c, x, y, z) -> z [1]
   help(false, c, x, y, z) -> towerIter(s(c), 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(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   tower(x, y) -> towerIter(0, x, y, s(0)) [1]
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) [1]
   help(true, c, x, y, z) -> z [1]
   help(false, c, x, y, z) -> towerIter(s(c), 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
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
help :: true:false -> 0:s -> 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_4
   help_5

(c) The following functions are completely defined:

   exp_2
   times_2
   ge_2
   plus_2

Due to the following rules being added:
none

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(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   tower(x, y) -> towerIter(0, x, y, s(0)) [1]
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) [1]
   help(true, c, x, y, z) -> z [1]
   help(false, c, x, y, z) -> towerIter(s(c), 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
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
help :: true:false -> 0:s -> 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(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> plus(y, 0) [2]
   times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2]
   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]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   tower(x, y) -> towerIter(0, x, y, s(0)) [1]
   towerIter(c, 0, y, z) -> help(true, c, 0, y, z) [2]
   towerIter(0, s(x''), y, z) -> help(false, 0, s(x''), y, z) [2]
   towerIter(s(x1), s(y''), y, z) -> help(ge(x1, y''), s(x1), s(y''), y, z) [2]
   help(true, c, x, y, z) -> z [1]
   help(false, c, x, y, 0) -> towerIter(s(c), x, y, s(0)) [2]
   help(false, c, x, y, s(y1)) -> towerIter(s(c), x, y, times(y, exp(y, y1))) [2]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
help :: true:false -> 0:s -> 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
   true => 1
   false => 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
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = 1 + x, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z :|: z2 = y, z >= 0, c >= 0, z3 = z, x >= 0, y >= 0, z' = 1, z'' = c, z1 = x
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + c, x, y, times(y, exp(y, y1))) :|: y1 >= 0, z2 = y, c >= 0, x >= 0, y >= 0, z'' = c, z1 = x, z' = 0, z3 = 1 + y1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + c, x, y, 1 + 0) :|: z2 = y, c >= 0, z3 = 0, x >= 0, y >= 0, z'' = c, z1 = x, z' = 0
   plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(x1, y''), 1 + x1, 1 + y'', y, z) :|: z1 = y, x1 >= 0, z >= 0, z2 = z, y >= 0, y'' >= 0, z' = 1 + x1, z'' = 1 + y''
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, c, 0, y, z) :|: z'' = 0, z1 = y, z >= 0, c >= 0, z2 = z, y >= 0, z' = c
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + x'', y, z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z'' = 1 + x'', x'' >= 0, z' = 0


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

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

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

   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
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0


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

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

   { ge }
   { plus }
   { times }
   { exp }
   { help, 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
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {ge}, {plus}, {times}, {exp}, {help,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
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(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
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {ge}, {plus}, {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 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
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(ge(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]

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

(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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

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

(32)
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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z'*z'' + z''

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

(34)
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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {times}, {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: ?, size: O(n^2) [2*z'*z'' + z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 4*z' + 2*z'*z'' + z''

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

(36)
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
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z'']

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

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

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

   exp(z', z'') -{ 7 + 6*z' }-> s5 :|: s5 >= 0, s5 <= 1 + 0 + 2 * ((1 + 0) * z'), z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s4 :|: s2 >= 0, s2 <= z'' + 2 * (z'' * (z' - 2)), s3 >= 0, s3 <= z'' + s2, s4 >= 0, s4 <= z'' + s3, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z'']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   exp(z', z'') -{ 7 + 6*z' }-> s5 :|: s5 >= 0, s5 <= 1 + 0 + 2 * ((1 + 0) * z'), z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s4 :|: s2 >= 0, s2 <= z'' + 2 * (z'' * (z' - 2)), s3 >= 0, s3 <= z'' + s2, s4 >= 0, s4 <= z'' + s3, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z'']
  exp: runtime: ?, size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   exp(z', z'') -{ 7 + 6*z' }-> s5 :|: s5 >= 0, s5 <= 1 + 0 + 2 * ((1 + 0) * z'), z' >= 0, z'' = 1 + 0
   exp(z', z'') -{ 2 }-> times(z', times(z', exp(z', z'' - 2))) :|: z' >= 0, z'' - 2 >= 0
   exp(z', z'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z3 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, times(z2, exp(z2, z3 - 1))) :|: z3 - 1 >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z' = 0
   help(z', z'', z1, z2, z3) -{ 2 }-> towerIter(1 + z'', z1, z2, 1 + 0) :|: z'' >= 0, z3 = 0, z1 >= 0, z2 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 4*z' + 2*z'*z'' + -1*z'' }-> s4 :|: s2 >= 0, s2 <= z'' + 2 * (z'' * (z' - 2)), s3 >= 0, s3 <= z'' + s2, s4 >= 0, s4 <= z'' + s3, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, z', z'', 1 + 0) :|: z' >= 0, z'' >= 0
   towerIter(z', z'', z1, z2) -{ 3 + z'' }-> help(s', 1 + (z' - 1), 1 + (z'' - 1), z1, z2) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z2 >= 0, z1 >= 0, z'' - 1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(1, z', 0, z1, z2) :|: z'' = 0, z2 >= 0, z' >= 0, z1 >= 0
   towerIter(z', z'', z1, z2) -{ 2 }-> help(0, 0, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z1 >= 0, z'' - 1 >= 0, z' = 0

Function symbols to be analyzed: {exp}, {help,towerIter}, {tower}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [4 + 4*z' + 2*z'*z'' + z''], size: O(n^2) [2*z'*z'' + z'']
  exp: runtime: INF, size: INF

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

(43) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(44)
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(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   ge(x, 0) -> true [1]
   ge(0, s(x)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   tower(x, y) -> towerIter(0, x, y, s(0)) [1]
   towerIter(c, x, y, z) -> help(ge(c, x), c, x, y, z) [1]
   help(true, c, x, y, z) -> z [1]
   help(false, c, x, y, z) -> towerIter(s(c), 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
times :: 0:s -> 0:s -> 0:s
exp :: 0:s -> 0:s -> 0:s
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
tower :: 0:s -> 0:s -> 0:s
towerIter :: 0:s -> 0:s -> 0:s -> 0:s -> 0:s
help :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s

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

(45) 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
   true => 1
   false => 0

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

(46)
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'') -{ 1 }-> 1 + 0 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = 1 + x, z' = 0
   help(z', z'', z1, z2, z3) -{ 1 }-> z :|: z2 = y, z >= 0, c >= 0, z3 = z, x >= 0, y >= 0, z' = 1, z'' = c, z1 = x
   help(z', z'', z1, z2, z3) -{ 1 }-> towerIter(1 + c, x, y, exp(y, z)) :|: z2 = y, z >= 0, c >= 0, z3 = z, x >= 0, y >= 0, z'' = c, z1 = x, z' = 0
   plus(z', z'') -{ 1 }-> x :|: x >= 0, z'' = x, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   tower(z', z'') -{ 1 }-> towerIter(0, x, y, 1 + 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   towerIter(z', z'', z1, z2) -{ 1 }-> help(ge(c, x), c, x, y, z) :|: z1 = y, z >= 0, c >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = c

Only complete derivations are relevant for the runtime complexity.

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   tower(z0, z1) -> towerIter(0, z0, z1, s(0))
   towerIter(z0, z1, z2, z3) -> help(ge(z0, z1), z0, z1, z2, z3)
   help(true, z0, z1, z2, z3) -> z3
   help(false, z0, z1, z2, z3) -> towerIter(s(z0), z1, z2, exp(z2, z3))
Tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, 0) -> c4
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(z0, 0) -> c6
   GE(0, s(z0)) -> c7
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWER(z0, z1) -> c9(TOWERITER(0, z0, z1, s(0)))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(true, z0, z1, z2, z3) -> c11
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
S tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, 0) -> c4
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(z0, 0) -> c6
   GE(0, s(z0)) -> c7
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWER(z0, z1) -> c9(TOWERITER(0, z0, z1, s(0)))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(true, z0, z1, z2, z3) -> c11
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, exp_2, ge_2, tower_2, towerIter_4, help_5

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, TOWER_2, TOWERITER_4, HELP_5

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_2, c6, c7, c8_1, c9_1, c10_2, c11, c12_2


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TOWER(z0, z1) -> c9(TOWERITER(0, z0, z1, s(0)))
Removed 6 trailing nodes:
   GE(z0, 0) -> c6
   GE(0, s(z0)) -> c7
   TIMES(0, z0) -> c2
   HELP(true, z0, z1, z2, z3) -> c11
   PLUS(0, z0) -> c
   EXP(z0, 0) -> c4

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   tower(z0, z1) -> towerIter(0, z0, z1, s(0))
   towerIter(z0, z1, z2, z3) -> help(ge(z0, z1), z0, z1, z2, z3)
   help(true, z0, z1, z2, z3) -> z3
   help(false, z0, z1, z2, z3) -> towerIter(s(z0), z1, z2, exp(z2, z3))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, exp_2, ge_2, tower_2, towerIter_4, help_5

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, TOWERITER_4, HELP_5

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c10_2, c12_2


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

(51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   tower(z0, z1) -> towerIter(0, z0, z1, s(0))
   towerIter(z0, z1, z2, z3) -> help(ge(z0, z1), z0, z1, z2, z3)
   help(true, z0, z1, z2, z3) -> z3
   help(false, z0, z1, z2, z3) -> towerIter(s(z0), z1, z2, exp(z2, z3))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, TOWERITER_4, HELP_5

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c10_2, c12_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(z0, z1, z2, z3) -> c10(HELP(ge(z0, z1), z0, z1, z2, z3), GE(z0, z1)) by 
   TOWERITER(z0, 0, x2, x3) -> c10(HELP(true, z0, 0, x2, x3), GE(z0, 0))
   TOWERITER(0, s(z0), x2, x3) -> c10(HELP(false, 0, s(z0), x2, x3), GE(0, s(z0)))
   TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(z0, 0, x2, x3) -> c10(HELP(true, z0, 0, x2, x3), GE(z0, 0))
   TOWERITER(0, s(z0), x2, x3) -> c10(HELP(false, 0, s(z0), x2, x3), GE(0, s(z0)))
   TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(z0, 0, x2, x3) -> c10(HELP(true, z0, 0, x2, x3), GE(z0, 0))
   TOWERITER(0, s(z0), x2, x3) -> c10(HELP(false, 0, s(z0), x2, x3), GE(0, s(z0)))
   TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c12_2, c10_2


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

(55) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TOWERITER(0, s(z0), x2, x3) -> c10(HELP(false, 0, s(z0), x2, x3), GE(0, s(z0)))
Removed 1 trailing nodes:
   TOWERITER(z0, 0, x2, x3) -> c10(HELP(true, z0, 0, x2, x3), GE(z0, 0))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c12_2, c10_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TOWERITER(s(z0), s(z1), x2, x3) -> c10(HELP(ge(z0, z1), s(z0), s(z1), x2, x3), GE(s(z0), s(z1))) by 
   TOWERITER(s(z0), s(0), x2, x3) -> c10(HELP(true, s(z0), s(0), x2, x3), GE(s(z0), s(0)))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(HELP(true, s(z0), s(0), x2, x3), GE(s(z0), s(0)))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(HELP(true, s(z0), s(0), x2, x3), GE(s(z0), s(0)))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c12_2, c10_2, c10_1


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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c12_2, c10_2, c10_1


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

(61) 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), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   ge(z0, 0) -> true
And the Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(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(GE(x_1, x_2)) = 0
   POL(HELP(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(TOWERITER(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(exp(x_1, x_2)) = [1] + x_1 + x_2
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(plus(x_1, x_2)) = x_2
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = [1]
   POL(true) = 0

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   PLUS_2, TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4

Compound Symbols:   c1_1, c3_2, c5_2, c8_1, c12_2, c10_2, c10_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c1(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, HELP_5, TOWERITER_4, PLUS_2

Compound Symbols:   c3_2, c5_2, c8_1, c12_2, c10_2, c10_1, c1_1


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

(65) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(false, z0, z1, z2, z3) -> c12(TOWERITER(s(z0), z1, z2, exp(z2, z3)), EXP(z2, z3)) by 
   HELP(false, s(0), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_2, c10_1, c1_1, c12_2


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

(67) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   TOWERITER(s(0), s(s(z0)), x2, x3) -> c10(HELP(false, s(0), s(s(z0)), x2, x3), GE(s(0), s(s(z0))))
   TOWERITER(s(z0), s(0), x2, x3) -> c10(GE(s(z0), s(0)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_2, c10_1, c1_1, c12_2


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

(69) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(EXP(x1, x2))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(EXP(x1, x2))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_2, c10_1, c1_1, c12_2, c_1


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

(71) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(EXP(x1, x2))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_2, c10_1, c1_1, c12_2, c_1


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

(73) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1))))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_2, c10_1, c1_1, c12_2, c_1


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

(75) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TOWERITER(s(s(z0)), s(s(z1)), x2, x3) -> c10(HELP(ge(z0, z1), s(s(z0)), s(s(z1)), x2, x3), GE(s(s(z0)), s(s(z1)))) by 
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0), GE(s(s(0)), s(s(x0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0), GE(s(s(0)), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0), GE(s(s(0)), s(s(x0))))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_1, c1_1, c12_2, c_1, c10_2


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

(77) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_1, c1_1, c12_2, c_1, c10_2, c2_1


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

(79) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
K tuples:
   TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1)))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, TOWERITER_4, PLUS_2, HELP_5

Compound Symbols:   c3_2, c5_2, c8_1, c10_1, c1_1, c12_2, c_1, c10_2, c2_1


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

(81) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TOWERITER(s(x0), s(x1), x2, x3) -> c10(GE(s(x0), s(x1))) by 
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c12_2, c_1, c10_2, c2_1, c10_1


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

(83) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(false, s(s(x0)), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(x0))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3)) by 
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c12(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)), EXP(x1, x2))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c10_2, c2_1, c10_1, c12_2


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

(85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c10_2, c2_1, c10_1, c12_2, c4_1


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

(87) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c10_2, c2_1, c10_1, c12_2, c4_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(x0), x1), s(s(s(x0))), s(s(x1)), x2, y0), GE(s(s(s(x0))), s(s(x1)))) by 
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0), GE(s(s(s(0))), s(s(x0))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0), GE(s(s(s(0))), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0), GE(s(s(s(0))), s(s(x0))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c2_1, c10_1, c12_2, c4_1, c10_2


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

(91) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1


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

(93) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1


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

(95) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace TOWERITER(s(s(s(x0))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(x0))), s(s(x1)))) by 
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   GE(s(z0), s(z1)) -> c8(GE(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, GE_2, PLUS_2, HELP_5, TOWERITER_4

Compound Symbols:   c3_2, c5_2, c8_1, c1_1, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1


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

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c8(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, PLUS_2, HELP_5, TOWERITER_4, GE_2

Compound Symbols:   c3_2, c5_2, c1_1, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1, c8_1


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

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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, GE_2, PLUS_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1, c8_1, c1_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(EXP(x1, x2)) by 
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, GE_2, PLUS_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1, c8_1, c1_1


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

(103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c8(GE(s(y0), s(y1))) by 
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, PLUS_2, GE_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c10_1, c12_2, c4_1, c10_2, c6_1, c1_1, c8_1


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

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(GE(s(s(0)), s(s(x0))))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c10(GE(s(s(0)), s(s(x0))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0))))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, PLUS_2, GE_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c12_2, c4_1, c10_2, c6_1, c10_1, c1_1, c8_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(GE(s(s(s(0))), s(s(x0)))) by 
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, PLUS_2, GE_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c12_2, c4_1, c10_2, c6_1, c10_1, c1_1, c8_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(GE(s(s(s(s(x0)))), s(s(x1)))) by 
   TOWERITER(s(s(s(s(z0)))), s(s(s(y1))), z2, z3) -> c10(GE(s(s(s(s(z0)))), s(s(s(y1)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
   TOWERITER(s(s(s(s(z0)))), s(s(s(y1))), z2, z3) -> c10(GE(s(s(s(s(z0)))), s(s(s(y1)))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0))))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
   TOWERITER(s(s(s(s(z0)))), s(s(s(y1))), z2, z3) -> c10(GE(s(s(s(s(z0)))), s(s(s(y1)))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, PLUS_2, GE_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c12_2, c4_1, c10_2, c6_1, c10_1, c1_1, c8_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c10(GE(s(s(s(0))), s(s(x0)))) by 
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c10(GE(s(s(s(0))), s(s(s(y1)))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> times(z0, exp(z0, z1))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
   TOWERITER(s(s(s(s(z0)))), s(s(s(y1))), z2, z3) -> c10(GE(s(s(s(s(z0)))), s(s(s(y1)))))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c10(GE(s(s(s(0))), s(s(s(y1)))))
S tuples:
   TIMES(s(z0), z1) -> c3(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   EXP(z0, s(z1)) -> c5(TIMES(z0, exp(z0, z1)), EXP(z0, z1))
   HELP(false, s(s(s(x0))), s(s(x1)), x2, x3) -> c12(TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, exp(x2, x3)), EXP(x2, x3))
   TOWERITER(s(s(s(s(x0)))), s(s(x1)), x2, y0) -> c10(HELP(ge(s(s(x0)), x1), s(s(s(s(x0)))), s(s(x1)), x2, y0), GE(s(s(s(s(x0)))), s(s(x1))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c8(GE(s(s(y0)), s(s(y1))))
K tuples:
   HELP(false, s(0), s(s(x0)), x1, x2) -> c(TOWERITER(s(s(0)), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(0)), s(s(x0)), x1, y0) -> c2(HELP(ge(0, x0), s(s(0)), s(s(x0)), x1, y0))
   HELP(false, s(s(0)), s(s(x0)), x1, x2) -> c4(TOWERITER(s(s(s(0))), s(s(x0)), x1, exp(x1, x2)))
   TOWERITER(s(s(s(0))), s(s(x0)), x1, y0) -> c6(HELP(ge(s(0), x0), s(s(s(0))), s(s(x0)), x1, y0))
   HELP(false, s(s(0)), s(s(z0)), z1, s(y1)) -> c4(EXP(z1, s(y1)))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c6(GE(s(s(s(0))), s(s(s(y1)))))
   TOWERITER(s(s(s(s(z0)))), s(s(s(y1))), z2, z3) -> c10(GE(s(s(s(s(z0)))), s(s(s(y1)))))
   TOWERITER(s(s(s(0))), s(s(s(y1))), z1, z2) -> c10(GE(s(s(s(0))), s(s(s(y1)))))
Defined Rule Symbols:   times_2, plus_2, exp_2, ge_2

Defined Pair Symbols:   TIMES_2, EXP_2, HELP_5, TOWERITER_4, PLUS_2, GE_2

Compound Symbols:   c3_2, c5_2, c_1, c2_1, c12_2, c4_1, c10_2, c6_1, c1_1, c8_1, c10_1